From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Thu, 7 Dec 2006 15:43:42 +0000 (+0000)
Subject: new theorems added. does not comile well yet :(( problems found in
X-Git-Tag: make_still_working~6604
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=9408f2869ddbbbae68dcb23ebf6c358c61888d0d;p=helm.git

new theorems added. does not comile well yet :(( problems found in
- csubc/csuba.ml (exception "List.nth" raised by matitaGui.ml at line 244)
- ty3/pr3.ml (matitac.opt returns with exit code 3 without evident error)
- sn3/props.ma (assertion failure raised by discrimination_tree.ml at line 62)
---

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/plist.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/plist.ma
deleted file mode 100644
index 3a0b28571..000000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/plist.ma
+++ /dev/null
@@ -1,38 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/Base/ext/plist".
-
-include "ext/preamble.ma".
-
-inductive PList: Set \def
-| PNil: PList
-| PCons: nat \to (nat \to (PList \to PList)).
-
-definition PConsTail:
- PList \to (nat \to (nat \to PList))
-\def
- let rec PConsTail (hds: PList) on hds: (nat \to (nat \to PList)) \def 
-(\lambda (h0: nat).(\lambda (d0: nat).(match hds with [PNil \Rightarrow 
-(PCons h0 d0 PNil) | (PCons h d hds0) \Rightarrow (PCons h d (PConsTail hds0 
-h0 d0))]))) in PConsTail.
-
-definition Ss:
- PList \to PList
-\def
- let rec Ss (hds: PList) on hds: PList \def (match hds with [PNil \Rightarrow 
-PNil | (PCons h d hds0) \Rightarrow (PCons h (S d) (Ss hds0))]) in Ss.
-
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/makefile b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/makefile
index ffdb70ded..db1724d0c 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/makefile
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/makefile
@@ -9,25 +9,31 @@ CLEANO=$(RT_BASEDIR)matitaclean.opt $(OPTIONS)
 
 devel:=$(shell basename `pwd`)
 
+ifneq "$(SRC)" ""
+  XXX="SRC=$(SRC)"
+endif
+
 all: preall
-	$(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) build $(devel)
+	$(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) build $(devel)
 clean: preall
-	$(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) clean $(devel)
+	$(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) clean $(devel)
 cleanall: preall
-	$(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MCLEAN) all
+	$(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MCLEAN) all
 
-all.opt opt: preall
-	$(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) build $(devel)
-clean.opt: preall
-	$(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) clean $(devel)
-cleanall.opt: preall
-	$(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MCLEANO) all
+all.opt opt: preall.opt
+	$(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) build $(devel)
+clean.opt: preall.opt
+	$(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) clean $(devel)
+cleanall.opt: preall.opt
+	$(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MCLEANO) all
 
 %.mo: preall
-	$(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) $@
-%.mo.opt: preall
-	$(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) $@
+	$(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) $@
+%.mo.opt: preall.opt
+	$(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) $@
 	
 preall:
-	$(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) init $(devel)
+	$(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) init $(devel)
 
+preall.opt:
+	$(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) init $(devel)
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/plist/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/plist/defs.ma
new file mode 100644
index 000000000..9bd72de7a
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/plist/defs.ma
@@ -0,0 +1,45 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/Base/plist/defs".
+
+include "ext/preamble.ma".
+
+inductive PList: Set \def
+| PNil: PList
+| PCons: nat \to (nat \to (PList \to PList)).
+
+definition PConsTail:
+ PList \to (nat \to (nat \to PList))
+\def
+ let rec PConsTail (hds: PList) on hds: (nat \to (nat \to PList)) \def 
+(\lambda (h0: nat).(\lambda (d0: nat).(match hds with [PNil \Rightarrow 
+(PCons h0 d0 PNil) | (PCons h d hds0) \Rightarrow (PCons h d (PConsTail hds0 
+h0 d0))]))) in PConsTail.
+
+definition Ss:
+ PList \to PList
+\def
+ let rec Ss (hds: PList) on hds: PList \def (match hds with [PNil \Rightarrow 
+PNil | (PCons h d hds0) \Rightarrow (PCons h (S d) (Ss hds0))]) in Ss.
+
+definition papp:
+ PList \to (PList \to PList)
+\def
+ let rec papp (a: PList) on a: (PList \to PList) \def (\lambda (b: 
+PList).(match a with [PNil \Rightarrow b | (PCons h d a0) \Rightarrow (PCons 
+h d (papp a0 b))])) in papp.
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/plist/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/plist/props.ma
new file mode 100644
index 000000000..ae5665fc6
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/plist/props.ma
@@ -0,0 +1,33 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/Base/plist/props".
+
+include "plist/defs.ma".
+
+theorem papp_ss:
+ \forall (is1: PList).(\forall (is2: PList).(eq PList (papp (Ss is1) (Ss 
+is2)) (Ss (papp is1 is2))))
+\def
+ \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2: 
+PList).(eq PList (papp (Ss p) (Ss is2)) (Ss (papp p is2))))) (\lambda (is2: 
+PList).(refl_equal PList (Ss is2))) (\lambda (n: nat).(\lambda (n0: 
+nat).(\lambda (p: PList).(\lambda (H: ((\forall (is2: PList).(eq PList (papp 
+(Ss p) (Ss is2)) (Ss (papp p is2)))))).(\lambda (is2: PList).(eq_ind_r PList 
+(Ss (papp p is2)) (\lambda (p0: PList).(eq PList (PCons n (S n0) p0) (PCons n 
+(S n0) (Ss (papp p is2))))) (refl_equal PList (PCons n (S n0) (Ss (papp p 
+is2)))) (papp (Ss p) (Ss is2)) (H is2))))))) is1).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/theory.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/theory.ma
index 008ba2f22..9ecfdb7ed 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/theory.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/theory.ma
@@ -28,5 +28,7 @@ include "blt/defs.ma".
 
 include "blt/props.ma".
 
-include "ext/plist.ma".
+include "plist/defs.ma".
+
+include "plist/props.ma".
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/types/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/types/defs.ma
index 888d92669..e2bfeaa7c 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/types/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/types/defs.ma
@@ -18,127 +18,129 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/Base/types/defs".
 
 include "ext/preamble.ma".
 
-inductive and3 (P0:Prop) (P1:Prop) (P2:Prop): Prop \def
+inductive and3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def
 | and3_intro: P0 \to (P1 \to (P2 \to (and3 P0 P1 P2))).
 
-inductive or3 (P0:Prop) (P1:Prop) (P2:Prop): Prop \def
+inductive or3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def
 | or3_intro0: P0 \to (or3 P0 P1 P2)
 | or3_intro1: P1 \to (or3 P0 P1 P2)
 | or3_intro2: P2 \to (or3 P0 P1 P2).
 
-inductive or4 (P0:Prop) (P1:Prop) (P2:Prop) (P3:Prop): Prop \def
+inductive or4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def
 | or4_intro0: P0 \to (or4 P0 P1 P2 P3)
 | or4_intro1: P1 \to (or4 P0 P1 P2 P3)
 | or4_intro2: P2 \to (or4 P0 P1 P2 P3)
 | or4_intro3: P3 \to (or4 P0 P1 P2 P3).
 
-inductive ex3 (A0:Set) (P0:A0 \to Prop) (P1:A0 \to Prop) (P2:A0 \to Prop): 
-Prop \def
+inductive ex3 (A0: Set) (P0: (A0 \to Prop)) (P1: (A0 \to Prop)) (P2: (A0 \to 
+Prop)): Prop \def
 | ex3_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to (ex3 A0 
 P0 P1 P2)))).
 
-inductive ex4 (A0:Set) (P0:A0 \to Prop) (P1:A0 \to Prop) (P2:A0 \to Prop) 
-(P3:A0 \to Prop): Prop \def
+inductive ex4 (A0: Set) (P0: (A0 \to Prop)) (P1: (A0 \to Prop)) (P2: (A0 \to 
+Prop)) (P3: (A0 \to Prop)): Prop \def
 | ex4_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to ((P3 x0) 
 \to (ex4 A0 P0 P1 P2 P3))))).
 
-inductive ex_2 (A0:Set) (A1:Set) (P0:A0 \to (A1 \to Prop)): Prop \def
+inductive ex_2 (A0: Set) (A1: Set) (P0: (A0 \to (A1 \to Prop))): Prop \def
 | ex_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to (ex_2 A0 A1 
 P0))).
 
-inductive ex2_2 (A0:Set) (A1:Set) (P0:A0 \to (A1 \to Prop)) (P1:A0 \to (A1 
-\to Prop)): Prop \def
+inductive ex2_2 (A0: Set) (A1: Set) (P0: (A0 \to (A1 \to Prop))) (P1: (A0 \to 
+(A1 \to Prop))): Prop \def
 | ex2_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) 
 \to (ex2_2 A0 A1 P0 P1)))).
 
-inductive ex3_2 (A0:Set) (A1:Set) (P0:A0 \to (A1 \to Prop)) (P1:A0 \to (A1 
-\to Prop)) (P2:A0 \to (A1 \to Prop)): Prop \def
+inductive ex3_2 (A0: Set) (A1: Set) (P0: (A0 \to (A1 \to Prop))) (P1: (A0 \to 
+(A1 \to Prop))) (P2: (A0 \to (A1 \to Prop))): Prop \def
 | ex3_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) 
 \to ((P2 x0 x1) \to (ex3_2 A0 A1 P0 P1 P2))))).
 
-inductive ex4_2 (A0:Set) (A1:Set) (P0:A0 \to (A1 \to Prop)) (P1:A0 \to (A1 
-\to Prop)) (P2:A0 \to (A1 \to Prop)) (P3:A0 \to (A1 \to Prop)): Prop \def
+inductive ex4_2 (A0: Set) (A1: Set) (P0: (A0 \to (A1 \to Prop))) (P1: (A0 \to 
+(A1 \to Prop))) (P2: (A0 \to (A1 \to Prop))) (P3: (A0 \to (A1 \to Prop))): 
+Prop \def
 | ex4_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) 
 \to ((P2 x0 x1) \to ((P3 x0 x1) \to (ex4_2 A0 A1 P0 P1 P2 P3)))))).
 
-inductive ex_3 (A0:Set) (A1:Set) (A2:Set) (P0:A0 \to (A1 \to (A2 \to Prop))): 
-Prop \def
+inductive ex_3 (A0: Set) (A1: Set) (A2: Set) (P0: (A0 \to (A1 \to (A2 \to 
+Prop)))): Prop \def
 | ex_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1 
 x2) \to (ex_3 A0 A1 A2 P0)))).
 
-inductive ex2_3 (A0:Set) (A1:Set) (A2:Set) (P0:A0 \to (A1 \to (A2 \to Prop))) 
-(P1:A0 \to (A1 \to (A2 \to Prop))): Prop \def
+inductive ex2_3 (A0: Set) (A1: Set) (A2: Set) (P0: (A0 \to (A1 \to (A2 \to 
+Prop)))) (P1: (A0 \to (A1 \to (A2 \to Prop)))): Prop \def
 | ex2_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 
 x1 x2) \to ((P1 x0 x1 x2) \to (ex2_3 A0 A1 A2 P0 P1))))).
 
-inductive ex3_3 (A0:Set) (A1:Set) (A2:Set) (P0:A0 \to (A1 \to (A2 \to Prop))) 
-(P1:A0 \to (A1 \to (A2 \to Prop))) (P2:A0 \to (A1 \to (A2 \to Prop))): Prop 
-\def
+inductive ex3_3 (A0: Set) (A1: Set) (A2: Set) (P0: (A0 \to (A1 \to (A2 \to 
+Prop)))) (P1: (A0 \to (A1 \to (A2 \to Prop)))) (P2: (A0 \to (A1 \to (A2 \to 
+Prop)))): Prop \def
 | ex3_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 
 x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to (ex3_3 A0 A1 A2 P0 P1 
 P2)))))).
 
-inductive ex4_3 (A0:Set) (A1:Set) (A2:Set) (P0:A0 \to (A1 \to (A2 \to Prop))) 
-(P1:A0 \to (A1 \to (A2 \to Prop))) (P2:A0 \to (A1 \to (A2 \to Prop))) (P3:A0 
-\to (A1 \to (A2 \to Prop))): Prop \def
+inductive ex4_3 (A0: Set) (A1: Set) (A2: Set) (P0: (A0 \to (A1 \to (A2 \to 
+Prop)))) (P1: (A0 \to (A1 \to (A2 \to Prop)))) (P2: (A0 \to (A1 \to (A2 \to 
+Prop)))) (P3: (A0 \to (A1 \to (A2 \to Prop)))): Prop \def
 | ex4_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 
 x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to (ex4_3 A0 
 A1 A2 P0 P1 P2 P3))))))).
 
-inductive ex3_4 (A0:Set) (A1:Set) (A2:Set) (A3:Set) (P0:A0 \to (A1 \to (A2 
-\to (A3 \to Prop)))) (P1:A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P2:A0 \to 
-(A1 \to (A2 \to (A3 \to Prop)))): Prop \def
+inductive ex3_4 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (P0: (A0 \to (A1 \to 
+(A2 \to (A3 \to Prop))))) (P1: (A0 \to (A1 \to (A2 \to (A3 \to Prop))))) (P2: 
+(A0 \to (A1 \to (A2 \to (A3 \to Prop))))): Prop \def
 | ex3_4_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall 
 (x3: A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to 
 (ex3_4 A0 A1 A2 A3 P0 P1 P2))))))).
 
-inductive ex4_4 (A0:Set) (A1:Set) (A2:Set) (A3:Set) (P0:A0 \to (A1 \to (A2 
-\to (A3 \to Prop)))) (P1:A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P2:A0 \to 
-(A1 \to (A2 \to (A3 \to Prop)))) (P3:A0 \to (A1 \to (A2 \to (A3 \to Prop)))): 
-Prop \def
+inductive ex4_4 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (P0: (A0 \to (A1 \to 
+(A2 \to (A3 \to Prop))))) (P1: (A0 \to (A1 \to (A2 \to (A3 \to Prop))))) (P2: 
+(A0 \to (A1 \to (A2 \to (A3 \to Prop))))) (P3: (A0 \to (A1 \to (A2 \to (A3 
+\to Prop))))): Prop \def
 | ex4_4_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall 
 (x3: A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to 
 ((P3 x0 x1 x2 x3) \to (ex4_4 A0 A1 A2 A3 P0 P1 P2 P3)))))))).
 
-inductive ex4_5 (A0:Set) (A1:Set) (A2:Set) (A3:Set) (A4:Set) (P0:A0 \to (A1 
-\to (A2 \to (A3 \to (A4 \to Prop))))) (P1:A0 \to (A1 \to (A2 \to (A3 \to (A4 
-\to Prop))))) (P2:A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P3:A0 \to 
-(A1 \to (A2 \to (A3 \to (A4 \to Prop))))): Prop \def
+inductive ex4_5 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (P0: (A0 
+\to (A1 \to (A2 \to (A3 \to (A4 \to Prop)))))) (P1: (A0 \to (A1 \to (A2 \to 
+(A3 \to (A4 \to Prop)))))) (P2: (A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
+Prop)))))) (P3: (A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop)))))): Prop \def
 | ex4_5_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall 
 (x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 x1 x2 x3 x4) \to 
 ((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to (ex4_5 A0 A1 A2 A3 A4 P0 P1 
 P2 P3))))))))).
 
-inductive ex5_5 (A0:Set) (A1:Set) (A2:Set) (A3:Set) (A4:Set) (P0:A0 \to (A1 
-\to (A2 \to (A3 \to (A4 \to Prop))))) (P1:A0 \to (A1 \to (A2 \to (A3 \to (A4 
-\to Prop))))) (P2:A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P3:A0 \to 
-(A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P4:A0 \to (A1 \to (A2 \to (A3 \to 
-(A4 \to Prop))))): Prop \def
+inductive ex5_5 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (P0: (A0 
+\to (A1 \to (A2 \to (A3 \to (A4 \to Prop)))))) (P1: (A0 \to (A1 \to (A2 \to 
+(A3 \to (A4 \to Prop)))))) (P2: (A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
+Prop)))))) (P3: (A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop)))))) (P4: (A0 
+\to (A1 \to (A2 \to (A3 \to (A4 \to Prop)))))): Prop \def
 | ex5_5_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall 
 (x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 x1 x2 x3 x4) \to 
 ((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to ((P4 x0 x1 x2 x3 x4) \to 
 (ex5_5 A0 A1 A2 A3 A4 P0 P1 P2 P3 P4)))))))))).
 
-inductive ex6_6 (A0:Set) (A1:Set) (A2:Set) (A3:Set) (A4:Set) (A5:Set) (P0:A0 
-\to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) (P1:A0 \to (A1 \to (A2 
-\to (A3 \to (A4 \to (A5 \to Prop)))))) (P2:A0 \to (A1 \to (A2 \to (A3 \to (A4 
-\to (A5 \to Prop)))))) (P3:A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to 
-Prop)))))) (P4:A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) 
-(P5:A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))): Prop \def
+inductive ex6_6 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (A5: Set) 
+(P0: (A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop))))))) (P1: (A0 \to 
+(A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop))))))) (P2: (A0 \to (A1 \to (A2 
+\to (A3 \to (A4 \to (A5 \to Prop))))))) (P3: (A0 \to (A1 \to (A2 \to (A3 \to 
+(A4 \to (A5 \to Prop))))))) (P4: (A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 
+\to Prop))))))) (P5: (A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to 
+Prop))))))): Prop \def
 | ex6_6_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall 
 (x3: A3).(\forall (x4: A4).(\forall (x5: A5).((P0 x0 x1 x2 x3 x4 x5) \to ((P1 
 x0 x1 x2 x3 x4 x5) \to ((P2 x0 x1 x2 x3 x4 x5) \to ((P3 x0 x1 x2 x3 x4 x5) 
 \to ((P4 x0 x1 x2 x3 x4 x5) \to ((P5 x0 x1 x2 x3 x4 x5) \to (ex6_6 A0 A1 A2 
 A3 A4 A5 P0 P1 P2 P3 P4 P5)))))))))))).
 
-inductive ex6_7 (A0:Set) (A1:Set) (A2:Set) (A3:Set) (A4:Set) (A5:Set) 
-(A6:Set) (P0:A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to 
-Prop))))))) (P1:A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to 
-Prop))))))) (P2:A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to 
-Prop))))))) (P3:A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to 
-Prop))))))) (P4:A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to 
-Prop))))))) (P5:A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to 
-Prop))))))): Prop \def
+inductive ex6_7 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (A5: Set) 
+(A6: Set) (P0: (A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to 
+Prop)))))))) (P1: (A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to 
+Prop)))))))) (P2: (A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to 
+Prop)))))))) (P3: (A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to 
+Prop)))))))) (P4: (A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to 
+Prop)))))))) (P5: (A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to 
+Prop)))))))): Prop \def
 | ex6_7_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall 
 (x3: A3).(\forall (x4: A4).(\forall (x5: A5).(\forall (x6: A6).((P0 x0 x1 x2 
 x3 x4 x5 x6) \to ((P1 x0 x1 x2 x3 x4 x5 x6) \to ((P2 x0 x1 x2 x3 x4 x5 x6) 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta.ma
index 675dbfe7c..8642561d2 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta.ma
@@ -18,4407 +18,3 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta".
 
 include "LambdaDelta/theory.ma".
 
-theorem bind_dec_not:
- \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) (not (eq B b1 b2))))
-\def
- \lambda (b1: B).(\lambda (b2: B).(let H_x \def (terms_props__bind_dec b1 b2) 
-in (let H \def H_x in (or_ind (eq B b1 b2) ((eq B b1 b2) \to (\forall (P: 
-Prop).P)) (or (eq B b1 b2) ((eq B b1 b2) \to False)) (\lambda (H0: (eq B b1 
-b2)).(or_introl (eq B b1 b2) ((eq B b1 b2) \to False) H0)) (\lambda (H0: 
-(((eq B b1 b2) \to (\forall (P: Prop).P)))).(or_intror (eq B b1 b2) ((eq B b1 
-b2) \to False) (\lambda (H1: (eq B b1 b2)).(H0 H1 False)))) H)))).
-
-definition TApp:
- TList \to (T \to TList)
-\def
- let rec TApp (ts: TList) on ts: (T \to TList) \def (\lambda (v: T).(match ts 
-with [TNil \Rightarrow (TCons v TNil) | (TCons t ts0) \Rightarrow (TCons t 
-(TApp ts0 v))])) in TApp.
-
-definition tslen:
- TList \to nat
-\def
- let rec tslen (ts: TList) on ts: nat \def (match ts with [TNil \Rightarrow O 
-| (TCons _ ts0) \Rightarrow (S (tslen ts0))]) in tslen.
-
-definition tslt:
- TList \to (TList \to Prop)
-\def
- \lambda (ts1: TList).(\lambda (ts2: TList).(lt (tslen ts1) (tslen ts2))).
-
-theorem tslt_wf__q_ind:
- \forall (P: ((TList \to Prop))).(((\forall (n: nat).((\lambda (P0: ((TList 
-\to Prop))).(\lambda (n0: nat).(\forall (ts: TList).((eq nat (tslen ts) n0) 
-\to (P0 ts))))) P n))) \to (\forall (ts: TList).(P ts)))
-\def
- let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts: 
-TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to 
-Prop))).(\lambda (H: ((\forall (n: nat).(\forall (ts: TList).((eq nat (tslen 
-ts) n) \to (P ts)))))).(\lambda (ts: TList).(H (tslen ts) ts (refl_equal nat 
-(tslen ts)))))).
-
-theorem tslt_wf_ind:
- \forall (P: ((TList \to Prop))).(((\forall (ts2: TList).(((\forall (ts1: 
-TList).((tslt ts1 ts2) \to (P ts1)))) \to (P ts2)))) \to (\forall (ts: 
-TList).(P ts)))
-\def
- let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts: 
-TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to 
-Prop))).(\lambda (H: ((\forall (ts2: TList).(((\forall (ts1: TList).((lt 
-(tslen ts1) (tslen ts2)) \to (P ts1)))) \to (P ts2))))).(\lambda (ts: 
-TList).(tslt_wf__q_ind (\lambda (t: TList).(P t)) (\lambda (n: 
-nat).(lt_wf_ind n (Q (\lambda (t: TList).(P t))) (\lambda (n0: nat).(\lambda 
-(H0: ((\forall (m: nat).((lt m n0) \to (Q (\lambda (t: TList).(P t)) 
-m))))).(\lambda (ts0: TList).(\lambda (H1: (eq nat (tslen ts0) n0)).(let H2 
-\def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall (m: nat).((lt m n1) \to 
-(\forall (ts1: TList).((eq nat (tslen ts1) m) \to (P ts1)))))) H0 (tslen ts0) 
-H1) in (H ts0 (\lambda (ts1: TList).(\lambda (H3: (lt (tslen ts1) (tslen 
-ts0))).(H2 (tslen ts1) H3 ts1 (refl_equal nat (tslen ts1))))))))))))) ts)))).
-
-theorem theads_tapp:
- \forall (k: K).(\forall (vs: TList).(\forall (v: T).(\forall (t: T).(eq T 
-(THeads k (TApp vs v) t) (THeads k vs (THead k v t))))))
-\def
- \lambda (k: K).(\lambda (vs: TList).(TList_ind (\lambda (t: TList).(\forall 
-(v: T).(\forall (t0: T).(eq T (THeads k (TApp t v) t0) (THeads k t (THead k v 
-t0)))))) (\lambda (v: T).(\lambda (t: T).(refl_equal T (THead k v t)))) 
-(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (v: T).(\forall 
-(t1: T).(eq T (THeads k (TApp t0 v) t1) (THeads k t0 (THead k v 
-t1))))))).(\lambda (v: T).(\lambda (t1: T).(eq_ind_r T (THeads k t0 (THead k 
-v t1)) (\lambda (t2: T).(eq T (THead k t t2) (THead k t (THeads k t0 (THead k 
-v t1))))) (refl_equal T (THead k t (THeads k t0 (THead k v t1)))) (THeads k 
-(TApp t0 v) t1) (H v t1))))))) vs)).
-
-theorem tcons_tapp_ex:
- \forall (ts1: TList).(\forall (t1: T).(ex2_2 TList T (\lambda (ts2: 
-TList).(\lambda (t2: T).(eq TList (TCons t1 ts1) (TApp ts2 t2)))) (\lambda 
-(ts2: TList).(\lambda (_: T).(eq nat (tslen ts1) (tslen ts2))))))
-\def
- \lambda (ts1: TList).(TList_ind (\lambda (t: TList).(\forall (t1: T).(ex2_2 
-TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t) (TApp 
-ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t) (tslen 
-ts2))))))) (\lambda (t1: T).(ex2_2_intro TList T (\lambda (ts2: 
-TList).(\lambda (t2: T).(eq TList (TCons t1 TNil) (TApp ts2 t2)))) (\lambda 
-(ts2: TList).(\lambda (_: T).(eq nat O (tslen ts2)))) TNil t1 (refl_equal 
-TList (TApp TNil t1)) (refl_equal nat (tslen TNil)))) (\lambda (t: 
-T).(\lambda (t0: TList).(\lambda (H: ((\forall (t1: T).(ex2_2 TList T 
-(\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t0) (TApp ts2 
-t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) (tslen 
-ts2)))))))).(\lambda (t1: T).(let H_x \def (H t) in (let H0 \def H_x in 
-(ex2_2_ind TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t 
-t0) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) 
-(tslen ts2)))) (ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq 
-TList (TCons t1 (TCons t t0)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda 
-(_: T).(eq nat (S (tslen t0)) (tslen ts2))))) (\lambda (x0: TList).(\lambda 
-(x1: T).(\lambda (H1: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H2: (eq 
-nat (tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t2: 
-TList).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t3: T).(eq TList (TCons 
-t1 t2) (TApp ts2 t3)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (S 
-(tslen t0)) (tslen ts2)))))) (eq_ind_r nat (tslen x0) (\lambda (n: 
-nat).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons 
-t1 (TApp x0 x1)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq 
-nat (S n) (tslen ts2)))))) (ex2_2_intro TList T (\lambda (ts2: 
-TList).(\lambda (t2: T).(eq TList (TCons t1 (TApp x0 x1)) (TApp ts2 t2)))) 
-(\lambda (ts2: TList).(\lambda (_: T).(eq nat (S (tslen x0)) (tslen ts2)))) 
-(TCons t1 x0) x1 (refl_equal TList (TApp (TCons t1 x0) x1)) (refl_equal nat 
-(tslen (TCons t1 x0)))) (tslen t0) H2) (TCons t t0) H1))))) H0))))))) ts1).
-
-theorem tlist_ind_rew:
- \forall (P: ((TList \to Prop))).((P TNil) \to (((\forall (ts: 
-TList).(\forall (t: T).((P ts) \to (P (TApp ts t)))))) \to (\forall (ts: 
-TList).(P ts))))
-\def
- \lambda (P: ((TList \to Prop))).(\lambda (H: (P TNil)).(\lambda (H0: 
-((\forall (ts: TList).(\forall (t: T).((P ts) \to (P (TApp ts 
-t))))))).(\lambda (ts: TList).(tslt_wf_ind (\lambda (t: TList).(P t)) 
-(\lambda (ts2: TList).(TList_ind (\lambda (t: TList).(((\forall (ts1: 
-TList).((tslt ts1 t) \to (P ts1)))) \to (P t))) (\lambda (_: ((\forall (ts1: 
-TList).((tslt ts1 TNil) \to (P ts1))))).H) (\lambda (t: T).(\lambda (t0: 
-TList).(\lambda (_: ((((\forall (ts1: TList).((tslt ts1 t0) \to (P ts1)))) 
-\to (P t0)))).(\lambda (H2: ((\forall (ts1: TList).((tslt ts1 (TCons t t0)) 
-\to (P ts1))))).(let H_x \def (tcons_tapp_ex t0 t) in (let H3 \def H_x in 
-(ex2_2_ind TList T (\lambda (ts3: TList).(\lambda (t2: T).(eq TList (TCons t 
-t0) (TApp ts3 t2)))) (\lambda (ts3: TList).(\lambda (_: T).(eq nat (tslen t0) 
-(tslen ts3)))) (P (TCons t t0)) (\lambda (x0: TList).(\lambda (x1: 
-T).(\lambda (H4: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H5: (eq nat 
-(tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t1: TList).(P 
-t1)) (H0 x0 x1 (H2 x0 (eq_ind nat (tslen t0) (\lambda (n: nat).(lt n (tslen 
-(TCons t t0)))) (le_n (tslen (TCons t t0))) (tslen x0) H5))) (TCons t t0) 
-H4))))) H3))))))) ts2)) ts)))).
-
-theorem lifts_tapp:
- \forall (h: nat).(\forall (d: nat).(\forall (v: T).(\forall (vs: TList).(eq 
-TList (lifts h d (TApp vs v)) (TApp (lifts h d vs) (lift h d v))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (v: T).(\lambda (vs: 
-TList).(TList_ind (\lambda (t: TList).(eq TList (lifts h d (TApp t v)) (TApp 
-(lifts h d t) (lift h d v)))) (refl_equal TList (TCons (lift h d v) TNil)) 
-(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq TList (lifts h d (TApp 
-t0 v)) (TApp (lifts h d t0) (lift h d v)))).(eq_ind_r TList (TApp (lifts h d 
-t0) (lift h d v)) (\lambda (t1: TList).(eq TList (TCons (lift h d t) t1) 
-(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v))))) (refl_equal TList 
-(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v)))) (lifts h d (TApp t0 
-v)) H)))) vs)))).
-
-theorem pr2_change:
- \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1: 
-T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Bind b) v1) t1 t2) \to 
-(\forall (v2: T).(pr2 (CHead c (Bind b) v2) t1 t2))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda 
-(v1: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind 
-b) v1) t1 t2)).(\lambda (v2: T).(insert_eq C (CHead c (Bind b) v1) (\lambda 
-(c0: C).(pr2 c0 t1 t2)) (pr2 (CHead c (Bind b) v2) t1 t2) (\lambda (y: 
-C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: 
-T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) v1)) \to (pr2 (CHead c (Bind 
-b) v2) t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda 
-(H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) v1))).(pr2_free 
-(CHead c (Bind b) v2) t3 t4 H2)))))) (\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind 
-Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 
-t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: (eq C c0 
-(CHead c (Bind b) v1))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1 
-(CHead d (Bind Abbr) u))) H2 (CHead c (Bind b) v1) H5) in (nat_ind (\lambda 
-(n: nat).((getl n (CHead c (Bind b) v1) (CHead d (Bind Abbr) u)) \to ((subst0 
-n u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t)))) (\lambda (H7: (getl O 
-(CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda (H8: (subst0 O u t4 
-t)).(let H9 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda 
-(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) 
-(CHead d (Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d 
-(Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) 
-H7))) in ((let H10 \def (f_equal C B (\lambda (e: C).(match e in C return 
-(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) 
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) 
-(CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 
-(getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H11 
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) 
-with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d 
-(Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) 
-u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in 
-(\lambda (H12: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H14 \def (eq_ind 
-T u (\lambda (t0: T).(subst0 O t0 t4 t)) H8 v1 H11) in (let H15 \def 
-(eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abbr))) H Abbr H12) in (eq_ind B 
-Abbr (\lambda (b0: B).(pr2 (CHead c (Bind b0) v2) t3 t)) (let H16 \def (match 
-(H15 (refl_equal B Abbr)) in False return (\lambda (_: False).(pr2 (CHead c 
-(Bind Abbr) v2) t3 t)) with []) in H16) b H12)))))) H10)) H9)))) (\lambda 
-(i0: nat).(\lambda (_: (((getl i0 (CHead c (Bind b) v1) (CHead d (Bind Abbr) 
-u)) \to ((subst0 i0 u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))).(\lambda 
-(H7: (getl (S i0) (CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda 
-(H8: (subst0 (S i0) u t4 t)).(pr2_delta (CHead c (Bind b) v2) d u (S i0) 
-(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c 
-(CHead d (Bind Abbr) u) v1 i0 H7) v2) t3 t4 H3 t H8))))) i H6 H4))))))))))))) 
-y t1 t2 H1))) H0)))))))).
-
-theorem pr3_gen_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u1: 
-T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind b) u1 t1) x) \to (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind b) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 (CHead c (Bind b) u1) t1 t2)))) (pr3 (CHead c (Bind 
-b) u1) t1 (lift (S O) O x)))))))))
-\def
- \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall 
-(c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind 
-b0) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
-(THead (Bind b0) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b0) u1) t1 t2)))) (pr3 
-(CHead c (Bind b0) u1) t1 (lift (S O) O x)))))))))) (\lambda (_: (not (eq B 
-Abbr Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: 
-T).(\lambda (H0: (pr3 c (THead (Bind Abbr) u1 t1) x)).(let H1 \def 
-(pr3_gen_abbr c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) 
-u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (or (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) 
-t1 (lift (S O) O x))) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 
-t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind 
-Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) 
-t1 (lift (S O) O x))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T x 
-(THead (Bind Abbr) x0 x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: (pr3 
-(CHead c (Bind Abbr) u1) t1 x1)).(or_introl (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S 
-O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
-(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) x0 x1 
-H3 H4 H5))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S 
-O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
-(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 
-(CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) H2)) H1)))))))) (\lambda (H: 
-(not (eq B Abst Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: 
-T).(\lambda (x: T).(\lambda (_: (pr3 c (THead (Bind Abst) u1 t1) x)).(let H1 
-\def (match (H (refl_equal B Abst)) in False return (\lambda (_: False).(or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 (CHead c (Bind Abst) u1) t1 t2)))) (pr3 (CHead c 
-(Bind Abst) u1) t1 (lift (S O) O x)))) with []) in H1))))))) (\lambda (_: 
-(not (eq B Void Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: 
-T).(\lambda (x: T).(\lambda (H0: (pr3 c (THead (Bind Void) u1 t1) x)).(let H1 
-\def (pr3_gen_void c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
-(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t2)))))) (pr3 (CHead c 
-(Bind Void) u1) t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) 
-u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda 
-(H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) 
-u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) 
-u) t1 t2))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
-(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead 
-c (Bind b0) u) t1 t2))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 
-t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H3: (eq T x (THead (Bind Void) x0 
-x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: ((\forall (b0: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(or_introl (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind Void) u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S 
-O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
-(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 t2))) x0 x1 
-H3 H4 (H5 Void u1)))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Void) u1) t1 
-(lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 
-t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x)) H2)) H1)))))))) b).
-
-theorem pr3_iso_appls_abbr:
- \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
-(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).(let u1 \def (THeads (Flat 
-Appl) vs (TLRef i)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
-(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) vs (lift (S i) O w)) 
-u2))))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda 
-(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind 
-(\lambda (t: TList).(let u1 \def (THeads (Flat Appl) t (TLRef i)) in (\forall 
-(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to 
-(pr3 c (THeads (Flat Appl) t (lift (S i) O w)) u2)))))) (\lambda (u2: 
-T).(\lambda (H0: (pr3 c (TLRef i) u2)).(\lambda (H1: (((iso (TLRef i) u2) \to 
-(\forall (P: Prop).P)))).(let H2 \def (pr3_gen_lref c u2 i H0) in (or_ind (eq 
-T u2 (TLRef i)) (ex3_3 C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_: 
-T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u: 
-T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(v: T).(eq T u2 (lift (S i) O v)))))) (pr3 c (lift (S i) O w) u2) (\lambda 
-(H3: (eq T u2 (TLRef i))).(let H4 \def (eq_ind T u2 (\lambda (t: T).((iso 
-(TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H3) in (eq_ind_r T 
-(TLRef i) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (H4 (iso_refl (TLRef 
-i)) (pr3 c (lift (S i) O w) (TLRef i))) u2 H3))) (\lambda (H3: (ex3_3 C T T 
-(\lambda (d0: C).(\lambda (u: T).(\lambda (_: T).(getl i c (CHead d0 (Bind 
-Abbr) u))))) (\lambda (d0: C).(\lambda (u: T).(\lambda (v: T).(pr3 d0 u v)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T u2 (lift (S i) O 
-v))))))).(ex3_3_ind C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_: 
-T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u: 
-T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(v: T).(eq T u2 (lift (S i) O v))))) (pr3 c (lift (S i) O w) u2) (\lambda 
-(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H4: (getl i c (CHead x0 
-(Bind Abbr) x1))).(\lambda (H5: (pr3 x0 x1 x2)).(\lambda (H6: (eq T u2 (lift 
-(S i) O x2))).(let H7 \def (eq_ind T u2 (\lambda (t: T).((iso (TLRef i) t) 
-\to (\forall (P: Prop).P))) H1 (lift (S i) O x2) H6) in (eq_ind_r T (lift (S 
-i) O x2) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (let H8 \def (eq_ind C 
-(CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H (CHead x0 (Bind 
-Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) 
-H4)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return 
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow 
-c0])) (CHead d (Bind Abbr) w) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d 
-(Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) H4)) in ((let H10 \def (f_equal 
-C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
-\Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) w) (CHead 
-x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind 
-Abbr) x1) H4)) in (\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 
-(\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 w H10) in (let H13 
-\def (eq_ind_r T x1 (\lambda (t: T).(pr3 x0 t x2)) H5 w H10) in (let H14 \def 
-(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) w))) H12 d 
-H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c0: C).(pr3 c0 w x2)) H13 d 
-H11) in (pr3_lift c d (S i) O (getl_drop Abbr c d w i H14) w x2 H15))))))) 
-H9))) u2 H6)))))))) H3)) H2))))) (\lambda (t: T).(\lambda (t0: 
-TList).(\lambda (H0: ((\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (TLRef 
-i)) u2) \to ((((iso (THeads (Flat Appl) t0 (TLRef i)) u2) \to (\forall (P: 
-Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (lift (S i) O w)) 
-u2)))))).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads 
-(Flat Appl) t0 (TLRef i))) u2)).(\lambda (H2: (((iso (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (TLRef i))) u2) \to (\forall (P: Prop).P)))).(let H3 
-\def (pr3_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) u2 H1) in (or3_ind 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 
-t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2)))) (ex4_4 T 
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 
-c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (H4: (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: 
-T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))))).(ex3_2_ind T T (\lambda 
-(u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
-(THeads (Flat Appl) t0 (TLRef i)) t2))) (pr3 c (THead (Flat Appl) t (THeads 
-(Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H5: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c t 
-x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(let H8 \def 
-(eq_ind T u2 (\lambda (t1: T).((iso (THead (Flat Appl) t (THeads (Flat Appl) 
-t0 (TLRef i))) t1) \to (\forall (P: Prop).P))) H2 (THead (Flat Appl) x0 x1) 
-H5) in (eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(pr3 c (THead 
-(Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) t1)) (H8 (iso_head t 
-x0 (THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) (pr3 c (THead (Flat 
-Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) (THead (Flat Appl) x0 x1))) 
-u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t 
-u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T 
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 
-c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O 
-w))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H6: (pr3 c t 
-x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
-Abst) x0 x1))).(\lambda (H8: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c 
-(Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t 
-(THead (Bind Abst) x0 x1)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift 
-(S i) O w))) c (pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead 
-(Bind Abst) x0 x1) (H0 (THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso 
-(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (P: 
-Prop).(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H9 P)))) t Appl) (THead 
-(Bind Abbr) t x1) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) 
-(THead (Bind Abbr) t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) 
-x0 x1)) (THead (Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 
-(pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t 
-x1) c (pr3_head_12 c t x2 H6 (Bind Abbr) x1 x3 (H8 Abbr x2)) u2 H5)))))))))) 
-H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind 
-B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
-(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift 
-(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
-(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat 
-Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda 
-(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not 
-(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
-(THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c (THead (Bind x0) x5 (THead 
-(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H8: (pr3 c t x4)).(\lambda 
-(H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t 
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat 
-Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Bind x0) 
-x1 (THead (Flat Appl) (lift (S O) O t) x2)) (THead (Flat Appl) t (THeads 
-(Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t (THead (Bind 
-x0) x1 x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c 
-(pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead (Bind x0) x1 
-x2) (H0 (THead (Bind x0) x1 x2) H6 (\lambda (H11: (iso (THeads (Flat Appl) t0 
-(TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (P: 
-Prop).(iso_flats_lref_bind_false Appl x0 i x1 x2 t0 H11 P)))) t Appl) (THead 
-(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr3_pr2 c (THead (Flat 
-Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift 
-(S O) O t) x2)) (pr2_free c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) 
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr0_upsilon x0 
-H5 t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2))))) (THead (Bind 
-x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr3_head_12 c x1 x1 
-(pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift (S O) O t) x2) (THead 
-(Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead c (Bind x0) x1) (lift 
-(S O) O t) (lift (S O) O x4) (pr3_lift (CHead c (Bind x0) x1) c (S O) O 
-(drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H8) (Flat Appl) x2 x2 
-(pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) 
-u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) 
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c (pr3_head_12 
-c x1 x5 H9 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat 
-Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 
-(lift (S O) O x4) Appl)) u2 H7)))))))))))))) H4)) H3)))))))) vs)))))).
-
-theorem pr3_iso_appl_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: 
-T).(\forall (t: T).(let u1 \def (THead (Flat Appl) v1 (THead (Bind b) v2 t)) 
-in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
-(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) v2 (THead (Flat Appl) 
-(lift (S O) O v1) t)) u2))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v1: T).(\lambda 
-(v2: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H0: (pr3 c 
-(THead (Flat Appl) v1 (THead (Bind b) v2 t)) u2)).(\lambda (H1: (((iso (THead 
-(Flat Appl) v1 (THead (Bind b) v2 t)) u2) \to (\forall (P: Prop).P)))).(let 
-H2 \def (pr3_gen_appl c v1 (THead (Bind b) v2 t) u2 H0) in (or3_ind (ex3_2 T 
-T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 c (THead (Bind b) v2 t) t2)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1 
-t2)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) 
-(\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 
-z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
-T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat 
-Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b0: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c (THead (Bind b) v2 
-(THead (Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (H3: (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 c (THead (Bind b) v2 t) t2))))).(ex3_2_ind T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
-(THead (Bind b) v2 t) t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) 
-(lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq 
-T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v1 x0)).(\lambda (_: 
-(pr3 c (THead (Bind b) v2 t) x1)).(let H7 \def (eq_ind T u2 (\lambda (t0: 
-T).((iso (THead (Flat Appl) v1 (THead (Bind b) v2 t)) t0) \to (\forall (P: 
-Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl) 
-x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S 
-O) O v1) t)) t0)) (H7 (iso_head v1 x0 (THead (Bind b) v2 t) x1 (Flat Appl)) 
-(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead 
-(Flat Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1 
-t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))) 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
-(THead (Bind b) v2 t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: 
-T).(pr3 (CHead c (Bind b0) u) z1 t2))))))) (pr3 c (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c (THead (Bind Abbr) 
-x2 x3) u2)).(\lambda (H5: (pr3 c v1 x2)).(\lambda (H6: (pr3 c (THead (Bind b) 
-v2 t) (THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b0: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b0) u) x1 x3))))).(pr3_t (THead (Bind Abbr) x2 x3) 
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) c (let H_x \def 
-(pr3_gen_bind b H c v2 t (THead (Bind Abst) x0 x1) H6) in (let H8 \def H_x in 
-(or_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst) 
-x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) 
-(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind Abst) x0 x1))) (pr3 c 
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind 
-Abbr) x2 x3)) (\lambda (H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
-T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: 
-T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) 
-(lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3)) (\lambda (x4: T).(\lambda 
-(x5: T).(\lambda (H10: (eq T (THead (Bind Abst) x0 x1) (THead (Bind b) x4 
-x5))).(\lambda (H11: (pr3 c v2 x4)).(\lambda (H12: (pr3 (CHead c (Bind b) v2) 
-t x5)).(let H13 \def (f_equal T B (\lambda (e: T).(match e in T return 
-(\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow 
-Abst | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with 
-[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (THead (Bind Abst) 
-x0 x1) (THead (Bind b) x4 x5) H10) in ((let H14 \def (f_equal T T (\lambda 
-(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 
-| (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind 
-Abst) x0 x1) (THead (Bind b) x4 x5) H10) in ((let H15 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) \Rightarrow t0])) 
-(THead (Bind Abst) x0 x1) (THead (Bind b) x4 x5) H10) in (\lambda (H16: (eq T 
-x0 x4)).(\lambda (H17: (eq B Abst b)).(let H18 \def (eq_ind_r T x5 (\lambda 
-(t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H12 x1 H15) in (let H19 \def 
-(eq_ind_r T x4 (\lambda (t0: T).(pr3 c v2 t0)) H11 x0 H16) in (let H20 \def 
-(eq_ind_r B b (\lambda (b0: B).(pr3 (CHead c (Bind b0) v2) t x1)) H18 Abst 
-H17) in (let H21 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H 
-Abst H17) in (eq_ind B Abst (\lambda (b0: B).(pr3 c (THead (Bind b0) v2 
-(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3))) (let H22 
-\def (match (H21 (refl_equal B Abst)) in False return (\lambda (_: 
-False).(pr3 c (THead (Bind Abst) v2 (THead (Flat Appl) (lift (S O) O v1) t)) 
-(THead (Bind Abbr) x2 x3))) with []) in H22) b H17)))))))) H14)) H13))))))) 
-H9)) (\lambda (H9: (pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind 
-Abst) x0 x1)))).(pr3_t (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O 
-x2) (lift (S O) O (THead (Bind Abst) x0 x1)))) (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O v1) t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift 
-(S O) O v1) t) (THead (Flat Appl) (lift (S O) O x2) (lift (S O) O (THead 
-(Bind Abst) x0 x1))) (Bind b) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O 
-v1) (lift (S O) O x2) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop 
-(Bind b) O c c (drop_refl c) v2) v1 x2 H5) t (lift (S O) O (THead (Bind Abst) 
-x0 x1)) H9 Appl)) (THead (Bind Abbr) x2 x3) (eq_ind T (lift (S O) O (THead 
-(Flat Appl) x2 (THead (Bind Abst) x0 x1))) (\lambda (t0: T).(pr3 c (THead 
-(Bind b) v2 t0) (THead (Bind Abbr) x2 x3))) (pr3_sing c (THead (Bind Abbr) x2 
-x1) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst) 
-x0 x1)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 
-(THead (Bind Abst) x0 x1)))) (THead (Bind Abbr) x2 x1) (pr0_zeta b H (THead 
-(Flat Appl) x2 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x1) (pr0_beta 
-x0 x2 x2 (pr0_refl x2) x1 x1 (pr0_refl x1)) v2)) (THead (Bind Abbr) x2 x3) 
-(pr3_head_12 c x2 x2 (pr3_refl c x2) (Bind Abbr) x1 x3 (H7 Abbr x2))) (THead 
-(Flat Appl) (lift (S O) O x2) (lift (S O) O (THead (Bind Abst) x0 x1))) 
-(lift_flat Appl x2 (THead (Bind Abst) x0 x1) (S O) O)))) H8))) u2 H4))))))))) 
-H3)) (\lambda (H3: (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead 
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) 
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c v1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind 
-B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 z1)))))))) (\lambda 
-(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: 
-T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) 
-O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) 
-y2) z1 z2))))))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O 
-v1) t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0 
-Abst))).(\lambda (H5: (pr3 c (THead (Bind b) v2 t) (THead (Bind x0) x1 
-x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) 
-O x4) x3)) u2)).(\lambda (H7: (pr3 c v1 x4)).(\lambda (H8: (pr3 c x1 
-x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t (THead (Bind 
-x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O v1) t)) c (let H_x \def (pr3_gen_bind b H c v2 t 
-(THead (Bind x0) x1 x2) H5) in (let H10 \def H_x in (or_ind (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) (pr3 (CHead c (Bind 
-b) v2) t (lift (S O) O (THead (Bind x0) x1 x2))) (pr3 c (THead (Bind b) v2 
-(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat 
-Appl) (lift (S O) O x4) x3))) (\lambda (H11: (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 (CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 (CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat Appl) 
-(lift (S O) O x4) x3))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H12: (eq 
-T (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7))).(\lambda (H13: (pr3 c v2 
-x6)).(\lambda (H14: (pr3 (CHead c (Bind b) v2) t x7)).(let H15 \def (f_equal 
-T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) 
-\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) \Rightarrow (match 
-k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
-\Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in 
-((let H16 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ t0 
-_) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in 
-((let H17 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x2 | (TLRef _) \Rightarrow x2 | (THead _ _ 
-t0) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in 
-(\lambda (H18: (eq T x1 x6)).(\lambda (H19: (eq B x0 b)).(let H20 \def 
-(eq_ind_r T x7 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H14 x2 H17) 
-in (let H21 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c v2 t0)) H13 x1 H18) 
-in (let H22 \def (eq_ind B x0 (\lambda (b0: B).(pr3 (CHead c (Bind b0) x5) x2 
-x3)) H9 b H19) in (let H23 \def (eq_ind B x0 (\lambda (b0: B).(not (eq B b0 
-Abst))) H4 b H19) in (eq_ind_r B b (\lambda (b0: B).(pr3 c (THead (Bind b) v2 
-(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind b0) x5 (THead (Flat 
-Appl) (lift (S O) O x4) x3)))) (pr3_head_21 c v2 x5 (pr3_t x1 v2 c H21 x5 H8) 
-(Bind b) (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat Appl) (lift (S 
-O) O x4) x3) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O 
-x4) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c 
-(drop_refl c) v2) v1 x4 H7) t x3 (pr3_t x2 t (CHead c (Bind b) v2) H20 x3 
-(pr3_pr3_pr3_t c v2 x1 H21 x2 x3 (Bind b) (pr3_pr3_pr3_t c x1 x5 H8 x2 x3 
-(Bind b) H22))) Appl)) x0 H19)))))))) H16)) H15))))))) H11)) (\lambda (H11: 
-(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind x0) x1 x2)))).(pr3_t 
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead 
-(Bind x0) x1 x2)))) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) 
-t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat 
-Appl) (lift (S O) O x4) (lift (S O) O (THead (Bind x0) x1 x2))) (Bind b) 
-(pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O x4) (pr3_lift 
-(CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) v2) 
-v1 x4 H7) t (lift (S O) O (THead (Bind x0) x1 x2)) H11 Appl)) (THead (Bind 
-x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (eq_ind T (lift (S O) O 
-(THead (Flat Appl) x4 (THead (Bind x0) x1 x2))) (\lambda (t0: T).(pr3 c 
-(THead (Bind b) v2 t0) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O 
-x4) x3)))) (pr3_sing c (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O 
-x4) x2)) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x4 (THead (Bind 
-x0) x1 x2)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) 
-x4 (THead (Bind x0) x1 x2)))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S 
-O) O x4) x2)) (pr0_zeta b H (THead (Flat Appl) x4 (THead (Bind x0) x1 x2)) 
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr0_upsilon x0 
-H4 x4 x4 (pr0_refl x4) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2)) v2)) (THead 
-(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (pr3_head_12 c x1 x5 
-H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat Appl) 
-(lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H9 (lift (S 
-O) O x4) Appl))) (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead 
-(Bind x0) x1 x2))) (lift_flat Appl x4 (THead (Bind x0) x1 x2) (S O) O)))) 
-H10))) u2 H6))))))))))))) H3)) H2)))))))))).
-
-theorem pr3_iso_appls_appl_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (v: T).(\forall (u: 
-T).(\forall (t: T).(\forall (vs: TList).(let u1 \def (THeads (Flat Appl) vs 
-(THead (Flat Appl) v (THead (Bind b) u t))) in (\forall (c: C).(\forall (u2: 
-T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c 
-(THeads (Flat Appl) vs (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) 
-t))) u2)))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v: T).(\lambda 
-(u: T).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: 
-TList).(let u1 \def (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind 
-b) u t))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 
-u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead 
-(Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))) (\lambda (c: 
-C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) v (THead (Bind b) 
-u t)) u2)).(\lambda (H1: (((iso (THead (Flat Appl) v (THead (Bind b) u t)) 
-u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b H v u t c u2 H0 H1))))) 
-(\lambda (t0: T).(\lambda (t1: TList).(\lambda (H0: ((\forall (c: C).(\forall 
-(u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u 
-t))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind 
-b) u t))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1 
-(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))).(\lambda 
-(c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t0 (THeads 
-(Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t)))) u2)).(\lambda 
-(H2: (((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v 
-(THead (Bind b) u t)))) u2) \to (\forall (P: Prop).P)))).(let H3 \def 
-(pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind 
-b) u t))) u2 H1) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
-T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead 
-(Flat Appl) v (THead (Bind b) u t))) t2)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
-Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: 
-B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2)))))))) (ex6_6 B T T T 
-T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
-c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead 
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) 
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c 
-(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat 
-Appl) (lift (S O) O v) t)))) u2) (\lambda (H4: (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
-(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) 
-t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
-(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
-Appl) v (THead (Bind b) u t))) t2))) (pr3 c (THead (Flat Appl) t0 (THeads 
-(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) 
-u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T u2 (THead (Flat 
-Appl) x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat 
-Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) x1)).(let H8 \def 
-(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl) 
-t1 (THead (Flat Appl) v (THead (Bind b) u t)))) t2) \to (\forall (P: 
-Prop).P))) H2 (THead (Flat Appl) x0 x1) H5) in (eq_ind_r T (THead (Flat Appl) 
-x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 
-(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) t2)) (H8 
-(iso_head t0 x0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u 
-t))) x1 (Flat Appl)) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 
-(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) (THead (Flat 
-Appl) x0 x1))) u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
-Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: 
-B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))))).(ex4_4_ind T T 
-T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c 
-(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
-Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: 
-B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))) (pr3 c (THead 
-(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) 
-(lift (S O) O v) t)))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: 
-T).(\lambda (x3: T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) 
-u2)).(\lambda (H6: (pr3 c t0 x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t1 
-(THead (Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 
-x1))).(\lambda (H8: ((\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind 
-b0) u0) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0 
-(THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) 
-t)))) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Flat 
-Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S 
-O) O v) t)))) c (pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u 
-(THead (Flat Appl) (lift (S O) O v) t))) (THead (Bind Abst) x0 x1) (H0 c 
-(THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso (THeads (Flat Appl) t1 (THead 
-(Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 x1))).(\lambda (P: 
-Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 (THead (Bind b) u t) 
-t1 H9 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c (THead (Flat Appl) 
-t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr2_free c (THead 
-(Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr0_beta 
-x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) 
-x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c t0 x2 H6 (Bind Abbr) x1 x3 
-(H8 Abbr x2)) u2 H5)))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda 
-(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
-(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead 
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) 
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind 
-B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u 
-t))) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b0) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))) (pr3 c 
-(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat 
-Appl) (lift (S O) O v) t)))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda 
-(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not 
-(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t1 (THead (Flat 
-Appl) v (THead (Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c 
-(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda 
-(H8: (pr3 c t0 x4)).(\lambda (H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c 
-(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S 
-O) O x4) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u 
-(THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t (THead (Bind x0) x1 (THead 
-(Flat Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) 
-t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t 
-(THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0 (THeads 
-(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c 
-(pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) 
-(lift (S O) O v) t))) (THead (Bind x0) x1 x2) (H0 c (THead (Bind x0) x1 x2) 
-H6 (\lambda (H11: (iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead 
-(Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (P: 
-Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind b) u t) t1 
-H11 P)))) t0 Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) 
-x2)) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind 
-x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat 
-Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) 
-(lift (S O) O t0) x2)) (pr0_upsilon x0 H5 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl 
-x1) x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S 
-O) O x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat 
-Appl) (lift (S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2) 
-(pr3_head_12 (CHead c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4) 
-(pr3_lift (CHead c (Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c 
-(drop_refl c) x1) t0 x4 H8) (Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind 
-x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 
-(THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat 
-Appl) (lift (S O) O x4) x2)) c (pr3_head_12 c x1 x5 H9 (Bind x0) (THead (Flat 
-Appl) (lift (S O) O x4) x2) (THead (Flat Appl) (lift (S O) O x4) x3) 
-(pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 (lift (S O) O x4) Appl)) u2 
-H7)))))))))))))) H4)) H3))))))))) vs)))))).
-
-theorem pr3_iso_appls_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (vs: TList).(\forall (u: 
-T).(\forall (t: T).(let u1 \def (THeads (Flat Appl) vs (THead (Bind b) u t)) 
-in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
-(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) 
-(lifts (S O) O vs) t)) u2))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (vs: 
-TList).(tlist_ind_rew (\lambda (t: TList).(\forall (u: T).(\forall (t0: 
-T).(let u1 \def (THeads (Flat Appl) t (THead (Bind b) u t0)) in (\forall (c: 
-C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: 
-Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t) 
-t0)) u2))))))))) (\lambda (u: T).(\lambda (t: T).(\lambda (c: C).(\lambda 
-(u2: T).(\lambda (H0: (pr3 c (THead (Bind b) u t) u2)).(\lambda (_: (((iso 
-(THead (Bind b) u t) u2) \to (\forall (P: Prop).P)))).H0)))))) (\lambda (ts: 
-TList).(\lambda (t: T).(\lambda (H0: ((\forall (u: T).(\forall (t0: 
-T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) ts (THead 
-(Bind b) u t0)) u2) \to ((((iso (THeads (Flat Appl) ts (THead (Bind b) u t0)) 
-u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat 
-Appl) (lifts (S O) O ts) t0)) u2))))))))).(\lambda (u: T).(\lambda (t0: 
-T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THeads (Flat Appl) 
-(TApp ts t) (THead (Bind b) u t0)) u2)).(\lambda (H2: (((iso (THeads (Flat 
-Appl) (TApp ts t) (THead (Bind b) u t0)) u2) \to (\forall (P: 
-Prop).P)))).(eq_ind_r TList (TApp (lifts (S O) O ts) (lift (S O) O t)) 
-(\lambda (t1: TList).(pr3 c (THead (Bind b) u (THeads (Flat Appl) t1 t0)) 
-u2)) (eq_ind_r T (THeads (Flat Appl) (lifts (S O) O ts) (THead (Flat Appl) 
-(lift (S O) O t) t0)) (\lambda (t1: T).(pr3 c (THead (Bind b) u t1) u2)) (let 
-H3 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind b) u t0)) 
-(\lambda (t1: T).(pr3 c t1 u2)) H1 (THeads (Flat Appl) ts (THead (Flat Appl) 
-t (THead (Bind b) u t0))) (theads_tapp (Flat Appl) ts t (THead (Bind b) u 
-t0))) in (let H4 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind 
-b) u t0)) (\lambda (t1: T).((iso t1 u2) \to (\forall (P: Prop).P))) H2 
-(THeads (Flat Appl) ts (THead (Flat Appl) t (THead (Bind b) u t0))) 
-(theads_tapp (Flat Appl) ts t (THead (Bind b) u t0))) in (TList_ind (\lambda 
-(t1: TList).(((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall 
-(u3: T).((pr3 c0 (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to 
-((((iso (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to (\forall (P: 
-Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O 
-t1) t2)) u3)))))))) \to ((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) t 
-(THead (Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat 
-Appl) t (THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c 
-(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t1) (THead (Flat Appl) 
-(lift (S O) O t) t0))) u2))))) (\lambda (_: ((\forall (u0: T).(\forall (t1: 
-T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) TNil (THead 
-(Bind b) u0 t1)) u3) \to ((((iso (THeads (Flat Appl) TNil (THead (Bind b) u0 
-t1)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads 
-(Flat Appl) (lifts (S O) O TNil) t1)) u3))))))))).(\lambda (H6: (pr3 c 
-(THeads (Flat Appl) TNil (THead (Flat Appl) t (THead (Bind b) u t0))) 
-u2)).(\lambda (H7: (((iso (THeads (Flat Appl) TNil (THead (Flat Appl) t 
-(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b 
-H t u t0 c u2 H6 H7)))) (\lambda (t1: T).(\lambda (ts0: TList).(\lambda (_: 
-((((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall (u3: T).((pr3 
-c0 (THeads (Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads 
-(Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to 
-(pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O ts0) t2)) 
-u3)))))))) \to ((pr3 c (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead 
-(Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) ts0 (THead (Flat Appl) t 
-(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead 
-(Bind b) u (THeads (Flat Appl) (lifts (S O) O ts0) (THead (Flat Appl) (lift 
-(S O) O t) t0))) u2)))))).(\lambda (H5: ((\forall (u0: T).(\forall (t2: 
-T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) (TCons t1 
-ts0) (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads (Flat Appl) (TCons t1 
-ts0) (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 
-(THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O (TCons t1 ts0)) t2)) 
-u3))))))))).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t1 ts0) (THead 
-(Flat Appl) t (THead (Bind b) u t0))) u2)).(\lambda (H7: (((iso (THeads (Flat 
-Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) u2) \to 
-(\forall (P: Prop).P)))).(H5 u (THead (Flat Appl) (lift (S O) O t) t0) c u2 
-(pr3_iso_appls_appl_bind b H t u t0 (TCons t1 ts0) c u2 H6 H7) (\lambda (H8: 
-(iso (THeads (Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl) 
-(lift (S O) O t) t0))) u2)).(\lambda (P: Prop).(H7 (iso_trans (THeads (Flat 
-Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) (THeads 
-(Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl) (lift (S O) O 
-t) t0))) (iso_head t1 t1 (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead 
-(Bind b) u t0))) (THeads (Flat Appl) ts0 (THead (Bind b) u (THead (Flat Appl) 
-(lift (S O) O t) t0))) (Flat Appl)) u2 H8) P)))))))))) ts H0 H3 H4))) (THeads 
-(Flat Appl) (TApp (lifts (S O) O ts) (lift (S O) O t)) t0) (theads_tapp (Flat 
-Appl) (lifts (S O) O ts) (lift (S O) O t) t0)) (lifts (S O) O (TApp ts t)) 
-(lifts_tapp (S O) O t ts))))))))))) vs))).
-
-theorem pr3_iso_beta:
- \forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 \def (THead (Flat 
-Appl) v (THead (Bind Abst) w t)) in (\forall (c: C).(\forall (u2: T).((pr3 c 
-u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind 
-Abbr) v t) u2))))))))
-\def
- \lambda (v: T).(\lambda (w: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: 
-T).(\lambda (H: (pr3 c (THead (Flat Appl) v (THead (Bind Abst) w t)) 
-u2)).(\lambda (H0: (((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) u2) 
-\to (\forall (P: Prop).P)))).(let H1 \def (pr3_gen_appl c v (THead (Bind 
-Abst) w t) u2 H) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
-T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v 
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) w t) t2)))) 
-(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: 
-T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) 
-w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind 
-b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c v u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c 
-(THead (Bind Abbr) v t) u2) (\lambda (H2: (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
-(THead (Bind Abst) w t) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: 
-T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) 
-w t) t2))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H3: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v 
-x0)).(\lambda (_: (pr3 c (THead (Bind Abst) w t) x1)).(let H6 \def (eq_ind T 
-u2 (\lambda (t0: T).((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) t0) 
-\to (\forall (P: Prop).P))) H0 (THead (Flat Appl) x0 x1) H3) in (eq_ind_r T 
-(THead (Flat Appl) x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind Abbr) v t) 
-t0)) (H6 (iso_head v x0 (THead (Bind Abst) w t) x1 (Flat Appl)) (pr3 c (THead 
-(Bind Abbr) v t) (THead (Flat Appl) x0 x1))) u2 H3))))))) H2)) (\lambda (H2: 
-(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: 
-T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) 
-w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind 
-b) u) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v 
-u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Bind Abbr) v t) 
-u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H3: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H4: (pr3 c v 
-x2)).(\lambda (H5: (pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) x0 
-x1))).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) 
-u) x1 x3))))).(let H7 \def (pr3_gen_abst c w t (THead (Bind Abst) x0 x1) H5) 
-in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst) 
-x0 x1) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w 
-u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda 
-(x4: T).(\lambda (x5: T).(\lambda (H8: (eq T (THead (Bind Abst) x0 x1) (THead 
-(Bind Abst) x4 x5))).(\lambda (H9: (pr3 c w x4)).(\lambda (H10: ((\forall (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x5))))).(let H11 \def (f_equal 
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) 
-(THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in ((let H12 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) 
-\Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in 
-(\lambda (H13: (eq T x0 x4)).(let H14 \def (eq_ind_r T x5 (\lambda (t0: 
-T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) H10 x1 
-H12) in (let H15 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c w t0)) H9 x0 
-H13) in (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) v t) c 
-(pr3_head_12 c v x2 H4 (Bind Abbr) t x3 (pr3_t x1 t (CHead c (Bind Abbr) x2) 
-(H14 Abbr x2) x3 (H6 Abbr x2))) u2 H3))))) H11))))))) H7)))))))))) H2)) 
-(\lambda (H2: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead (Bind b) y1 
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
-T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat 
-Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
-(THead (Bind Abst) w t) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda 
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) 
-u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2))))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: B).(\lambda 
-(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: 
-T).(\lambda (H3: (not (eq B x0 Abst))).(\lambda (H4: (pr3 c (THead (Bind 
-Abst) w t) (THead (Bind x0) x1 x2))).(\lambda (H5: (pr3 c (THead (Bind x0) x5 
-(THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (_: (pr3 c v 
-x4)).(\lambda (_: (pr3 c x1 x5)).(\lambda (H8: (pr3 (CHead c (Bind x0) x5) x2 
-x3)).(let H9 \def (pr3_gen_abst c w t (THead (Bind x0) x1 x2) H4) in 
-(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 
-x2) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w 
-u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda 
-(x6: T).(\lambda (x7: T).(\lambda (H10: (eq T (THead (Bind x0) x1 x2) (THead 
-(Bind Abst) x6 x7))).(\lambda (H11: (pr3 c w x6)).(\lambda (H12: ((\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x7))))).(let H13 \def 
-(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with 
-[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) 
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) 
-\Rightarrow b | (Flat _) \Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead 
-(Bind Abst) x6 x7) H10) in ((let H14 \def (f_equal T T (\lambda (e: T).(match 
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) 
-\Rightarrow x1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind x0) x1 x2) 
-(THead (Bind Abst) x6 x7) H10) in ((let H15 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 | 
-(TLRef _) \Rightarrow x2 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind x0) 
-x1 x2) (THead (Bind Abst) x6 x7) H10) in (\lambda (H16: (eq T x1 
-x6)).(\lambda (H17: (eq B x0 Abst)).(let H18 \def (eq_ind_r T x7 (\lambda 
-(t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) 
-H12 x2 H15) in (let H19 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c w t0)) 
-H11 x1 H16) in (let H20 \def (eq_ind B x0 (\lambda (b: B).(pr3 (CHead c (Bind 
-b) x5) x2 x3)) H8 Abst H17) in (let H21 \def (eq_ind B x0 (\lambda (b: 
-B).(pr3 c (THead (Bind b) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)) 
-H5 Abst H17) in (let H22 \def (eq_ind B x0 (\lambda (b: B).(not (eq B b 
-Abst))) H3 Abst H17) in (let H23 \def (match (H22 (refl_equal B Abst)) in 
-False return (\lambda (_: False).(pr3 c (THead (Bind Abbr) v t) u2)) with []) 
-in H23))))))))) H14)) H13))))))) H9)))))))))))))) H2)) H1)))))))).
-
-theorem pr3_iso_appls_beta:
- \forall (us: TList).(\forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 
-\def (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) w t))) in 
-(\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
-(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) us (THead (Bind Abbr) 
-v t)) u2)))))))))
-\def
- \lambda (us: TList).(TList_ind (\lambda (t: TList).(\forall (v: T).(\forall 
-(w: T).(\forall (t0: T).(let u1 \def (THeads (Flat Appl) t (THead (Flat Appl) 
-v (THead (Bind Abst) w t0))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 
-u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat 
-Appl) t (THead (Bind Abbr) v t0)) u2)))))))))) (\lambda (v: T).(\lambda (w: 
-T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H: (pr3 c 
-(THead (Flat Appl) v (THead (Bind Abst) w t)) u2)).(\lambda (H0: (((iso 
-(THead (Flat Appl) v (THead (Bind Abst) w t)) u2) \to (\forall (P: 
-Prop).P)))).(pr3_iso_beta v w t c u2 H H0)))))))) (\lambda (t: T).(\lambda 
-(t0: TList).(\lambda (H: ((\forall (v: T).(\forall (w: T).(\forall (t1: 
-T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (THead 
-(Flat Appl) v (THead (Bind Abst) w t1))) u2) \to ((((iso (THeads (Flat Appl) 
-t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) u2) \to (\forall (P: 
-Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1)) 
-u2)))))))))).(\lambda (v: T).(\lambda (w: T).(\lambda (t1: T).(\lambda (c: 
-C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) t (THeads (Flat 
-Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) u2)).(\lambda (H1: 
-(((iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Flat Appl) v 
-(THead (Bind Abst) w t1)))) u2) \to (\forall (P: Prop).P)))).(let H2 \def 
-(pr3_gen_appl c t (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind 
-Abst) w t1))) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
-T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) 
-t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2)))) (ex4_4 T T T T 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c 
-(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
-Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
-(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c 
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) 
-(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
-(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
-Appl) v (THead (Bind Abst) w t1))) t2))))).(ex3_2_ind T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
-(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2))) 
-(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) 
-u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat 
-Appl) x0 x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat 
-Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) x1)).(let H7 \def 
-(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t (THeads (Flat Appl) 
-t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) t2) \to (\forall (P: 
-Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl) 
-x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 
-(THead (Bind Abbr) v t1))) t2)) (H7 (iso_head t x0 (THeads (Flat Appl) t0 
-(THead (Flat Appl) v (THead (Bind Abst) w t1))) x1 (Flat Appl)) (pr3 c (THead 
-(Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) (THead (Flat 
-Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
-Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T T T 
-T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c 
-(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
-Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Flat 
-Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c 
-(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t x2)).(\lambda (H6: (pr3 
-c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) 
-(THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b: B).(\forall (u: 
-T).(pr3 (CHead c (Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) 
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c 
-(pr3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads 
-(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind Abst) x0 x1) (H v w t1 
-c (THead (Bind Abst) x0 x1) H6 (\lambda (H8: (iso (THeads (Flat Appl) t0 
-(THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) x0 
-x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 
-(THead (Bind Abst) w t1) t0 H8 P)))) t Appl) (THead (Bind Abbr) t x1) 
-(pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) 
-t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead 
-(Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 (pr0_refl x1))))) u2 
-(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t x1) c (pr3_head_12 c t 
-x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3: 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) 
-w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c 
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 
-z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
-Appl) v (THead (Bind Abst) w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda 
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) 
-u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead 
-(Bind Abbr) v t1))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: 
-T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq 
-B x0 Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v 
-(THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c 
-(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda 
-(H7: (pr3 c t x4)).(\lambda (H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c 
-(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S 
-O) O x4) x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) 
-v t1))) c (pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) 
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c 
-(pr3_t (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads 
-(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind x0) x1 x2) (H v w t1 c 
-(THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads (Flat Appl) t0 (THead 
-(Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda 
-(P: Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind Abst) 
-w t1) t0 H10 P)))) t Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) 
-O t) x2)) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead 
-(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr2_free c (THead 
-(Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) 
-(lift (S O) O t) x2)) (pr0_upsilon x0 H4 t t (pr0_refl t) x1 x1 (pr0_refl x1) 
-x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O 
-x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl) 
-(lift (S O) O t) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 
-(CHead c (Bind x0) x1) (lift (S O) O t) (lift (S O) O x4) (pr3_lift (CHead c 
-(Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H7) 
-(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift 
-(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S 
-O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c 
-(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) 
-(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) 
-x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))))))) us).
-
-theorem csuba_gen_abst_rev:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c 
-(CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: 
-(csuba g c (CHead d1 (Bind Abst) u))).(let H0 \def (match H in csuba return 
-(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 c) 
-\to ((eq C c1 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq C c 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) with 
-[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1: 
-(eq C (CSort n) (CHead d1 (Bind Abst) u))).(eq_ind C (CSort n) (\lambda (c0: 
-C).((eq C (CSort n) (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq 
-C c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda 
-(H2: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H3 \def (eq_ind C (CSort 
-n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abst) 
-u) H2) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) H3))) c H0 H1))) | (csuba_head 
-c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) c)).(\lambda 
-(H2: (eq C (CHead c2 k u0) (CHead d1 (Bind Abst) u))).(eq_ind C (CHead c1 k 
-u0) (\lambda (c0: C).((eq C (CHead c2 k u0) (CHead d1 (Bind Abst) u)) \to 
-((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abst) 
-u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda (H3: (eq C (CHead c2 k 
-u0) (CHead d1 (Bind Abst) u))).(let H4 \def (f_equal C T (\lambda (e: 
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | 
-(CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) 
-in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda 
-(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) 
-(CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) in ((let H6 \def (f_equal C C 
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 
-(Bind Abst) u) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind Abst)) \to 
-((eq T u0 u) \to ((csuba g c1 c0) \to (ex2 C (\lambda (d2: C).(eq C (CHead c1 
-k u0) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))))) 
-(\lambda (H7: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda (k0: 
-K).((eq T u0 u) \to ((csuba g c1 d1) \to (ex2 C (\lambda (d2: C).(eq C (CHead 
-c1 k0 u0) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))) 
-(\lambda (H8: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((csuba g c1 d1) \to 
-(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst) 
-u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda (H9: (csuba g c1 
-d1)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) u) (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 
-(Bind Abst) u)) H9)) u0 (sym_eq T u0 u H8))) k (sym_eq K k (Bind Abst) H7))) 
-c2 (sym_eq C c2 d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a 
-H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) 
-c)).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst) 
-u))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2 
-(Bind Abbr) u0) (CHead d1 (Bind Abst) u)) \to ((csuba g c1 c2) \to ((arity g 
-c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c0 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) (\lambda 
-(H5: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst) u))).(let H6 \def 
-(eq_ind C (CHead c2 (Bind Abbr) u0) (\lambda (e: C).(match e in C return 
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) 
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow 
-True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) 
-\Rightarrow False])])) I (CHead d1 (Bind Abst) u) H5) in (False_ind ((csuba g 
-c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C 
-(\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1)))))) H6))) c H3 H4 H0 H1 H2)))]) in (H0 
-(refl_equal C c) (refl_equal C (CHead d1 (Bind Abst) u)))))))).
-
-theorem csuba_gen_void_rev:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c 
-(CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind 
-Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: 
-(csuba g c (CHead d1 (Bind Void) u))).(let H0 \def (match H in csuba return 
-(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 c) 
-\to ((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c 
-(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) with 
-[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1: 
-(eq C (CSort n) (CHead d1 (Bind Void) u))).(eq_ind C (CSort n) (\lambda (c0: 
-C).((eq C (CSort n) (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq 
-C c0 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda 
-(H2: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H3 \def (eq_ind C (CSort 
-n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Void) 
-u) H2) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind 
-Void) u))) (\lambda (d2: C).(csuba g d2 d1))) H3))) c H0 H1))) | (csuba_head 
-c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) c)).(\lambda 
-(H2: (eq C (CHead c2 k u0) (CHead d1 (Bind Void) u))).(eq_ind C (CHead c1 k 
-u0) (\lambda (c0: C).((eq C (CHead c2 k u0) (CHead d1 (Bind Void) u)) \to 
-((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Void) 
-u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda (H3: (eq C (CHead c2 k 
-u0) (CHead d1 (Bind Void) u))).(let H4 \def (f_equal C T (\lambda (e: 
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | 
-(CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3) 
-in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda 
-(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) 
-(CHead c2 k u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \def (f_equal C C 
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 
-(Bind Void) u) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind Void)) \to 
-((eq T u0 u) \to ((csuba g c1 c0) \to (ex2 C (\lambda (d2: C).(eq C (CHead c1 
-k u0) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))))) 
-(\lambda (H7: (eq K k (Bind Void))).(eq_ind K (Bind Void) (\lambda (k0: 
-K).((eq T u0 u) \to ((csuba g c1 d1) \to (ex2 C (\lambda (d2: C).(eq C (CHead 
-c1 k0 u0) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))) 
-(\lambda (H8: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((csuba g c1 d1) \to 
-(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) t) (CHead d2 (Bind Void) 
-u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda (H9: (csuba g c1 
-d1)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u) (CHead d2 
-(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 
-(Bind Void) u)) H9)) u0 (sym_eq T u0 u H8))) k (sym_eq K k (Bind Void) H7))) 
-c2 (sym_eq C c2 d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a 
-H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) 
-c)).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void) 
-u))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2 
-(Bind Abbr) u0) (CHead d1 (Bind Void) u)) \to ((csuba g c1 c2) \to ((arity g 
-c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c0 
-(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) (\lambda 
-(H5: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void) u))).(let H6 \def 
-(eq_ind C (CHead c2 (Bind Abbr) u0) (\lambda (e: C).(match e in C return 
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) 
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow 
-True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) 
-\Rightarrow False])])) I (CHead d1 (Bind Void) u) H5) in (False_ind ((csuba g 
-c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C 
-(\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u))) 
-(\lambda (d2: C).(csuba g d2 d1)))))) H6))) c H3 H4 H0 H1 H2)))]) in (H0 
-(refl_equal C c) (refl_equal C (CHead d1 (Bind Void) u)))))))).
-
-theorem csuba_gen_abbr_rev:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g c 
-(CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-a))))))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda 
-(H: (csuba g c (CHead d1 (Bind Abbr) u1))).(let H0 \def (match H in csuba 
-return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C 
-c0 c) \to ((eq C c1 (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2: 
-C).(eq C c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead 
-d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))))))))) with [(csuba_sort n) \Rightarrow (\lambda 
-(H0: (eq C (CSort n) c)).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr) 
-u1))).(eq_ind C (CSort n) (\lambda (c0: C).((eq C (CSort n) (CHead d1 (Bind 
-Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))))) 
-(\lambda (H2: (eq C (CSort n) (CHead d1 (Bind Abbr) u1))).(let H3 \def 
-(eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_: 
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
-False])) I (CHead d1 (Bind Abbr) u1) H2) in (False_ind (or (ex2 C (\lambda 
-(d2: C).(eq C (CSort n) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g 
-d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C 
-(CSort n) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) H3))) c H0 H1))) | (csuba_head 
-c1 c2 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) c)).(\lambda 
-(H2: (eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1))).(eq_ind C (CHead c1 k 
-u) (\lambda (c0: C).((eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1)) \to 
-((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))) 
-(\lambda (H3: (eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1))).(let H4 \def 
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u) 
-(CHead d1 (Bind Abbr) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: 
-C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | 
-(CHead _ k0 _) \Rightarrow k0])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) 
-in ((let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda 
-(_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) 
-(CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in (eq_ind C d1 (\lambda (c0: 
-C).((eq K k (Bind Abbr)) \to ((eq T u u1) \to ((csuba g c1 c0) \to (or (ex2 C 
-(\lambda (d2: C).(eq C (CHead c1 k u) (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(eq C (CHead c1 k u) (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))))))) 
-(\lambda (H7: (eq K k (Bind Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0: 
-K).((eq T u u1) \to ((csuba g c1 d1) \to (or (ex2 C (\lambda (d2: C).(eq C 
-(CHead c1 k0 u) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C 
-(CHead c1 k0 u) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))))))) (\lambda (H8: (eq T u 
-u1)).(eq_ind T u1 (\lambda (t: T).((csuba g c1 d1) \to (or (ex2 C (\lambda 
-(d2: C).(eq C (CHead c1 (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(eq C (CHead c1 (Bind Abbr) t) (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-a)))))))) (\lambda (H9: (csuba g c1 d1)).(or_introl (ex2 C (\lambda (d2: 
-C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C 
-(\lambda (d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind Abbr) u1)) 
-H9))) u (sym_eq T u u1 H8))) k (sym_eq K k (Bind Abbr) H7))) c2 (sym_eq C c2 
-d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u H2) 
-\Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) c)).(\lambda (H4: 
-(eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1))).(eq_ind C (CHead 
-c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2 (Bind Abbr) u) (CHead d1 
-(Bind Abbr) u1)) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to 
-((arity g c2 u a) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g a0))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 a0))))))))))) 
-(\lambda (H5: (eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1))).(let 
-H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) 
-with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 
-(Bind Abbr) u) (CHead d1 (Bind Abbr) u1) H5) in ((let H7 \def (f_equal C C 
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind Abbr) u) 
-(CHead d1 (Bind Abbr) u1) H5) in (eq_ind C d1 (\lambda (c0: C).((eq T u u1) 
-\to ((csuba g c1 c0) \to ((arity g c1 t (asucc g a)) \to ((arity g c0 u a) 
-\to (or (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: 
-A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a0: A).(arity g d1 u1 a0))))))))))) (\lambda (H8: (eq T u u1)).(eq_ind T u1 
-(\lambda (t0: T).((csuba g c1 d1) \to ((arity g c1 t (asucc g a)) \to ((arity 
-g d1 t0 a) \to (or (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) 
-t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: 
-A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a0: A).(arity g d1 u1 a0)))))))))) (\lambda (H9: (csuba g c1 d1)).(\lambda 
-(H10: (arity g c1 t (asucc g a))).(\lambda (H11: (arity g d1 u1 
-a)).(or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead 
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: 
-A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a0: A).(arity g d1 u1 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g 
-a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 
-a0)))) c1 t a (refl_equal C (CHead c1 (Bind Abst) t)) H9 H10 H11))))) u 
-(sym_eq T u u1 H8))) c2 (sym_eq C c2 d1 H7))) H6))) c H3 H4 H0 H1 H2)))]) in 
-(H0 (refl_equal C c) (refl_equal C (CHead d1 (Bind Abbr) u1)))))))).
-
-theorem csuba_gen_flat_rev:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall 
-(f: F).((csuba g c (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda 
-(f: F).(\lambda (H: (csuba g c (CHead d1 (Flat f) u1))).(let H0 \def (match H 
-in csuba return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 
-c1)).((eq C c0 c) \to ((eq C c1 (CHead d1 (Flat f) u1)) \to (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) with [(csuba_sort n) 
-\Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1: (eq C (CSort n) 
-(CHead d1 (Flat f) u1))).(eq_ind C (CSort n) (\lambda (c0: C).((eq C (CSort 
-n) (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(eq C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba 
-g d2 d1)))))) (\lambda (H2: (eq C (CSort n) (CHead d1 (Flat f) u1))).(let H3 
-\def (eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_: 
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
-False])) I (CHead d1 (Flat f) u1) H2) in (False_ind (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))) H3))) c H0 H1))) | (csuba_head c1 c2 H0 
-k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) c)).(\lambda (H2: (eq C 
-(CHead c2 k u) (CHead d1 (Flat f) u1))).(eq_ind C (CHead c1 k u) (\lambda 
-(c0: C).((eq C (CHead c2 k u) (CHead d1 (Flat f) u1)) \to ((csuba g c1 c2) 
-\to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Flat f) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (H3: 
-(eq C (CHead c2 k u) (CHead d1 (Flat f) u1))).(let H4 \def (f_equal C T 
-(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
-\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Flat 
-f) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return 
-(\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow 
-k0])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C C 
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u) (CHead d1 
-(Flat f) u1) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Flat f)) \to ((eq 
-T u u1) \to ((csuba g c1 c0) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(eq C (CHead c1 k u) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1)))))))) (\lambda (H7: (eq K k (Flat f))).(eq_ind K 
-(Flat f) (\lambda (k0: K).((eq T u u1) \to ((csuba g c1 d1) \to (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k0 u) (CHead d2 (Flat f) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (H8: 
-(eq T u u1)).(eq_ind T u1 (\lambda (t: T).((csuba g c1 d1) \to (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Flat f) t) (CHead d2 (Flat 
-f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (\lambda (H9: 
-(csuba g c1 d1)).(ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(eq C 
-(CHead c1 (Flat f) u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1))) c1 u1 (refl_equal C (CHead c1 (Flat f) u1)) H9)) u 
-(sym_eq T u u1 H8))) k (sym_eq K k (Flat f) H7))) c2 (sym_eq C c2 d1 H6))) 
-H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u H2) \Rightarrow 
-(\lambda (H3: (eq C (CHead c1 (Bind Abst) t) c)).(\lambda (H4: (eq C (CHead 
-c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(eq_ind C (CHead c1 (Bind Abst) t) 
-(\lambda (c0: C).((eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1)) \to 
-((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u a) \to 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Flat f) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) (\lambda (H5: 
-(eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind 
-C (CHead c2 (Bind Abbr) u) (\lambda (e: C).(match e in C return (\lambda (_: 
-C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match 
-k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat 
-_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind ((csuba 
-g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u a) \to (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 
-(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H6))) 
-c H3 H4 H0 H1 H2)))]) in (H0 (refl_equal C c) (refl_equal C (CHead d1 (Flat 
-f) u1))))))))).
-
-theorem csuba_gen_bind_rev:
- \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall 
-(v1: T).((csuba g c2 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))))))
-\def
- \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda 
-(v1: T).(\lambda (H: (csuba g c2 (CHead e1 (Bind b1) v1))).(let H0 \def 
-(match H in csuba return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csuba 
-? c c0)).((eq C c c2) \to ((eq C c0 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind 
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 
-e1)))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) 
-c2)).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind b1) v1))).(eq_ind C (CSort 
-n) (\lambda (c: C).((eq C (CSort n) (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind 
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 
-e1))))))) (\lambda (H2: (eq C (CSort n) (CHead e1 (Bind b1) v1))).(let H3 
-\def (eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_: 
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
-False])) I (CHead e1 (Bind b1) v1) H2) in (False_ind (ex2_3 B C T (\lambda 
-(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 (Bind b2) 
-v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))) 
-H3))) c2 H0 H1))) | (csuba_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C 
-(CHead c1 k u) c2)).(\lambda (H2: (eq C (CHead c0 k u) (CHead e1 (Bind b1) 
-v1))).(eq_ind C (CHead c1 k u) (\lambda (c: C).((eq C (CHead c0 k u) (CHead 
-e1 (Bind b1) v1)) \to ((csuba g c1 c0) \to (ex2_3 B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))) 
-(\lambda (H3: (eq C (CHead c0 k u) (CHead e1 (Bind b1) v1))).(let H4 \def 
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u) 
-(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: 
-C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | 
-(CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u) (CHead e1 (Bind b1) v1) H3) 
-in ((let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda 
-(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) 
-(CHead c0 k u) (CHead e1 (Bind b1) v1) H3) in (eq_ind C e1 (\lambda (c: 
-C).((eq K k (Bind b1)) \to ((eq T u v1) \to ((csuba g c1 c) \to (ex2_3 B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k u) 
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
-T).(csuba g e2 e1))))))))) (\lambda (H7: (eq K k (Bind b1))).(eq_ind K (Bind 
-b1) (\lambda (k0: K).((eq T u v1) \to ((csuba g c1 e1) \to (ex2_3 B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k0 u) 
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
-T).(csuba g e2 e1)))))))) (\lambda (H8: (eq T u v1)).(eq_ind T v1 (\lambda 
-(t: T).((csuba g c1 e1) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
-C).(\lambda (v2: T).(eq C (CHead c1 (Bind b1) t) (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) 
-(\lambda (H9: (csuba g c1 e1)).(let H10 \def (eq_ind T u (\lambda (t: T).(eq 
-C (CHead c1 k t) c2)) H1 v1 H8) in (let H11 \def (eq_ind K k (\lambda (k0: 
-K).(eq C (CHead c1 k0 v1) c2)) H10 (Bind b1) H7) in (let H12 \def (eq_ind_r C 
-c2 (\lambda (c: C).(csuba g c (CHead e1 (Bind b1) v1))) H (CHead c1 (Bind b1) 
-v1) H11) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda 
-(v2: T).(eq C (CHead c1 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda 
-(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))) b1 c1 v1 
-(refl_equal C (CHead c1 (Bind b1) v1)) H9))))) u (sym_eq T u v1 H8))) k 
-(sym_eq K k (Bind b1) H7))) c0 (sym_eq C c0 e1 H6))) H5)) H4))) c2 H1 H2 
-H0))) | (csuba_abst c1 c0 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C 
-(CHead c1 (Bind Abst) t) c2)).(\lambda (H4: (eq C (CHead c0 (Bind Abbr) u) 
-(CHead e1 (Bind b1) v1))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c: 
-C).((eq C (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1)) \to ((csuba g c1 
-c0) \to ((arity g c1 t (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind 
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 
-e1)))))))))) (\lambda (H5: (eq C (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) 
-v1))).(let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda 
-(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) 
-(CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def 
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with 
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return 
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
-Abbr])])) (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H8 
-\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) 
-with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 
-(Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in (eq_ind C e1 (\lambda (c: 
-C).((eq B Abbr b1) \to ((eq T u v1) \to ((csuba g c1 c) \to ((arity g c1 t 
-(asucc g a)) \to ((arity g c u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda 
-(e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) 
-v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 
-e1))))))))))) (\lambda (H9: (eq B Abbr b1)).(eq_ind B Abbr (\lambda (_: 
-B).((eq T u v1) \to ((csuba g c1 e1) \to ((arity g c1 t (asucc g a)) \to 
-((arity g e1 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda 
-(v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) v2))))) (\lambda 
-(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))))) (\lambda 
-(H10: (eq T u v1)).(eq_ind T v1 (\lambda (t0: T).((csuba g c1 e1) \to ((arity 
-g c1 t (asucc g a)) \to ((arity g e1 t0 a) \to (ex2_3 B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 
-(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g 
-e2 e1))))))))) (\lambda (H11: (csuba g c1 e1)).(\lambda (_: (arity g c1 t 
-(asucc g a))).(\lambda (_: (arity g e1 v1 a)).(let H14 \def (eq_ind_r C c2 
-(\lambda (c: C).(csuba g c (CHead e1 (Bind b1) v1))) H (CHead c1 (Bind Abst) 
-t) H3) in (let H15 \def (eq_ind_r B b1 (\lambda (b: B).(csuba g (CHead c1 
-(Bind Abst) t) (CHead e1 (Bind b) v1))) H14 Abbr H9) in (ex2_3_intro B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind 
-Abst) t) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csuba g e2 e1)))) Abst c1 t (refl_equal C (CHead c1 (Bind 
-Abst) t)) H11)))))) u (sym_eq T u v1 H10))) b1 H9)) c0 (sym_eq C c0 e1 H8))) 
-H7)) H6))) c2 H3 H4 H0 H1 H2)))]) in (H0 (refl_equal C c2) (refl_equal C 
-(CHead e1 (Bind b1) v1))))))))).
-
-theorem csuba_clear_trans:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c2 c1) \to 
-(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) 
-(\lambda (e2: C).(clear c2 e2))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c2 
-c1)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear 
-c0 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c 
-e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) 
-e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e2 e1)) 
-(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: 
-C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c4 
-e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c3 
-e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: 
-(clear (CHead c4 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c4 k0 u) 
-e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear 
-(CHead c3 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c4 (Bind 
-b) u) e1)).(eq_ind_r C (CHead c4 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda 
-(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind b) u) e2)))) 
-(ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind b) u))) (\lambda 
-(e2: C).(clear (CHead c3 (Bind b) u) e2)) (CHead c3 (Bind b) u) (csuba_head g 
-c3 c4 H0 (Bind b) u) (clear_bind b c3 u)) e1 (clear_gen_bind b c4 e1 u H3)))) 
-(\lambda (f: F).(\lambda (H3: (clear (CHead c4 (Flat f) u) e1)).(let H4 \def 
-(H1 e1 (clear_gen_flat f c4 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g 
-e2 e1)) (\lambda (e2: C).(clear c3 e2)) (ex2 C (\lambda (e2: C).(csuba g e2 
-e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f) u) e2))) (\lambda (x: 
-C).(\lambda (H5: (csuba g x e1)).(\lambda (H6: (clear c3 x)).(ex_intro2 C 
-(\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f) 
-u) e2)) x H5 (clear_flat c3 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: 
-C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall 
-(e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda 
-(e2: C).(clear c3 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H2: 
-(arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u 
-a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c4 (Bind Abbr) u) 
-e1)).(eq_ind_r C (CHead c4 (Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda 
-(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) 
-e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind Abbr) u))) 
-(\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) e2)) (CHead c3 (Bind Abst) 
-t) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abst c3 t)) e1 
-(clear_gen_bind Abbr c4 e1 u H4))))))))))))) c2 c1 H)))).
-
-theorem csuba_drop_abst_rev:
- \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i 
-O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g 
-c2 c1) \to (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1))))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: 
-C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: 
-G).(\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda (d2: C).(drop n O c2 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))))))))) 
-(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 
-(CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: 
-(csuba g c2 c1)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c2 c)) H0 
-(CHead d1 (Bind Abst) u) (drop_gen_refl c1 (CHead d1 (Bind Abst) u) H)) in 
-(let H_x \def (csuba_gen_abst_rev g d1 c2 u H1) in (let H2 \def H_x in 
-(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x: C).(\lambda (H3: 
-(eq C c2 (CHead x (Bind Abst) u))).(\lambda (H4: (csuba g x d1)).(eq_ind_r C 
-(CHead x (Bind Abst) u) (\lambda (c: C).(ex2 C (\lambda (d2: C).(drop O O c 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (ex_intro2 C 
-(\lambda (d2: C).(drop O O (CHead x (Bind Abst) u) (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1)) x (drop_refl (CHead x (Bind Abst) u)) H4) 
-c2 H3)))) H2))))))))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1: 
-C).(\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abst) u)) 
-\to (\forall (g: G).(\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda 
-(d2: C).(drop n O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
-d1)))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: 
-C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall 
-(g: G).(\forall (c2: C).((csuba g c2 c) \to (ex2 C (\lambda (d2: C).(drop (S 
-n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))))) 
-(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n) 
-O (CSort n0) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2: 
-C).(\lambda (_: (csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind 
-Abst) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (ex2 C (\lambda (d2: 
-C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
-d1))) (\lambda (H2: (eq C (CHead d1 (Bind Abst) u) (CSort n0))).(\lambda (_: 
-(eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (match H2 in eq 
-return (\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c (CSort n0)) \to (ex2 
-C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1)))))) with [refl_equal \Rightarrow (\lambda (H5: (eq C 
-(CHead d1 (Bind Abst) u) (CSort n0))).(let H6 \def (eq_ind C (CHead d1 (Bind 
-Abst) u) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with 
-[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n0) 
-H5) in (False_ind (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) H6)))]) in (H5 (refl_equal C 
-(CSort n0))))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u) 
-H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u: 
-T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall 
-(c2: C).((csuba g c2 c) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead 
-d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))))))))).(\lambda (k: 
-K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n) 
-O (CHead c k t) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2: 
-C).(\lambda (H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: K).((csuba 
-g c2 (CHead c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abst) u)) \to 
-(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda 
-(d2: C).(csuba g d2 d1)))))) (\lambda (b: B).(\lambda (H3: (csuba g c2 (CHead 
-c (Bind b) t))).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abst) 
-u))).(B_ind (\lambda (b0: B).((csuba g c2 (CHead c (Bind b0) t)) \to ((drop 
-(r (Bind b0) n) O c (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: 
-C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
-d1)))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) t))).(\lambda (H6: 
-(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abst) u))).(let H_x \def 
-(csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C 
-(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba 
-g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq 
-C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g c t a))))) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (H8: (ex2 C 
-(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba 
-g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) 
-(\lambda (d2: C).(csuba g d2 c)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x: 
-C).(\lambda (H9: (eq C c2 (CHead x (Bind Abbr) t))).(\lambda (H10: (csuba g x 
-c)).(eq_ind_r C (CHead x (Bind Abbr) t) (\lambda (c0: C).(ex2 C (\lambda (d2: 
-C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
-d1)))) (let H11 \def (H c d1 u H6 g x H10) in (ex2_ind C (\lambda (d2: 
-C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) 
-(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x0: C).(\lambda (H12: 
-(drop n O x (CHead x0 (Bind Abst) u))).(\lambda (H13: (csuba g x0 d1)).(let 
-H14 \def (refl_equal nat (r (Bind Abst) n)) in (let H15 \def (eq_ind nat n 
-(\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abst) u))) H12 (r (Bind Abst) 
-n) H14) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) 
-t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop 
-(Bind Abbr) n x (CHead x0 (Bind Abst) u) H15 t) H13)))))) H11)) c2 H9)))) 
-H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) (ex2 C 
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: 
-A).(\lambda (H9: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H10: (csuba g 
-x0 c)).(\lambda (_: (arity g x0 x1 (asucc g x2))).(\lambda (_: (arity g c t 
-x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c0: C).(ex2 C (\lambda 
-(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g 
-d2 d1)))) (let H13 \def (H c d1 u H6 g x0 H10) in (ex2_ind C (\lambda (d2: 
-C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) 
-(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x: C).(\lambda 
-(H14: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H15: (csuba g x 
-d1)).(let H16 \def (refl_equal nat (r (Bind Abst) n)) in (let H17 \def 
-(eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x (Bind Abst) u))) H14 
-(r (Bind Abst) n) H16) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead 
-x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
-d1)) x (drop_drop (Bind Abst) n x0 (CHead x (Bind Abst) u) H17 x1) H15)))))) 
-H13)) c2 H9)))))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind 
-Abst) t))).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) 
-u))).(let H_x \def (csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in 
-(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: 
-C).(csuba g d2 c)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x: C).(\lambda (H8: 
-(eq C c2 (CHead x (Bind Abst) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C 
-(CHead x (Bind Abst) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) 
-O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H10 
-\def (H c d1 u H6 g x H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead 
-d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda 
-(d2: C).(csuba g d2 d1))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead 
-x0 (Bind Abst) u))).(\lambda (H12: (csuba g x0 d1)).(let H13 \def (refl_equal 
-nat (r (Bind Abst) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: 
-nat).(drop n0 O x (CHead x0 (Bind Abst) u))) H11 (r (Bind Abst) n) H13) in 
-(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop (Bind Abst) 
-n x (CHead x0 (Bind Abst) u) H14 t) H12)))))) H10)) c2 H8)))) H7))))) 
-(\lambda (H5: (csuba g c2 (CHead c (Bind Void) t))).(\lambda (H6: (drop (r 
-(Bind Void) n) O c (CHead d1 (Bind Abst) u))).(let H_x \def 
-(csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda 
-(d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: C).(csuba g d2 c)) 
-(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda 
-(d2: C).(csuba g d2 d1))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x 
-(Bind Void) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C (CHead x (Bind 
-Void) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H10 \def (H c d1 u 
-H6 g x H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) 
-u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(drop (S n) O 
-(CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g 
-d2 d1))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead x0 (Bind Abst) 
-u))).(\lambda (H12: (csuba g x0 d1)).(let H13 \def (refl_equal nat (r (Bind 
-Abst) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x 
-(CHead x0 (Bind Abst) u))) H11 (r (Bind Abst) n) H13) in (ex_intro2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) 
-u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop (Bind Void) n x (CHead 
-x0 (Bind Abst) u) H14 t) H12)))))) H10)) c2 H8)))) H7))))) b H3 H4)))) 
-(\lambda (f: F).(\lambda (H3: (csuba g c2 (CHead c (Flat f) t))).(\lambda 
-(H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abst) u))).(let H_x \def 
-(csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T 
-(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 c))) (ex2 C (\lambda (d2: C).(drop (S n) 
-O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda 
-(x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) 
-x1))).(\lambda (H7: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Flat f) x1) 
-(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H8 \def (H0 d1 u H4 g x0 
-H7) in (ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) 
-u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(drop (S n) O 
-(CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g 
-d2 d1))) (\lambda (x: C).(\lambda (H9: (drop (S n) O x0 (CHead x (Bind Abst) 
-u))).(\lambda (H10: (csuba g x d1)).(ex_intro2 C (\lambda (d2: C).(drop (S n) 
-O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g 
-d2 d1)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u) H9 x1) H10)))) 
-H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abst) u) t n 
-H1)))))))))))) c1)))) i).
-
-theorem csuba_drop_abbr_rev:
- \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i 
-O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba 
-g c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-a)))))))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: 
-C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: 
-G).(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop n 
-O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: 
-T).(\lambda (H: (drop O O c1 (CHead d1 (Bind Abbr) u1))).(\lambda (g: 
-G).(\lambda (c2: C).(\lambda (H0: (csuba g c2 c1)).(let H1 \def (eq_ind C c1 
-(\lambda (c: C).(csuba g c2 c)) H0 (CHead d1 (Bind Abbr) u1) (drop_gen_refl 
-c1 (CHead d1 (Bind Abbr) u1) H)) in (let H_x \def (csuba_gen_abbr_rev g d1 c2 
-u1 H1) in (let H2 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead 
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H3: 
-(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: 
-C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O 
-O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x 
-(Bind Abbr) u1))).(\lambda (H5: (csuba g x d1)).(eq_ind_r C (CHead x (Bind 
-Abbr) u1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-a))))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) 
-u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x (Bind 
-Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: 
-C).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1)) x (drop_refl (CHead x (Bind Abbr) u1)) H5)) c2 
-H4)))) H3)) (\lambda (H3: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H4: (eq C c2 (CHead x0 (Bind 
-Abst) x1))).(\lambda (H5: (csuba g x0 d1)).(\lambda (H6: (arity g x0 x1 
-(asucc g x2))).(\lambda (H7: (arity g d1 u1 x2)).(eq_ind_r C (CHead x0 (Bind 
-Abst) x1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-a))))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Abst) 
-x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind 
-Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abst) x1) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_refl (CHead x0 (Bind Abst) x1)) H5 
-H6 H7)) c2 H4)))))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H: 
-((\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop n O c1 (CHead d1 
-(Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c1) \to 
-(or (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-a))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: 
-C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to (\forall 
-(g: G).(\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop 
-(S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
-c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))))))))))) (\lambda (n0: nat).(\lambda (d1: 
-C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) (CHead d1 (Bind 
-Abbr) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (csuba g c2 (CSort 
-n0))).(and3_ind (eq C (CHead d1 (Bind Abbr) u1) (CSort n0)) (eq nat (S n) O) 
-(eq nat O O) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) 
-(\lambda (H2: (eq C (CHead d1 (Bind Abbr) u1) (CSort n0))).(\lambda (_: (eq 
-nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (match H2 in eq return 
-(\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c (CSort n0)) \to (or (ex2 C 
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))) with 
-[refl_equal \Rightarrow (\lambda (H5: (eq C (CHead d1 (Bind Abbr) u1) (CSort 
-n0))).(let H6 \def (eq_ind C (CHead d1 (Bind Abbr) u1) (\lambda (e: C).(match 
-e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | 
-(CHead _ _ _) \Rightarrow True])) I (CSort n0) H5) in (False_ind (or (ex2 C 
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) H6)))]) in (H5 
-(refl_equal C (CSort n0))))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind 
-Abbr) u1) H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: 
-C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to (\forall 
-(g: G).(\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop 
-(S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
-c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))))))))))))).(\lambda (k: K).(\lambda (t: 
-T).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H1: (drop (S n) O (CHead c k 
-t) (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: 
-(csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: K).((csuba g c2 (CHead c k0 
-t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C 
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))) (\lambda (b: 
-B).(\lambda (H3: (csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop (r 
-(Bind b) n) O c (CHead d1 (Bind Abbr) u1))).(B_ind (\lambda (b0: B).((csuba g 
-c2 (CHead c (Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind 
-Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-a))))))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) t))).(\lambda (H6: 
-(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def 
-(csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C 
-(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba 
-g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq 
-C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g c t a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind 
-Abbr) t))) (\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq 
-C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C 
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x: 
-C).(\lambda (H9: (eq C c2 (CHead x (Bind Abbr) t))).(\lambda (H10: (csuba g x 
-c)).(eq_ind_r C (CHead x (Bind Abbr) t) (\lambda (c0: C).(or (ex2 C (\lambda 
-(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
-g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))))) (let H11 \def (H c d1 u1 H6 g 
-x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or 
-(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n 
-O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind 
-C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind 
-Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
-(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x0: 
-C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14: 
-(csuba g x0 d1)).(let H15 \def (refl_equal nat (r (Bind Abst) n)) in (let H16 
-\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) 
-u1))) H13 (r (Bind Abst) n) H15) in (or_introl (ex2 C (\lambda (d2: C).(drop 
-(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop (Bind Abbr) 
-n x (CHead x0 (Bind Abbr) u1) H16 t) H14))))))) H12)) (\lambda (H12: (ex4_3 C 
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or (ex2 C (\lambda (d2: C).(drop 
-(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) 
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x 
-(CHead x0 (Bind Abst) x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15: 
-(arity g x0 x1 (asucc g x2))).(\lambda (H16: (arity g d1 u1 x2)).(let H17 
-\def (refl_equal nat (r (Bind Abst) n)) in (let H18 \def (eq_ind nat n 
-(\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abst) x1))) H13 (r (Bind 
-Abst) n) H17) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x 
-(Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
-(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex4_3_intro C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
-(Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Abbr) n 
-x (CHead x0 (Bind Abst) x1) H18 t) H14 H15 H16))))))))))) H12)) H11)) c2 
-H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc 
-g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) 
-(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) 
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 
-(CHead x0 (Bind Abst) x1))).(\lambda (H10: (csuba g x0 c)).(\lambda (_: 
-(arity g x0 x1 (asucc g x2))).(\lambda (_: (arity g c t x2)).(eq_ind_r C 
-(CHead x0 (Bind Abst) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop 
-(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
-c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in 
-(or_ind (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))))) (\lambda (H14: (ex2 C (\lambda (d2: C).(drop n 
-O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind 
-C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind 
-Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
-(CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x: 
-C).(\lambda (H15: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H16: 
-(csuba g x d1)).(let H17 \def (refl_equal nat (r (Bind Abst) n)) in (let H18 
-\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x (Bind Abbr) 
-u1))) H15 (r (Bind Abst) n) H17) in (or_introl (ex2 C (\lambda (d2: C).(drop 
-(S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead 
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind 
-Abst) n x0 (CHead x (Bind Abbr) u1) H18 x1) H16))))))) H14)) (\lambda (H14: 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: 
-A).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Abst) x4))).(\lambda (H16: 
-(csuba g x3 d1)).(\lambda (H17: (arity g x3 x4 (asucc g x5))).(\lambda (H18: 
-(arity g d1 u1 x5)).(let H19 \def (refl_equal nat (r (Bind Abst) n)) in (let 
-H20 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x3 (Bind Abst) 
-x4))) H15 (r (Bind Abst) n) H19) in (or_intror (ex2 C (\lambda (d2: C).(drop 
-(S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
-n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x3 x4 x5 
-(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Abst) x4) H20 x1) H16 H17 
-H18))))))))))) H14)) H13)) c2 H9)))))))) H8)) H7))))) (\lambda (H5: (csuba g 
-c2 (CHead c (Bind Abst) t))).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead 
-d1 (Bind Abbr) u1))).(let H_x \def (csuba_gen_abst_rev g c c2 t H5) in (let 
-H7 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) 
-t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) 
-O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead 
-d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x 
-(Bind Abst) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C (CHead x (Bind 
-Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead 
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda 
-(d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n 
-O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead 
-x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
-n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H11: 
-(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr) 
-u1))).(\lambda (H13: (csuba g x0 d1)).(let H14 \def (refl_equal nat (r (Bind 
-Abst) n)) in (let H15 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x 
-(CHead x0 (Bind Abbr) u1))) H12 (r (Bind Abst) n) H14) in (or_introl (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O 
-(CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g 
-d2 d1)) x0 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abbr) u1) H15 t) 
-H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
-(Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x0: C).(\lambda (x1: 
-T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst) 
-x1))).(\lambda (H13: (csuba g x0 d1)).(\lambda (H14: (arity g x0 x1 (asucc g 
-x2))).(\lambda (H15: (arity g d1 u1 x2)).(let H16 \def (refl_equal nat (r 
-(Bind Abst) n)) in (let H17 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O 
-x (CHead x0 (Bind Abst) x1))) H12 (r (Bind Abst) n) H16) in (or_intror (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))) x0 x1 x2 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abst) x1) H17 t) 
-H13 H14 H15))))))))))) H11)) H10)) c2 H8)))) H7))))) (\lambda (H5: (csuba g 
-c2 (CHead c (Bind Void) t))).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead 
-d1 (Bind Abbr) u1))).(let H_x \def (csuba_gen_void_rev g c c2 t H5) in (let 
-H7 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) 
-t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) 
-O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead 
-d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x 
-(Bind Void) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C (CHead x (Bind 
-Void) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead 
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda 
-(d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n 
-O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead 
-x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
-n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H11: 
-(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr) 
-u1))).(\lambda (H13: (csuba g x0 d1)).(let H14 \def (refl_equal nat (r (Bind 
-Abst) n)) in (let H15 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x 
-(CHead x0 (Bind Abbr) u1))) H12 (r (Bind Abst) n) H14) in (or_introl (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O 
-(CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g 
-d2 d1)) x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abbr) u1) H15 t) 
-H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
-(Bind Void) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x0: C).(\lambda (x1: 
-T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst) 
-x1))).(\lambda (H13: (csuba g x0 d1)).(\lambda (H14: (arity g x0 x1 (asucc g 
-x2))).(\lambda (H15: (arity g d1 u1 x2)).(let H16 \def (refl_equal nat (r 
-(Bind Abst) n)) in (let H17 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O 
-x (CHead x0 (Bind Abst) x1))) H12 (r (Bind Abst) n) H16) in (or_intror (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))) x0 x1 x2 (drop_drop (Bind Void) n x (CHead x0 (Bind Abst) x1) H17 t) 
-H13 H14 H15))))))))))) H11)) H10)) c2 H8)))) H7))))) b H3 H4)))) (\lambda (f: 
-F).(\lambda (H3: (csuba g c2 (CHead c (Flat f) t))).(\lambda (H4: (drop (r 
-(Flat f) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def (csuba_gen_flat_rev 
-g c c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T (\lambda (d2: 
-C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O 
-c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead 
-d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: 
-(eq C c2 (CHead x0 (Flat f) x1))).(\lambda (H7: (csuba g x0 c)).(eq_ind_r C 
-(CHead x0 (Flat f) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S 
-n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 
-C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))))) (let H8 \def (H0 d1 u1 H4 g x0 H7) in (or_ind 
-(ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S 
-n) O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1)))).(ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S 
-n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) 
-(\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abbr) 
-u1))).(\lambda (H11: (csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop 
-(S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Flat f) n 
-x0 (CHead x (Bind Abbr) u1) H10 x1) H11))))) H9)) (\lambda (H9: (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop (S n) O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or (ex2 C (\lambda (d2: C).(drop 
-(S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) 
-(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H10: (drop (S n) 
-O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H11: (csuba g x2 d1)).(\lambda 
-(H12: (arity g x2 x3 (asucc g x4))).(\lambda (H13: (arity g d1 u1 
-x4)).(or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
-(Flat f) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))) x2 x3 x4 (drop_drop (Flat f) n x0 (CHead x2 (Bind 
-Abst) x3) H10 x1) H11 H12 H13))))))))) H9)) H8)) c2 H6))))) H5)))))) k H2 
-(drop_gen_drop k c (CHead d1 (Bind Abbr) u1) t n H1)))))))))))) c1)))) i).
-
-theorem csuba_getl_abst_rev:
- \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall 
-(i: nat).((getl i c1 (CHead d1 (Bind Abst) u)) \to (\forall (c2: C).((csuba g 
-c2 c1) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda 
-(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u))).(let H0 \def 
-(getl_gen_all c1 (CHead d1 (Bind Abst) u) i H) in (ex2_ind C (\lambda (e: 
-C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u))) 
-(\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda (d2: C).(getl i c2 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda (x: 
-C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind 
-Abst) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1 
-(Bind Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda 
-(d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
-d1)))))))) (\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda 
-(H4: (clear (CSort n) (CHead d1 (Bind Abst) u))).(clear_gen_sort (CHead d1 
-(Bind Abst) u) n H4 (\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda 
-(d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
-d1))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 
-(CHead d1 (Bind Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (ex2 C 
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: 
-(drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 
-(Bind Abst) u))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to 
-((clear (CHead x0 k0 t) (CHead d1 (Bind Abst) u)) \to (\forall (c2: 
-C).((csuba g c2 c1) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) (\lambda (b: B).(\lambda 
-(H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0 
-(Bind b) t) (CHead d1 (Bind Abst) u))).(let H7 \def (f_equal C C (\lambda (e: 
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | 
-(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t) 
-(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in ((let H8 \def 
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with 
-[(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 in K 
-return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
-\Rightarrow Abst])])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t) 
-(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in ((let H9 \def 
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind 
-Abst) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u) 
-t H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda 
-(c2: C).(\lambda (H12: (csuba g c2 c1)).(let H13 \def (eq_ind_r T t (\lambda 
-(t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def 
-(eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u))) H13 Abst 
-H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c 
-(Bind Abst) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abst_rev i c1 d1 u 
-H15 g c2 H12) in (ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(getl i 
-c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda 
-(x1: C).(\lambda (H17: (drop i O c2 (CHead x1 (Bind Abst) u))).(\lambda (H18: 
-(csuba g x1 d1)).(ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x1 (getl_intro i c2 (CHead x1 
-(Bind Abst) u) (CHead x1 (Bind Abst) u) H17 (clear_bind Abst x1 u)) H18)))) 
-H16)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead 
-x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind 
-Abst) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c 
-(CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) \to (ex2 C 
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop n 
-O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (ex2 C 
-(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1)))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead 
-x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g c2 x1)).(let H10 
-\def (eq_ind C x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead x0 (Flat f) t) 
-(drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def (clear_flat x0 
-(CHead d1 (Bind Abst) u) (clear_gen_flat f x0 (CHead d1 (Bind Abst) u) t H6) 
-f t) in (let H11 \def (csuba_clear_trans g (CHead x0 (Flat f) t) c2 H10 
-(CHead d1 (Bind Abst) u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g e2 
-(CHead d1 (Bind Abst) u))) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda 
-(d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
-d1))) (\lambda (x2: C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abst) 
-u))).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abst_rev g d1 x2 
-u H12) in (let H14 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: 
-C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
-(\lambda (x3: C).(\lambda (H15: (eq C x2 (CHead x3 (Bind Abst) u))).(\lambda 
-(H16: (csuba g x3 d1)).(let H17 \def (eq_ind C x2 (\lambda (c: C).(clear c2 
-c)) H13 (CHead x3 (Bind Abst) u) H15) in (ex_intro2 C (\lambda (d2: C).(getl 
-O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x3 
-(getl_intro O c2 (CHead x3 (Bind Abst) u) c2 (drop_refl c2) H17) H16))))) 
-H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1: 
-C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) 
-\to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda 
-(d2: C).(csuba g d2 d1))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O 
-x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g c2 
-x1)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B 
-C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind 
-b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead 
-x0 (Flat f) t))))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x2: B).(\lambda (x3: 
-C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) 
-x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 \def 
-(csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C 
-(\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) (\lambda (e2: 
-C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x5: C).(\lambda (H15: 
-(csuba g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: (clear c2 x5)).(let H_x 
-\def (csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let H17 \def H_x in 
-(ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
-T).(csuba g e2 x3)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x6: B).(\lambda (x7: 
-C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) 
-x8))).(\lambda (H19: (csuba g x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: 
-C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 
-x7 H19) in (ex2_ind C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(getl (S n) c2 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x9: 
-C).(\lambda (H22: (getl n x7 (CHead x9 (Bind Abst) u))).(\lambda (H23: (csuba 
-g x9 d1)).(ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) 
-u))) (\lambda (d2: C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20 
-(CHead x9 (Bind Abst) u) n H22) H23)))) H21)))))))) H17)))))) H14))))))) 
-H11)))))))) i) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))).
-
-theorem csuba_getl_abbr_rev:
- \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall 
-(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba 
-g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda 
-(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u1))).(let H0 \def 
-(getl_gen_all c1 (CHead d1 (Bind Abbr) u1) i H) in (ex2_ind C (\lambda (e: 
-C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u1))) 
-(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))))))) (\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: 
-(clear x (CHead d1 (Bind Abbr) u1))).(C_ind (\lambda (c: C).((drop i O c1 c) 
-\to ((clear c (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 
-c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))))) (\lambda 
-(n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) 
-(CHead d1 (Bind Abbr) u1))).(clear_gen_sort (CHead d1 (Bind Abbr) u1) n H4 
-(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear 
-x0 (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 c1) \to (or 
-(ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))))))))))).(\lambda (k: K).(\lambda 
-(t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear 
-(CHead x0 k t) (CHead d1 (Bind Abbr) u1))).(K_ind (\lambda (k0: K).((drop i O 
-c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind Abbr) u1)) 
-\to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i 
-c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))))))))) (\lambda (b: B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) 
-t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 (Bind Abbr) 
-u1))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda 
-(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) 
-(CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead 
-d1 (Bind Abbr) u1) t H6)) in ((let H8 \def (f_equal C B (\lambda (e: 
-C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | 
-(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with 
-[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d1 (Bind 
-Abbr) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) 
-u1) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e in C return 
-(\lambda (_: C).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t0) 
-\Rightarrow t0])) (CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b) t) 
-(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) in (\lambda (H10: (eq B 
-Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: C).(\lambda (H12: (csuba 
-g c2 c1)).(let H13 \def (eq_ind_r T t (\lambda (t0: T).(drop i O c1 (CHead x0 
-(Bind b) t0))) H5 u1 H9) in (let H14 \def (eq_ind_r B b (\lambda (b0: 
-B).(drop i O c1 (CHead x0 (Bind b0) u1))) H13 Abbr H10) in (let H15 \def 
-(eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c (Bind Abbr) u1))) H14 d1 
-H11) in (let H16 \def (csuba_drop_abbr_rev i c1 d1 u1 H15 g c2 H12) in 
-(or_ind (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or (ex2 C 
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H17: (ex2 C (\lambda 
-(d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(getl i c2 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))))) (\lambda (x1: C).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abbr) 
-u1))).(\lambda (H19: (csuba g x1 d1)).(or_introl (ex2 C (\lambda (d2: 
-C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead 
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x1 (getl_intro i c2 
-(CHead x1 (Bind Abbr) u1) (CHead x1 (Bind Abbr) u1) H18 (clear_bind Abbr x1 
-u1)) H19))))) H17)) (\lambda (H17: (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x1: 
-C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1 
-(Bind Abst) x2))).(\lambda (H19: (csuba g x1 d1)).(\lambda (H20: (arity g x1 
-x2 (asucc g x3))).(\lambda (H21: (arity g d1 u1 x3)).(or_intror (ex2 C 
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x1 x2 x3 
-(getl_intro i c2 (CHead x1 (Bind Abst) x2) (CHead x1 (Bind Abst) x2) H18 
-(clear_bind Abst x1 x2)) H19 H20 H21))))))))) H17)) H16)))))))))) H8)) 
-H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f) 
-t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr) 
-u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0 
-(Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda 
-(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i 
-c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: 
-C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) 
-\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))))) (\lambda 
-(x1: C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: 
-C).(\lambda (H9: (csuba g c2 x1)).(let H10 \def (eq_ind C x1 (\lambda (c: 
-C).(csuba g c2 c)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat 
-f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abbr) u1) 
-(clear_gen_flat f x0 (CHead d1 (Bind Abbr) u1) t H6) f t) in (let H11 \def 
-(csuba_clear_trans g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u1) 
-H_y) in (ex2_ind C (\lambda (e2: C).(csuba g e2 (CHead d1 (Bind Abbr) u1))) 
-(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead 
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) 
-(\lambda (x2: C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abbr) 
-u1))).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abbr_rev g d1 x2 
-u1 H12) in (let H14 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C x2 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H15: 
-(ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: 
-C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead 
-x3 (Bind Abbr) u1))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C 
-x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) u1) H16) in 
-(or_introl (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C 
-(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abbr) u1) c2 
-(drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(eq C x2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or (ex2 C (\lambda (d2: C).(getl 
-O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H16: 
-(eq C x2 (CHead x3 (Bind Abst) x4))).(\lambda (H17: (csuba g x3 d1)).(\lambda 
-(H18: (arity g x3 x4 (asucc g x5))).(\lambda (H19: (arity g d1 u1 x5)).(let 
-H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst) 
-x4) H16) in (or_intror (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O 
-c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))) x3 x4 x5 (getl_intro O c2 (CHead x3 (Bind Abst) 
-x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) H14)))))) H11)))))))) 
-(\lambda (n: nat).(\lambda (H8: ((\forall (x1: C).((drop n O x1 (CHead x0 
-(Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or (ex2 C (\lambda 
-(d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n 
-c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S 
-n) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g c2 
-x1)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B 
-C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind 
-b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead 
-x0 (Flat f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) 
-(\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1 
-(CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) 
-t))).(let H14 \def (csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) 
-H12) in (ex2_ind C (\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) 
-(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))))) (\lambda (x5: C).(\lambda (H15: (csuba g x5 (CHead x3 (Bind x2) 
-x4))).(\lambda (H16: (clear c2 x5)).(let H_x \def (csuba_gen_bind_rev g x2 x3 
-x5 x4 H15) in (let H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda 
-(e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
-B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 x3)))) (or (ex2 C (\lambda 
-(d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g 
-d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl 
-(S n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x6: B).(\lambda (x7: 
-C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) 
-x8))).(\lambda (H19: (csuba g x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: 
-C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 
-x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or 
-(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H22: 
-(ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: 
-C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S 
-n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))))) (\lambda (x9: C).(\lambda (H23: (getl n x7 
-(CHead x9 (Bind Abbr) u1))).(\lambda (H24: (csuba g x9 d1)).(or_introl (ex2 C 
-(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(getl (S n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: 
-C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) u1) n H23) 
-H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) 
-(\lambda (x9: C).(\lambda (x10: T).(\lambda (x11: A).(\lambda (H23: (getl n 
-x7 (CHead x9 (Bind Abst) x10))).(\lambda (H24: (csuba g x9 d1)).(\lambda 
-(H25: (arity g x9 x10 (asucc g x11))).(\lambda (H26: (arity g d1 u1 
-x11)).(or_intror (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S 
-n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))) x9 x10 x11 (getl_clear_bind x6 c2 x7 x8 H20 
-(CHead x9 (Bind Abst) x10) n H23) H24 H25 H26))))))))) H22)) H21)))))))) 
-H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) x H1 H2)))) 
-H0))))))).
-
-theorem sn3_gen_bind:
- \forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
-(THead (Bind b) u t)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t))))))
-\def
- \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: 
-(sn3 c (THead (Bind b) u t))).(insert_eq T (THead (Bind b) u t) (\lambda (t0: 
-T).(sn3 c t0)) (land (sn3 c u) (sn3 (CHead c (Bind b) u) t)) (\lambda (y: 
-T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T y (THead 
-(Bind b) u t0)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t0)))) (unintro 
-T u (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind b) t0 x)) \to 
-(land (sn3 c t0) (sn3 (CHead c (Bind b) t0) x))))) (sn3_ind c (\lambda (t0: 
-T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Bind b) x x0)) \to 
-(land (sn3 c x) (sn3 (CHead c (Bind b) x) x0)))))) (\lambda (t1: T).(\lambda 
-(H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 
-c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) 
-\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall 
-(x0: T).((eq T t2 (THead (Bind b) x x0)) \to (land (sn3 c x) (sn3 (CHead c 
-(Bind b) x) x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T 
-t1 (THead (Bind b) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0: 
-T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c 
-t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Bind b) x1 
-x2)) \to (land (sn3 c x1) (sn3 (CHead c (Bind b) x1) x2))))))))) H2 (THead 
-(Bind b) x x0) H3) in (let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall 
-(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to 
-(sn3 c t2))))) H1 (THead (Bind b) x x0) H3) in (conj (sn3 c x) (sn3 (CHead c 
-(Bind b) x) x0) (sn3_sing c x (\lambda (t2: T).(\lambda (H6: (((eq T x t2) 
-\to (\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x t2)).(let H8 \def (H4 
-(THead (Bind b) t2 x0) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind 
-b) t2 x0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | 
-(TLRef _) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind b) x 
-x0) (THead (Bind b) t2 x0) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: 
-T).(pr3 c x t0)) H7 x H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: 
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H6 x H9) in (H11 (refl_equal T 
-x) P)))))) (pr3_head_12 c x t2 H7 (Bind b) x0 x0 (pr3_refl (CHead c (Bind b) 
-t2) x0)) t2 x0 (refl_equal T (THead (Bind b) t2 x0))) in (and_ind (sn3 c t2) 
-(sn3 (CHead c (Bind b) t2) x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda 
-(_: (sn3 (CHead c (Bind b) t2) x0)).H9)) H8)))))) (sn3_sing (CHead c (Bind b) 
-x) x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P: 
-Prop).P)))).(\lambda (H7: (pr3 (CHead c (Bind b) x) x0 t2)).(let H8 \def (H4 
-(THead (Bind b) x t2) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind 
-b) x t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind b) x 
-x0) (THead (Bind b) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: 
-T).(pr3 (CHead c (Bind b) x) x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T 
-t2 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in 
-(H11 (refl_equal T x0) P)))))) (pr3_head_12 c x x (pr3_refl c x) (Bind b) x0 
-t2 H7) x t2 (refl_equal T (THead (Bind b) x t2))) in (and_ind (sn3 c x) (sn3 
-(CHead c (Bind b) x) t2) (sn3 (CHead c (Bind b) x) t2) (\lambda (_: (sn3 c 
-x)).(\lambda (H10: (sn3 (CHead c (Bind b) x) t2)).H10)) H8))))))))))))))) y 
-H0))))) H))))).
-
-theorem sn3_gen_head:
- \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
-(THead k u t)) \to (sn3 c u)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (c: C).(\forall (u: 
-T).(\forall (t: T).((sn3 c (THead k0 u t)) \to (sn3 c u)))))) (\lambda (b: 
-B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead 
-(Bind b) u t))).(let H_x \def (sn3_gen_bind b c u t H) in (let H0 \def H_x in 
-(and_ind (sn3 c u) (sn3 (CHead c (Bind b) u) t) (sn3 c u) (\lambda (H1: (sn3 
-c u)).(\lambda (_: (sn3 (CHead c (Bind b) u) t)).H1)) H0)))))))) (\lambda (f: 
-F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead 
-(Flat f) u t))).(let H_x \def (sn3_gen_flat f c u t H) in (let H0 \def H_x in 
-(and_ind (sn3 c u) (sn3 c t) (sn3 c u) (\lambda (H1: (sn3 c u)).(\lambda (_: 
-(sn3 c t)).H1)) H0)))))))) k).
-
-theorem sn3_gen_cflat:
- \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 (CHead 
-c (Flat f) u) t) \to (sn3 c t)))))
-\def
- \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: 
-(sn3 (CHead c (Flat f) u) t)).(sn3_ind (CHead c (Flat f) u) (\lambda (t0: 
-T).(sn3 c t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 
-t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to 
-(sn3 (CHead c (Flat f) u) t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T 
-t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to 
-(sn3 c t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) 
-\to (\forall (P: Prop).P)))).(\lambda (H3: (pr3 c t1 t2)).(H1 t2 H2 
-(pr3_cflat c t1 t2 H3 f u))))))))) t H))))).
-
-theorem sn3_cflat:
- \forall (c: C).(\forall (t: T).((sn3 c t) \to (\forall (f: F).(\forall (u: 
-T).(sn3 (CHead c (Flat f) u) t)))))
-\def
- \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(\lambda (f: 
-F).(\lambda (u: T).(sn3_ind c (\lambda (t0: T).(sn3 (CHead c (Flat f) u) t0)) 
-(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
-(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall 
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
-(sn3 (CHead c (Flat f) u) t2)))))).(sn3_pr2_intro (CHead c (Flat f) u) t1 
-(\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) \to (\forall (P: 
-Prop).P)))).(\lambda (H3: (pr2 (CHead c (Flat f) u) t1 t2)).(H1 t2 H2 
-(pr3_pr2 c t1 t2 (pr2_gen_cflat f c u t1 t2 H3)))))))))) t H))))).
-
-theorem sn3_shift:
- \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c 
-(THead (Bind b) v t)) \to (sn3 (CHead c (Bind b) v) t)))))
-\def
- \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: 
-(sn3 c (THead (Bind b) v t))).(let H_x \def (sn3_gen_bind b c v t H) in (let 
-H0 \def H_x in (and_ind (sn3 c v) (sn3 (CHead c (Bind b) v) t) (sn3 (CHead c 
-(Bind b) v) t) (\lambda (_: (sn3 c v)).(\lambda (H2: (sn3 (CHead c (Bind b) 
-v) t)).H2)) H0))))))).
-
-theorem sn3_change:
- \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1: 
-T).(\forall (t: T).((sn3 (CHead c (Bind b) v1) t) \to (\forall (v2: T).(sn3 
-(CHead c (Bind b) v2) t)))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda 
-(v1: T).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c (Bind b) v1) t)).(\lambda 
-(v2: T).(sn3_ind (CHead c (Bind b) v1) (\lambda (t0: T).(sn3 (CHead c (Bind 
-b) v2) t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) 
-\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to (sn3 
-(CHead c (Bind b) v1) t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 
-t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to 
-(sn3 (CHead c (Bind b) v2) t2)))))).(sn3_pr2_intro (CHead c (Bind b) v2) t1 
-(\lambda (t2: T).(\lambda (H3: (((eq T t1 t2) \to (\forall (P: 
-Prop).P)))).(\lambda (H4: (pr2 (CHead c (Bind b) v2) t1 t2)).(H2 t2 H3 
-(pr3_pr2 (CHead c (Bind b) v1) t1 t2 (pr2_change b H c v2 t1 t2 H4 
-v1)))))))))) t H0))))))).
-
-theorem sn3_cpr3_trans:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
-(k: K).(\forall (t: T).((sn3 (CHead c k u1) t) \to (sn3 (CHead c k u2) 
-t)))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 
-u2)).(\lambda (k: K).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c k u1) 
-t)).(sn3_ind (CHead c k u1) (\lambda (t0: T).(sn3 (CHead c k u2) t0)) 
-(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
-(P: Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u1) 
-t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u2) 
-t2)))))).(sn3_sing (CHead c k u2) t1 (\lambda (t2: T).(\lambda (H3: (((eq T 
-t1 t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 (CHead c k u2) t1 
-t2)).(H2 t2 H3 (pr3_pr3_pr3_t c u1 u2 H t1 t2 k H4))))))))) t H0))))))).
-
-theorem sn3_bind:
- \forall (b: B).(\forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: 
-T).((sn3 (CHead c (Bind b) u) t) \to (sn3 c (THead (Bind b) u t)))))))
-\def
- \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c 
-u)).(sn3_ind c (\lambda (t: T).(\forall (t0: T).((sn3 (CHead c (Bind b) t) 
-t0) \to (sn3 c (THead (Bind b) t t0))))) (\lambda (t1: T).(\lambda (_: 
-((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 
-t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to 
-(\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (t: T).((sn3 (CHead c 
-(Bind b) t2) t) \to (sn3 c (THead (Bind b) t2 t))))))))).(\lambda (t: 
-T).(\lambda (H2: (sn3 (CHead c (Bind b) t1) t)).(sn3_ind (CHead c (Bind b) 
-t1) (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (\lambda (t2: 
-T).(\lambda (H3: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: 
-Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind b) 
-t1) t3)))))).(\lambda (H4: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: 
-Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 c (THead (Bind b) 
-t1 t3))))))).(sn3_sing c (THead (Bind b) t1 t2) (\lambda (t3: T).(\lambda 
-(H5: (((eq T (THead (Bind b) t1 t2) t3) \to (\forall (P: Prop).P)))).(\lambda 
-(H6: (pr3 c (THead (Bind b) t1 t2) t3)).(let H_x \def (bind_dec_not b Abst) 
-in (let H7 \def H_x in (or_ind (eq B b Abst) (not (eq B b Abst)) (sn3 c t3) 
-(\lambda (H8: (eq B b Abst)).(let H9 \def (eq_ind B b (\lambda (b0: B).(pr3 c 
-(THead (Bind b0) t1 t2) t3)) H6 Abst H8) in (let H10 \def (eq_ind B b 
-(\lambda (b0: B).((eq T (THead (Bind b0) t1 t2) t3) \to (\forall (P: 
-Prop).P))) H5 Abst H8) in (let H11 \def (eq_ind B b (\lambda (b0: B).(\forall 
-(t4: T).((((eq T t2 t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind 
-b0) t1) t2 t4) \to (sn3 c (THead (Bind b0) t1 t4)))))) H4 Abst H8) in (let 
-H12 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T t2 t4) \to 
-(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) t2 t4) \to (sn3 
-(CHead c (Bind b0) t1) t4))))) H3 Abst H8) in (let H13 \def (eq_ind B b 
-(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P))) 
-\to ((pr3 c t1 t4) \to (\forall (t0: T).((sn3 (CHead c (Bind b0) t4) t0) \to 
-(sn3 c (THead (Bind b0) t4 t0)))))))) H1 Abst H8) in (let H14 \def 
-(pr3_gen_abst c t1 t2 t3 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: 
-T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall 
-(u0: T).(pr3 (CHead c (Bind b0) u0) t2 t4))))) (sn3 c t3) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H15: (eq T t3 (THead (Bind Abst) x0 
-x1))).(\lambda (H16: (pr3 c t1 x0)).(\lambda (H17: ((\forall (b0: B).(\forall 
-(u0: T).(pr3 (CHead c (Bind b0) u0) t2 x1))))).(let H18 \def (eq_ind T t3 
-(\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) t0) \to (\forall (P: 
-Prop).P))) H10 (THead (Bind Abst) x0 x1) H15) in (eq_ind_r T (THead (Bind 
-Abst) x0 x1) (\lambda (t0: T).(sn3 c t0)) (let H_x0 \def (term_dec t1 x0) in 
-(let H19 \def H_x0 in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: 
-Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H20: (eq T t1 x0)).(let 
-H21 \def (eq_ind_r T x0 (\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) 
-(THead (Bind Abst) t0 x1)) \to (\forall (P: Prop).P))) H18 t1 H20) in (let 
-H22 \def (eq_ind_r T x0 (\lambda (t0: T).(pr3 c t1 t0)) H16 t1 H20) in 
-(eq_ind T t1 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t0 x1))) (let H_x1 
-\def (term_dec t2 x1) in (let H23 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 
-x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abst) t1 x1)) (\lambda 
-(H24: (eq T t2 x1)).(let H25 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T 
-(THead (Bind Abst) t1 t2) (THead (Bind Abst) t1 t0)) \to (\forall (P: 
-Prop).P))) H21 t2 H24) in (let H26 \def (eq_ind_r T x1 (\lambda (t0: 
-T).(\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) 
-H17 t2 H24) in (eq_ind T t2 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t1 
-t0))) (H25 (refl_equal T (THead (Bind Abst) t1 t2)) (sn3 c (THead (Bind Abst) 
-t1 t2))) x1 H24)))) (\lambda (H24: (((eq T t2 x1) \to (\forall (P: 
-Prop).P)))).(H11 x1 H24 (H17 Abst t1))) H23))) x0 H20)))) (\lambda (H20: 
-(((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x1 \def (term_dec t2 x1) 
-in (let H21 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: 
-Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H22: (eq T t2 x1)).(let 
-H23 \def (eq_ind_r T x1 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: 
-T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) H17 t2 H22) in (eq_ind T t2 (\lambda 
-(t0: T).(sn3 c (THead (Bind Abst) x0 t0))) (H13 x0 H20 H16 t2 (sn3_cpr3_trans 
-c t1 x0 H16 (Bind Abst) t2 (sn3_sing (CHead c (Bind Abst) t1) t2 H12))) x1 
-H22))) (\lambda (H22: (((eq T t2 x1) \to (\forall (P: Prop).P)))).(H13 x0 H20 
-H16 x1 (sn3_cpr3_trans c t1 x0 H16 (Bind Abst) x1 (H12 x1 H22 (H17 Abst 
-t1))))) H21)))) H19))) t3 H15))))))) H14)))))))) (\lambda (H8: (not (eq B b 
-Abst))).(let H_x0 \def (pr3_gen_bind b H8 c t1 t2 t3 H6) in (let H9 \def H_x0 
-in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
-b) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) t2 t4)))) (pr3 (CHead c (Bind 
-b) t1) t2 (lift (S O) O t3)) (sn3 c t3) (\lambda (H10: (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 
-(CHead c (Bind b) t1) t2 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda 
-(t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 (CHead c (Bind b) 
-t1) t2 t4))) (sn3 c t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq 
-T t3 (THead (Bind b) x0 x1))).(\lambda (H12: (pr3 c t1 x0)).(\lambda (H13: 
-(pr3 (CHead c (Bind b) t1) t2 x1)).(let H14 \def (eq_ind T t3 (\lambda (t0: 
-T).((eq T (THead (Bind b) t1 t2) t0) \to (\forall (P: Prop).P))) H5 (THead 
-(Bind b) x0 x1) H11) in (eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: 
-T).(sn3 c t0)) (let H_x1 \def (term_dec t1 x0) in (let H15 \def H_x1 in 
-(or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: Prop).P)) (sn3 c (THead 
-(Bind b) x0 x1)) (\lambda (H16: (eq T t1 x0)).(let H17 \def (eq_ind_r T x0 
-(\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t0 x1)) \to 
-(\forall (P: Prop).P))) H14 t1 H16) in (let H18 \def (eq_ind_r T x0 (\lambda 
-(t0: T).(pr3 c t1 t0)) H12 t1 H16) in (eq_ind T t1 (\lambda (t0: T).(sn3 c 
-(THead (Bind b) t0 x1))) (let H_x2 \def (term_dec t2 x1) in (let H19 \def 
-H_x2 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: Prop).P)) (sn3 c 
-(THead (Bind b) t1 x1)) (\lambda (H20: (eq T t2 x1)).(let H21 \def (eq_ind_r 
-T x1 (\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t1 t0)) 
-\to (\forall (P: Prop).P))) H17 t2 H20) in (let H22 \def (eq_ind_r T x1 
-(\lambda (t0: T).(pr3 (CHead c (Bind b) t1) t2 t0)) H13 t2 H20) in (eq_ind T 
-t2 (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (H21 (refl_equal T (THead 
-(Bind b) t1 t2)) (sn3 c (THead (Bind b) t1 t2))) x1 H20)))) (\lambda (H20: 
-(((eq T t2 x1) \to (\forall (P: Prop).P)))).(H4 x1 H20 H13)) H19))) x0 
-H16)))) (\lambda (H16: (((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x2 
-\def (term_dec t2 x1) in (let H17 \def H_x2 in (or_ind (eq T t2 x1) ((eq T t2 
-x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind b) x0 x1)) (\lambda (H18: 
-(eq T t2 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t0: T).(pr3 (CHead c 
-(Bind b) t1) t2 t0)) H13 t2 H18) in (eq_ind T t2 (\lambda (t0: T).(sn3 c 
-(THead (Bind b) x0 t0))) (H1 x0 H16 H12 t2 (sn3_cpr3_trans c t1 x0 H12 (Bind 
-b) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3))) x1 H18))) (\lambda (H18: (((eq 
-T t2 x1) \to (\forall (P: Prop).P)))).(H1 x0 H16 H12 x1 (sn3_cpr3_trans c t1 
-x0 H12 (Bind b) x1 (H3 x1 H18 H13)))) H17)))) H15))) t3 H11))))))) H10)) 
-(\lambda (H10: (pr3 (CHead c (Bind b) t1) t2 (lift (S O) O 
-t3))).(sn3_gen_lift (CHead c (Bind b) t1) t3 (S O) O (sn3_pr3_trans (CHead c 
-(Bind b) t1) t2 (sn3_pr2_intro (CHead c (Bind b) t1) t2 (\lambda (t0: 
-T).(\lambda (H11: (((eq T t2 t0) \to (\forall (P: Prop).P)))).(\lambda (H12: 
-(pr2 (CHead c (Bind b) t1) t2 t0)).(H3 t0 H11 (pr3_pr2 (CHead c (Bind b) t1) 
-t2 t0 H12)))))) (lift (S O) O t3) H10) c (drop_drop (Bind b) O c c (drop_refl 
-c) t1))) H9)))) H7)))))))))) t H2)))))) u H)))).
-
-theorem sn3_beta:
- \forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c (THead (Bind Abbr) v 
-t)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead 
-(Bind Abst) w t))))))))
-\def
- \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead 
-(Bind Abbr) v t))).(insert_eq T (THead (Bind Abbr) v t) (\lambda (t0: T).(sn3 
-c t0)) (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead 
-(Bind Abst) w t))))) (\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t 
-(\lambda (t0: T).((eq T y (THead (Bind Abbr) v t0)) \to (\forall (w: T).((sn3 
-c w) \to (sn3 c (THead (Flat Appl) v (THead (Bind Abst) w t0))))))) (unintro 
-T v (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind Abbr) t0 x)) \to 
-(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) t0 (THead (Bind 
-Abst) w x)))))))) (sn3_ind c (\lambda (t0: T).(\forall (x: T).(\forall (x0: 
-T).((eq T t0 (THead (Bind Abbr) x x0)) \to (\forall (w: T).((sn3 c w) \to 
-(sn3 c (THead (Flat Appl) x (THead (Bind Abst) w x0))))))))) (\lambda (t1: 
-T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall 
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
-(\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Bind Abbr) x x0)) \to 
-(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) 
-w x0))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1 
-(THead (Bind Abbr) x x0))).(\lambda (w: T).(\lambda (H4: (sn3 c w)).(let H5 
-\def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to 
-(\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (\forall (x1: T).(\forall (x2: 
-T).((eq T t2 (THead (Bind Abbr) x1 x2)) \to (\forall (w0: T).((sn3 c w0) \to 
-(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) w0 x2)))))))))))) H2 (THead 
-(Bind Abbr) x x0) H3) in (let H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall 
-(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to 
-(sn3 c t2))))) H1 (THead (Bind Abbr) x x0) H3) in (sn3_ind c (\lambda (t0: 
-T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t0 x0)))) (\lambda (t2: 
-T).(\lambda (H7: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: 
-Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H8: ((\forall 
-(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to 
-(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t3 x0)))))))).(sn3_pr2_intro c 
-(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (\lambda (t3: T).(\lambda 
-(H9: (((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t3) \to (\forall 
-(P: Prop).P)))).(\lambda (H10: (pr2 c (THead (Flat Appl) x (THead (Bind Abst) 
-t2 x0)) t3)).(let H11 \def (pr2_gen_appl c x (THead (Bind Abst) t2 x0) t3 
-H10) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) t2 x0) t4)))) 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (THead (Bind Abst) t2 x0) (THead (Bind Abst) y1 z1)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
-(Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind 
-b) u) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c t3) 
-(\lambda (H12: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) t2 x0) 
-t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) t2 x0) t4))) (sn3 
-c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H13: (eq T t3 (THead (Flat 
-Appl) x1 x2))).(\lambda (H14: (pr2 c x x1)).(\lambda (H15: (pr2 c (THead 
-(Bind Abst) t2 x0) x2)).(let H16 \def (eq_ind T t3 (\lambda (t0: T).((eq T 
-(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P: 
-Prop).P))) H9 (THead (Flat Appl) x1 x2) H13) in (eq_ind_r T (THead (Flat 
-Appl) x1 x2) (\lambda (t0: T).(sn3 c t0)) (let H17 \def (pr2_gen_abst c t2 x0 
-x2 H15) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead 
-(Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t2 u2))) 
-(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead 
-c (Bind b) u) x0 t4))))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3: 
-T).(\lambda (x4: T).(\lambda (H18: (eq T x2 (THead (Bind Abst) x3 
-x4))).(\lambda (H19: (pr2 c t2 x3)).(\lambda (H20: ((\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) x0 x4))))).(let H21 \def (eq_ind T x2 
-(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) 
-(THead (Flat Appl) x1 t0)) \to (\forall (P: Prop).P))) H16 (THead (Bind Abst) 
-x3 x4) H18) in (eq_ind_r T (THead (Bind Abst) x3 x4) (\lambda (t0: T).(sn3 c 
-(THead (Flat Appl) x1 t0))) (let H_x \def (term_dec t2 x3) in (let H22 \def 
-H_x in (or_ind (eq T t2 x3) ((eq T t2 x3) \to (\forall (P: Prop).P)) (sn3 c 
-(THead (Flat Appl) x1 (THead (Bind Abst) x3 x4))) (\lambda (H23: (eq T t2 
-x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: T).((eq T (THead (Flat Appl) 
-x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) x1 (THead (Bind Abst) t0 
-x4))) \to (\forall (P: Prop).P))) H21 t2 H23) in (let H25 \def (eq_ind_r T x3 
-(\lambda (t0: T).(pr2 c t2 t0)) H19 t2 H23) in (eq_ind T t2 (\lambda (t0: 
-T).(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t0 x4)))) (let H_x0 \def 
-(term_dec x x1) in (let H26 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to 
-(\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t2 
-x4))) (\lambda (H27: (eq T x x1)).(let H28 \def (eq_ind_r T x1 (\lambda (t0: 
-T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) 
-t0 (THead (Bind Abst) t2 x4))) \to (\forall (P: Prop).P))) H24 x H27) in (let 
-H29 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H27) in (eq_ind 
-T x (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) t2 
-x4)))) (let H_x1 \def (term_dec x0 x4) in (let H30 \def H_x1 in (or_ind (eq T 
-x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x 
-(THead (Bind Abst) t2 x4))) (\lambda (H31: (eq T x0 x4)).(let H32 \def 
-(eq_ind_r T x4 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind 
-Abst) t2 x0)) (THead (Flat Appl) x (THead (Bind Abst) t2 t0))) \to (\forall 
-(P: Prop).P))) H28 x0 H31) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: 
-T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 
-H31) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead 
-(Bind Abst) t2 t0)))) (H32 (refl_equal T (THead (Flat Appl) x (THead (Bind 
-Abst) t2 x0))) (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)))) x4 
-H31)))) (\lambda (H31: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead 
-(Bind Abbr) x x4) (\lambda (H32: (eq T (THead (Bind Abbr) x x0) (THead (Bind 
-Abbr) x x4))).(\lambda (P: Prop).(let H33 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind 
-Abbr) x x0) (THead (Bind Abbr) x x4) H32) in (let H34 \def (eq_ind_r T x4 
-(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H31 x0 H33) in 
-(let H35 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H33) in (H34 (refl_equal T x0) 
-P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) 
-(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead 
-(Bind Abbr) x x4)) t2 (sn3_sing c t2 H7))) H30))) x1 H27)))) (\lambda (H27: 
-(((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind Abbr) x1 x4) 
-(\lambda (H28: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x1 
-x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) 
-\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
-(THead (Bind Abbr) x1 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind 
-Abbr) x x0) (THead (Bind Abbr) x1 x4) H28) in (\lambda (H31: (eq T x 
-x1)).(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (let H33 \def 
-(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) 
-H27 x H31) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 
-x H31) in (H33 (refl_equal T x) P)))))) H29)))) (pr3_head_12 c x x1 (pr3_pr2 
-c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20 
-Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) t2 (sn3_sing c t2 
-H7))) H26))) x3 H23)))) (\lambda (H23: (((eq T t2 x3) \to (\forall (P: 
-Prop).P)))).(let H_x0 \def (term_dec x x1) in (let H24 \def H_x0 in (or_ind 
-(eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) 
-x1 (THead (Bind Abst) x3 x4))) (\lambda (H25: (eq T x x1)).(let H26 \def 
-(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H25) in (eq_ind T x 
-(\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4)))) 
-(let H_x1 \def (term_dec x0 x4) in (let H27 \def H_x1 in (or_ind (eq T x0 x4) 
-((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x (THead 
-(Bind Abst) x3 x4))) (\lambda (H28: (eq T x0 x4)).(let H29 \def (eq_ind_r T 
-x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-x0 t0)))) H20 x0 H28) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat 
-Appl) x (THead (Bind Abst) x3 t0)))) (H8 x3 H23 (pr3_pr2 c t2 x3 H19)) x4 
-H28))) (\lambda (H28: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead 
-(Bind Abbr) x x4) (\lambda (H29: (eq T (THead (Bind Abbr) x x0) (THead (Bind 
-Abbr) x x4))).(\lambda (P: Prop).(let H30 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind 
-Abbr) x x0) (THead (Bind Abbr) x x4) H29) in (let H31 \def (eq_ind_r T x4 
-(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H28 x0 H30) in 
-(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (H31 (refl_equal T x0) 
-P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) 
-(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead 
-(Bind Abbr) x x4)) x3 (H7 x3 H23 (pr3_pr2 c t2 x3 H19)))) H27))) x1 H25))) 
-(\lambda (H25: (((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind 
-Abbr) x1 x4) (\lambda (H26: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
-x1 x4))).(\lambda (P: Prop).(let H27 \def (f_equal T T (\lambda (e: T).(match 
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) 
-\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
-(THead (Bind Abbr) x1 x4) H26) in ((let H28 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind 
-Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in (\lambda (H29: (eq T x 
-x1)).(let H30 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (let H31 \def 
-(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) 
-H25 x H29) in (let H32 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 
-x H29) in (H31 (refl_equal T x) P)))))) H27)))) (pr3_head_12 c x x1 (pr3_pr2 
-c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20 
-Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) x3 (H7 x3 H23 
-(pr3_pr2 c t2 x3 H19)))) H24)))) H22))) x2 H18))))))) H17)) t3 H13))))))) 
-H12)) (\lambda (H12: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T 
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind Abst) t2 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
-Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-z1 t4))))))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (x4: T).(\lambda (H13: (eq T (THead (Bind Abst) t2 x0) (THead 
-(Bind Abst) x1 x2))).(\lambda (H14: (eq T t3 (THead (Bind Abbr) x3 
-x4))).(\lambda (H15: (pr2 c x x3)).(\lambda (H16: ((\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H17 \def (eq_ind T t3 
-(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) 
-\to (\forall (P: Prop).P))) H9 (THead (Bind Abbr) x3 x4) H14) in (eq_ind_r T 
-(THead (Bind Abbr) x3 x4) (\lambda (t0: T).(sn3 c t0)) (let H18 \def (f_equal 
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) \Rightarrow t0])) 
-(THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in ((let H19 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) 
-\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in 
-(\lambda (_: (eq T t2 x1)).(let H21 \def (eq_ind_r T x2 (\lambda (t0: 
-T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t0 x4)))) H16 x0 
-H19) in (let H_x \def (term_dec x x3) in (let H22 \def H_x in (or_ind (eq T x 
-x3) ((eq T x x3) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abbr) x3 x4)) 
-(\lambda (H23: (eq T x x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: 
-T).(pr2 c x t0)) H15 x H23) in (eq_ind T x (\lambda (t0: T).(sn3 c (THead 
-(Bind Abbr) t0 x4))) (let H_x0 \def (term_dec x0 x4) in (let H25 \def H_x0 in 
-(or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead 
-(Bind Abbr) x x4)) (\lambda (H26: (eq T x0 x4)).(let H27 \def (eq_ind_r T x4 
-(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 
-t0)))) H21 x0 H26) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Bind Abbr) 
-x t0))) (sn3_sing c (THead (Bind Abbr) x x0) H6) x4 H26))) (\lambda (H26: 
-(((eq T x0 x4) \to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x x4) 
-(\lambda (H27: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x 
-x4))).(\lambda (P: Prop).(let H28 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) 
-\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
-(THead (Bind Abbr) x x4) H27) in (let H29 \def (eq_ind_r T x4 (\lambda (t0: 
-T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H26 x0 H28) in (let H30 \def 
-(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c 
-(Bind b) u) x0 t0)))) H21 x0 H28) in (H29 (refl_equal T x0) P)))))) (pr3_pr2 
-c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4 
-(Bind Abbr) (H21 Abbr x))))) H25))) x3 H23))) (\lambda (H23: (((eq T x x3) 
-\to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x3 x4) (\lambda (H24: (eq 
-T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x3 x4))).(\lambda (P: 
-Prop).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | 
-(THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
-x3 x4) H24) in ((let H26 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) 
-\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
-(THead (Bind Abbr) x3 x4) H24) in (\lambda (H27: (eq T x x3)).(let H28 \def 
-(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c 
-(Bind b) u) x0 t0)))) H21 x0 H26) in (let H29 \def (eq_ind_r T x3 (\lambda 
-(t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H23 x H27) in (let H30 
-\def (eq_ind_r T x3 (\lambda (t0: T).(pr2 c x t0)) H15 x H27) in (H29 
-(refl_equal T x) P)))))) H25)))) (pr3_head_12 c x x3 (pr3_pr2 c x x3 H15) 
-(Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x3) x0 x4 (H21 Abbr x3))))) 
-H22)))))) H18)) t3 H14)))))))))) H12)) (\lambda (H12: (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) 
-y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3) 
-(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda 
-(x5: T).(\lambda (x6: T).(\lambda (H13: (not (eq B x1 Abst))).(\lambda (H14: 
-(eq T (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3))).(\lambda (H15: (eq 
-T t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda 
-(_: (pr2 c x x5)).(\lambda (H17: (pr2 c x2 x6)).(\lambda (H18: (pr2 (CHead c 
-(Bind x1) x6) x3 x4)).(let H19 \def (eq_ind T t3 (\lambda (t0: T).((eq T 
-(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P: 
-Prop).P))) H9 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) 
-H15) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) 
-x4)) (\lambda (t0: T).(sn3 c t0)) (let H20 \def (f_equal T B (\lambda (e: 
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | 
-(TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return 
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
-Abst])])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in ((let H21 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in 
-((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ 
-t0) \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) 
-in (\lambda (H23: (eq T t2 x2)).(\lambda (H24: (eq B Abst x1)).(let H25 \def 
-(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H18 x0 
-H22) in (let H26 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H17 t2 
-H23) in (let H27 \def (eq_ind_r B x1 (\lambda (b: B).(pr2 (CHead c (Bind b) 
-x6) x0 x4)) H25 Abst H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b: 
-B).(not (eq B b Abst))) H13 Abst H24) in (eq_ind B Abst (\lambda (b: B).(sn3 
-c (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (let H29 
-\def (match (H28 (refl_equal B Abst)) in False return (\lambda (_: 
-False).(sn3 c (THead (Bind Abst) x6 (THead (Flat Appl) (lift (S O) O x5) 
-x4)))) with []) in H29) x1 H24)))))))) H21)) H20)) t3 H15)))))))))))))) H12)) 
-H11))))))))) w H4))))))))))) y H0))))) H)))).
-
-theorem nf3_appl_abbr:
- \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
-(CHead d (Bind Abbr) w)) \to (\forall (v: T).((sn3 c (THead (Flat Appl) v 
-(lift (S i) O w))) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda 
-(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (v: T).(\lambda (H0: (sn3 c 
-(THead (Flat Appl) v (lift (S i) O w)))).(insert_eq T (THead (Flat Appl) v 
-(lift (S i) O w)) (\lambda (t: T).(sn3 c t)) (sn3 c (THead (Flat Appl) v 
-(TLRef i))) (\lambda (y: T).(\lambda (H1: (sn3 c y)).(unintro T v (\lambda 
-(t: T).((eq T y (THead (Flat Appl) t (lift (S i) O w))) \to (sn3 c (THead 
-(Flat Appl) t (TLRef i))))) (sn3_ind c (\lambda (t: T).(\forall (x: T).((eq T 
-t (THead (Flat Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x 
-(TLRef i)))))) (\lambda (t1: T).(\lambda (H2: ((\forall (t2: T).((((eq T t1 
-t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c 
-t2)))))).(\lambda (H3: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).((eq T t2 (THead (Flat 
-Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x (TLRef 
-i)))))))))).(\lambda (x: T).(\lambda (H4: (eq T t1 (THead (Flat Appl) x (lift 
-(S i) O w)))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: 
-T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall 
-(x0: T).((eq T t2 (THead (Flat Appl) x0 (lift (S i) O w))) \to (sn3 c (THead 
-(Flat Appl) x0 (TLRef i))))))))) H3 (THead (Flat Appl) x (lift (S i) O w)) 
-H4) in (let H6 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: T).((((eq T t 
-t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (sn3 c t2))))) H2 
-(THead (Flat Appl) x (lift (S i) O w)) H4) in (sn3_pr2_intro c (THead (Flat 
-Appl) x (TLRef i)) (\lambda (t2: T).(\lambda (H7: (((eq T (THead (Flat Appl) 
-x (TLRef i)) t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr2 c (THead 
-(Flat Appl) x (TLRef i)) t2)).(let H9 \def (pr2_gen_appl c x (TLRef i) t2 H8) 
-in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x 
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq 
-T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) 
-(sn3 c t2) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T 
-T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H12: (pr2 c 
-x x0)).(\lambda (H13: (pr2 c (TLRef i) x1)).(let H14 \def (eq_ind T t2 
-(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: 
-Prop).P))) H7 (THead (Flat Appl) x0 x1) H11) in (eq_ind_r T (THead (Flat 
-Appl) x0 x1) (\lambda (t: T).(sn3 c t)) (let H15 \def (pr2_gen_lref c x1 i 
-H13) in (or_ind (eq T x1 (TLRef i)) (ex2_2 C T (\lambda (d0: C).(\lambda (u: 
-T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq 
-T x1 (lift (S i) O u))))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda (H16: 
-(eq T x1 (TLRef i))).(let H17 \def (eq_ind T x1 (\lambda (t: T).((eq T (THead 
-(Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P: 
-Prop).P))) H14 (TLRef i) H16) in (eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c 
-(THead (Flat Appl) x0 t))) (let H_x \def (term_dec x x0) in (let H18 \def H_x 
-in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c (THead 
-(Flat Appl) x0 (TLRef i))) (\lambda (H19: (eq T x x0)).(let H20 \def 
-(eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead 
-(Flat Appl) t (TLRef i))) \to (\forall (P: Prop).P))) H17 x H19) in (let H21 
-\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H19) in (eq_ind T x 
-(\lambda (t: T).(sn3 c (THead (Flat Appl) t (TLRef i)))) (H20 (refl_equal T 
-(THead (Flat Appl) x (TLRef i))) (sn3 c (THead (Flat Appl) x (TLRef i)))) x0 
-H19)))) (\lambda (H19: (((eq T x x0) \to (\forall (P: Prop).P)))).(H5 (THead 
-(Flat Appl) x0 (lift (S i) O w)) (\lambda (H20: (eq T (THead (Flat Appl) x 
-(lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)))).(\lambda (P: 
-Prop).(let H21 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | 
-(THead _ t _) \Rightarrow t])) (THead (Flat Appl) x (lift (S i) O w)) (THead 
-(Flat Appl) x0 (lift (S i) O w)) H20) in (let H22 \def (eq_ind_r T x0 
-(\lambda (t: T).((eq T x t) \to (\forall (P0: Prop).P0))) H19 x H21) in (let 
-H23 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H21) in (H22 
-(refl_equal T x) P)))))) (pr3_pr2 c (THead (Flat Appl) x (lift (S i) O w)) 
-(THead (Flat Appl) x0 (lift (S i) O w)) (pr2_head_1 c x x0 H12 (Flat Appl) 
-(lift (S i) O w))) x0 (refl_equal T (THead (Flat Appl) x0 (lift (S i) O 
-w))))) H18))) x1 H16))) (\lambda (H16: (ex2_2 C T (\lambda (d0: C).(\lambda 
-(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
-T).(eq T x1 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda 
-(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
-T).(eq T x1 (lift (S i) O u)))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda 
-(x2: C).(\lambda (x3: T).(\lambda (H17: (getl i c (CHead x2 (Bind Abbr) 
-x3))).(\lambda (H18: (eq T x1 (lift (S i) O x3))).(let H19 \def (eq_ind T x1 
-(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 
-t)) \to (\forall (P: Prop).P))) H14 (lift (S i) O x3) H18) in (eq_ind_r T 
-(lift (S i) O x3) (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H20 
-\def (eq_ind C (CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H 
-(CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 
-(Bind Abbr) x3) H17)) in (let H21 \def (f_equal C C (\lambda (e: C).(match e 
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) 
-\Rightarrow c0])) (CHead d (Bind Abbr) w) (CHead x2 (Bind Abbr) x3) 
-(getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 (Bind Abbr) x3) H17)) in 
-((let H22 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: 
-C).T) with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d 
-(Bind Abbr) w) (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) 
-i H (CHead x2 (Bind Abbr) x3) H17)) in (\lambda (H23: (eq C d x2)).(let H24 
-\def (eq_ind_r T x3 (\lambda (t: T).(getl i c (CHead x2 (Bind Abbr) t))) H20 
-w H22) in (eq_ind T w (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 (lift (S 
-i) O t)))) (let H25 \def (eq_ind_r C x2 (\lambda (c0: C).(getl i c (CHead c0 
-(Bind Abbr) w))) H24 d H23) in (let H_x \def (term_dec x x0) in (let H26 \def 
-H_x in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c 
-(THead (Flat Appl) x0 (lift (S i) O w))) (\lambda (H27: (eq T x x0)).(let H28 
-\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H27) in (eq_ind T x 
-(\lambda (t: T).(sn3 c (THead (Flat Appl) t (lift (S i) O w)))) (sn3_sing c 
-(THead (Flat Appl) x (lift (S i) O w)) H6) x0 H27))) (\lambda (H27: (((eq T x 
-x0) \to (\forall (P: Prop).P)))).(H6 (THead (Flat Appl) x0 (lift (S i) O w)) 
-(\lambda (H28: (eq T (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat 
-Appl) x0 (lift (S i) O w)))).(\lambda (P: Prop).(let H29 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t])) 
-(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O 
-w)) H28) in (let H30 \def (eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to 
-(\forall (P0: Prop).P0))) H27 x H29) in (let H31 \def (eq_ind_r T x0 (\lambda 
-(t: T).(pr2 c x t)) H12 x H29) in (H30 (refl_equal T x) P)))))) (pr3_pr2 c 
-(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O 
-w)) (pr2_head_1 c x x0 H12 (Flat Appl) (lift (S i) O w))))) H26)))) x3 
-H22)))) H21))) x1 H18)))))) H16)) H15)) t2 H11))))))) H10)) (\lambda (H10: 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 
-t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind 
-b) u) z1 t3))))))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: 
-T).(\lambda (x3: T).(\lambda (H11: (eq T (TLRef i) (THead (Bind Abst) x0 
-x1))).(\lambda (H12: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c 
-x x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) 
-u) x1 x3))))).(let H15 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat 
-Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 (THead (Bind Abbr) x2 
-x3) H12) in (eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) 
-(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 
-x1) H11) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H16)) t2 
-H12)))))))))) H10)) (\lambda (H10: (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind 
-B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq 
-T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) 
-(sn3 c t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 
-Abst))).(\lambda (H12: (eq T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda 
-(H13: (eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) 
-x3)))).(\lambda (_: (pr2 c x x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: 
-(pr2 (CHead c (Bind x0) x5) x2 x3)).(let H17 \def (eq_ind T t2 (\lambda (t: 
-T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 
-(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) H13) in 
-(eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) 
-(\lambda (t: T).(sn3 c t)) (let H18 \def (eq_ind T (TLRef i) (\lambda (ee: 
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
-(THead (Bind x0) x1 x2) H12) in (False_ind (sn3 c (THead (Bind x0) x5 (THead 
-(Flat Appl) (lift (S O) O x4) x3))) H18)) t2 H13)))))))))))))) H10)) 
-H9))))))))))))) y H1)))) H0))))))).
-
-theorem sn3_appl_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: 
-T).((sn3 c u) \to (\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) u) 
-(THead (Flat Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v 
-(THead (Bind b) u t))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda 
-(u: T).(\lambda (H0: (sn3 c u)).(sn3_ind c (\lambda (t: T).(\forall (t0: 
-T).(\forall (v: T).((sn3 (CHead c (Bind b) t) (THead (Flat Appl) (lift (S O) 
-O v) t0)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t t0))))))) 
-(\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
-(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall 
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
-(\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) t2) (THead (Flat 
-Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t2 
-t))))))))))).(\lambda (t: T).(\lambda (v: T).(\lambda (H3: (sn3 (CHead c 
-(Bind b) t1) (THead (Flat Appl) (lift (S O) O v) t))).(insert_eq T (THead 
-(Flat Appl) (lift (S O) O v) t) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) 
-t0)) (sn3 c (THead (Flat Appl) v (THead (Bind b) t1 t))) (\lambda (y: 
-T).(\lambda (H4: (sn3 (CHead c (Bind b) t1) y)).(unintro T t (\lambda (t0: 
-T).((eq T y (THead (Flat Appl) (lift (S O) O v) t0)) \to (sn3 c (THead (Flat 
-Appl) v (THead (Bind b) t1 t0))))) (unintro T v (\lambda (t0: T).(\forall (x: 
-T).((eq T y (THead (Flat Appl) (lift (S O) O t0) x)) \to (sn3 c (THead (Flat 
-Appl) t0 (THead (Bind b) t1 x)))))) (sn3_ind (CHead c (Bind b) t1) (\lambda 
-(t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat Appl) (lift 
-(S O) O x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) t1 x0))))))) 
-(\lambda (t2: T).(\lambda (H5: ((\forall (t3: T).((((eq T t2 t3) \to (\forall 
-(P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind 
-b) t1) t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t2 t3) \to (\forall 
-(P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (\forall (x: 
-T).(\forall (x0: T).((eq T t3 (THead (Flat Appl) (lift (S O) O x) x0)) \to 
-(sn3 c (THead (Flat Appl) x (THead (Bind b) t1 x0))))))))))).(\lambda (x: 
-T).(\lambda (x0: T).(\lambda (H7: (eq T t2 (THead (Flat Appl) (lift (S O) O 
-x) x0))).(let H8 \def (eq_ind T t2 (\lambda (t0: T).(\forall (t3: T).((((eq T 
-t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t0 t3) \to 
-(\forall (x1: T).(\forall (x2: T).((eq T t3 (THead (Flat Appl) (lift (S O) O 
-x1) x2)) \to (sn3 c (THead (Flat Appl) x1 (THead (Bind b) t1 x2)))))))))) H6 
-(THead (Flat Appl) (lift (S O) O x) x0) H7) in (let H9 \def (eq_ind T t2 
-(\lambda (t0: T).(\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) 
-\to ((pr3 (CHead c (Bind b) t1) t0 t3) \to (sn3 (CHead c (Bind b) t1) t3))))) 
-H5 (THead (Flat Appl) (lift (S O) O x) x0) H7) in (sn3_pr2_intro c (THead 
-(Flat Appl) x (THead (Bind b) t1 x0)) (\lambda (t3: T).(\lambda (H10: (((eq T 
-(THead (Flat Appl) x (THead (Bind b) t1 x0)) t3) \to (\forall (P: 
-Prop).P)))).(\lambda (H11: (pr2 c (THead (Flat Appl) x (THead (Bind b) t1 
-x0)) t3)).(let H12 \def (pr2_gen_appl c x (THead (Bind b) t1 x0) t3 H11) in 
-(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat 
-Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4)))) (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
-Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) 
-u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead 
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
-b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2)))))))) (sn3 c t3) 
-(\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) 
-t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))) (sn3 c 
-t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H14: (eq T t3 (THead (Flat 
-Appl) x1 x2))).(\lambda (H15: (pr2 c x x1)).(\lambda (H16: (pr2 c (THead 
-(Bind b) t1 x0) x2)).(let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T 
-(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) 
-H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat Appl) x1 x2) 
-(\lambda (t0: T).(sn3 c t0)) (let H_x \def (pr3_gen_bind b H c t1 x0 x2) in 
-(let H18 \def (H_x (pr3_pr2 c (THead (Bind b) t1 x0) x2 H16)) in (or_ind 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4)))) (pr3 (CHead c (Bind 
-b) t1) x0 (lift (S O) O x2)) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (H19: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 t4)))) (\lambda 
-(u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 
-(CHead c (Bind b) t1) x0 t4))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda 
-(x3: T).(\lambda (x4: T).(\lambda (H20: (eq T x2 (THead (Bind b) x3 
-x4))).(\lambda (H21: (pr3 c t1 x3)).(\lambda (H22: (pr3 (CHead c (Bind b) t1) 
-x0 x4)).(let H23 \def (eq_ind T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) 
-x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 t0)) \to (\forall (P: 
-Prop).P))) H17 (THead (Bind b) x3 x4) H20) in (eq_ind_r T (THead (Bind b) x3 
-x4) (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 t0))) (let H_x0 \def 
-(term_dec t1 x3) in (let H24 \def H_x0 in (or_ind (eq T t1 x3) ((eq T t1 x3) 
-\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind b) x3 
-x4))) (\lambda (H25: (eq T t1 x3)).(let H26 \def (eq_ind_r T x3 (\lambda (t0: 
-T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 
-(THead (Bind b) t0 x4))) \to (\forall (P: Prop).P))) H23 t1 H25) in (let H27 
-\def (eq_ind_r T x3 (\lambda (t0: T).(pr3 c t1 t0)) H21 t1 H25) in (eq_ind T 
-t1 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 (THead (Bind b) t0 x4)))) 
-(let H_x1 \def (term_dec x0 x4) in (let H28 \def H_x1 in (or_ind (eq T x0 x4) 
-((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead 
-(Bind b) t1 x4))) (\lambda (H29: (eq T x0 x4)).(let H30 \def (eq_ind_r T x4 
-(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead 
-(Flat Appl) x1 (THead (Bind b) t1 t0))) \to (\forall (P: Prop).P))) H26 x0 
-H29) in (let H31 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) 
-t1) x0 t0)) H22 x0 H29) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat 
-Appl) x1 (THead (Bind b) t1 t0)))) (let H_x2 \def (term_dec x x1) in (let H32 
-\def H_x2 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 
-c (THead (Flat Appl) x1 (THead (Bind b) t1 x0))) (\lambda (H33: (eq T x 
-x1)).(let H34 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) 
-x (THead (Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to 
-(\forall (P: Prop).P))) H30 x H33) in (let H35 \def (eq_ind_r T x1 (\lambda 
-(t0: T).(pr2 c x t0)) H15 x H33) in (eq_ind T x (\lambda (t0: T).(sn3 c 
-(THead (Flat Appl) t0 (THead (Bind b) t1 x0)))) (H34 (refl_equal T (THead 
-(Flat Appl) x (THead (Bind b) t1 x0))) (sn3 c (THead (Flat Appl) x (THead 
-(Bind b) t1 x0)))) x1 H33)))) (\lambda (H33: (((eq T x x1) \to (\forall (P: 
-Prop).P)))).(H8 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H34: (eq T 
-(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
-x0))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map 
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
-lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
-(lift (S O) O x1) x0) H34) in (let H36 \def (eq_ind_r T x1 (\lambda (t0: 
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H33 x (lift_inj x x1 (S O) O 
-H35)) in (let H37 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x 
-(lift_inj x x1 (S O) O H35)) in (H36 (refl_equal T x) P)))))) (pr3_flat 
-(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c 
-(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 
-(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl) x1 x0 
-(refl_equal T (THead (Flat Appl) (lift (S O) O x1) x0)))) H32))) x4 H29)))) 
-(\lambda (H29: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H8 (THead (Flat 
-Appl) (lift (S O) O x1) x4) (\lambda (H30: (eq T (THead (Flat Appl) (lift (S 
-O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P: 
-Prop).(let H31 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat 
-\to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
-lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
-(lift (S O) O x1) x4) H30) in ((let H32 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat 
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H30) in 
-(\lambda (H33: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H34 \def 
-(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) 
-H29 x0 H32) in (let H35 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T (THead 
-(Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 (THead (Bind b) 
-t1 t0))) \to (\forall (P0: Prop).P0))) H26 x0 H32) in (let H36 \def (eq_ind_r 
-T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) H22 x0 H32) in (let 
-H37 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead 
-(Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to (\forall 
-(P0: Prop).P0))) H35 x (lift_inj x x1 (S O) O H33)) in (let H38 \def 
-(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O 
-H33)) in (H34 (refl_equal T x0) P)))))))) H31)))) (pr3_flat (CHead c (Bind b) 
-t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S 
-O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) 
-x0 x4 H22 Appl) x1 x4 (refl_equal T (THead (Flat Appl) (lift (S O) O x1) 
-x4)))) H28))) x3 H25)))) (\lambda (H25: (((eq T t1 x3) \to (\forall (P: 
-Prop).P)))).(H2 x3 H25 H21 x4 x1 (sn3_cpr3_trans c t1 x3 H21 (Bind b) (THead 
-(Flat Appl) (lift (S O) O x1) x4) (let H_x1 \def (term_dec x0 x4) in (let H26 
-\def H_x1 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) 
-(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x4)) (\lambda 
-(H27: (eq T x0 x4)).(let H28 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead 
-c (Bind b) t1) x0 t0)) H22 x0 H27) in (eq_ind T x0 (\lambda (t0: T).(sn3 
-(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) t0))) (let H_x2 
-\def (term_dec x x1) in (let H29 \def H_x2 in (or_ind (eq T x x1) ((eq T x 
-x1) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) 
-(lift (S O) O x1) x0)) (\lambda (H30: (eq T x x1)).(let H31 \def (eq_ind_r T 
-x1 (\lambda (t0: T).(pr2 c x t0)) H15 x H30) in (eq_ind T x (\lambda (t0: 
-T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0))) 
-(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9) 
-x1 H30))) (\lambda (H30: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9 
-(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H31: (eq T (THead (Flat 
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
-x0))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map 
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
-lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
-(lift (S O) O x1) x0) H31) in (let H33 \def (eq_ind_r T x1 (\lambda (t0: 
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H30 x (lift_inj x x1 (S O) O 
-H32)) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x 
-(lift_inj x x1 (S O) O H32)) in (H33 (refl_equal T x) P)))))) (pr3_flat 
-(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c 
-(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 
-(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl))) 
-H29))) x4 H27))) (\lambda (H27: (((eq T x0 x4) \to (\forall (P: 
-Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x1) x4) (\lambda (H28: (eq T 
-(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
-x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map 
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
-lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
-(lift (S O) O x1) x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat 
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H28) in 
-(\lambda (H31: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H32 \def 
-(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) 
-H27 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c 
-(Bind b) t1) x0 t0)) H22 x0 H30) in (let H34 \def (eq_ind_r T x1 (\lambda 
-(t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H31)) in (H32 (refl_equal 
-T x0) P)))))) H29)))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift 
-(S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c 
-c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x4 H22 Appl))) H26)))))) 
-H24))) x2 H20))))))) H19)) (\lambda (H19: (pr3 (CHead c (Bind b) t1) x0 (lift 
-(S O) O x2))).(sn3_gen_lift (CHead c (Bind b) t1) (THead (Flat Appl) x1 x2) 
-(S O) O (eq_ind_r T (THead (Flat Appl) (lift (S O) O x1) (lift (S O) (s (Flat 
-Appl) O) x2)) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) t0)) (sn3_pr3_trans 
-(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x0) (let H_x0 \def 
-(term_dec x x1) in (let H20 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to 
-(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S 
-O) O x1) x0)) (\lambda (H21: (eq T x x1)).(let H22 \def (eq_ind_r T x1 
-(\lambda (t0: T).(pr2 c x t0)) H15 x H21) in (eq_ind T x (\lambda (t0: 
-T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0))) 
-(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9) 
-x1 H21))) (\lambda (H21: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9 
-(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H22: (eq T (THead (Flat 
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
-x0))).(\lambda (P: Prop).(let H23 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map 
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
-lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t4))]) in lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
-(lift (S O) O x1) x0) H22) in (let H24 \def (eq_ind_r T x1 (\lambda (t0: 
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H21 x (lift_inj x x1 (S O) O 
-H23)) in (let H25 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x 
-(lift_inj x x1 (S O) O H23)) in (H24 (refl_equal T x) P)))))) (pr3_flat 
-(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c 
-(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 
-(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl))) 
-H20))) (THead (Flat Appl) (lift (S O) O x1) (lift (S O) O x2)) (pr3_thin_dx 
-(CHead c (Bind b) t1) x0 (lift (S O) O x2) H19 (lift (S O) O x1) Appl)) (lift 
-(S O) O (THead (Flat Appl) x1 x2)) (lift_head (Flat Appl) x1 x2 (S O) O)) c 
-(drop_drop (Bind b) O c c (drop_refl c) t1))) H18))) t3 H14))))))) H13)) 
-(\lambda (H13: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: 
-T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: 
-T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))))).(ex4_4_ind T T T T (\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
-b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x 
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: 
-T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))) 
-(sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (H14: (eq T (THead (Bind b) t1 x0) (THead (Bind Abst) x1 
-x2))).(\lambda (H15: (eq T t3 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c 
-x x3)).(\lambda (H17: ((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind 
-b0) u0) x2 x4))))).(let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T (THead 
-(Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) H10 
-(THead (Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4) 
-(\lambda (t0: T).(sn3 c t0)) (let H19 \def (f_equal T B (\lambda (e: 
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | 
-(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return 
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
-b])])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in ((let H20 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in 
-((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ 
-t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) 
-in (\lambda (_: (eq T t1 x1)).(\lambda (H23: (eq B b Abst)).(let H24 \def 
-(eq_ind_r T x2 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead 
-c (Bind b0) u0) t0 x4)))) H17 x0 H21) in (let H25 \def (eq_ind B b (\lambda 
-(b0: B).((eq T (THead (Flat Appl) x (THead (Bind b0) t1 x0)) (THead (Bind 
-Abbr) x3 x4)) \to (\forall (P: Prop).P))) H18 Abst H23) in (let H26 \def 
-(eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T (THead (Flat Appl) 
-(lift (S O) O x) x0) t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind 
-b0) t1) (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (sn3 (CHead c (Bind 
-b0) t1) t4))))) H9 Abst H23) in (let H27 \def (eq_ind B b (\lambda (b0: 
-B).(\forall (t4: T).((((eq T (THead (Flat Appl) (lift (S O) O x) x0) t4) \to 
-(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) (THead (Flat Appl) 
-(lift (S O) O x) x0) t4) \to (\forall (x5: T).(\forall (x6: T).((eq T t4 
-(THead (Flat Appl) (lift (S O) O x5) x6)) \to (sn3 c (THead (Flat Appl) x5 
-(THead (Bind b0) t1 x6)))))))))) H8 Abst H23) in (let H28 \def (eq_ind B b 
-(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P))) 
-\to ((pr3 c t1 t4) \to (\forall (t0: T).(\forall (v0: T).((sn3 (CHead c (Bind 
-b0) t4) (THead (Flat Appl) (lift (S O) O v0) t0)) \to (sn3 c (THead (Flat 
-Appl) v0 (THead (Bind b0) t4 t0)))))))))) H2 Abst H23) in (let H29 \def 
-(eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H Abst H23) in (let H30 
-\def (match (H29 (refl_equal B Abst)) in False return (\lambda (_: 
-False).(sn3 c (THead (Bind Abbr) x3 x4))) with []) in H30)))))))))) H20)) 
-H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 B T T T T T (\lambda (b0: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) 
-t1 x0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-t3 (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
-y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 
-z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 
-Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind 
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b0) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b0: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2))))))) (sn3 c t3) (\lambda (x1: 
-B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: 
-T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15: (eq T 
-(THead (Bind b) t1 x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T t3 
-(THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda 
-(H17: (pr2 c x x5)).(\lambda (H18: (pr2 c x2 x6)).(\lambda (H19: (pr2 (CHead 
-c (Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t0: T).((eq T 
-(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) 
-H10 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) H16) in 
-(eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) 
-(\lambda (t0: T).(sn3 c t0)) (let H21 \def (f_equal T B (\lambda (e: 
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | 
-(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return 
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
-b])])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in ((let H22 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in 
-((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ 
-t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in 
-(\lambda (H24: (eq T t1 x2)).(\lambda (H25: (eq B b x1)).(let H26 \def 
-(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H19 x0 
-H23) in (let H27 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H18 t1 
-H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b0: B).(pr2 (CHead c (Bind b0) 
-x6) x0 x4)) H26 b H25) in (eq_ind B b (\lambda (b0: B).(sn3 c (THead (Bind 
-b0) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (sn3_pr3_trans c (THead 
-(Bind b) t1 (THead (Flat Appl) (lift (S O) O x5) x4)) (sn3_bind b c t1 
-(sn3_sing c t1 H1) (THead (Flat Appl) (lift (S O) O x5) x4) (let H_x \def 
-(term_dec x x5) in (let H29 \def H_x in (or_ind (eq T x x5) ((eq T x x5) \to 
-(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S 
-O) O x5) x4)) (\lambda (H30: (eq T x x5)).(let H31 \def (eq_ind_r T x5 
-(\lambda (t0: T).(pr2 c x t0)) H17 x H30) in (eq_ind T x (\lambda (t0: 
-T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x4))) (let 
-H_x0 \def (term_dec x0 x4) in (let H32 \def H_x0 in (or_ind (eq T x0 x4) ((eq 
-T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat 
-Appl) (lift (S O) O x) x4)) (\lambda (H33: (eq T x0 x4)).(let H34 \def 
-(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0 
-H33) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat 
-Appl) (lift (S O) O x) t0))) (sn3_sing (CHead c (Bind b) t1) (THead (Flat 
-Appl) (lift (S O) O x) x0) H9) x4 H33))) (\lambda (H33: (((eq T x0 x4) \to 
-(\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x) x4) (\lambda 
-(H34: (eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift 
-(S O) O x) x4))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat 
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x) x4) H34) in 
-(let H36 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: 
-Prop).P0))) H33 x0 H35) in (let H37 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2 
-(CHead c (Bind b) x6) x0 t0)) H28 x0 H35) in (H36 (refl_equal T x0) P)))))) 
-(pr3_pr3_pr3_t c t1 x6 (pr3_pr2 c t1 x6 H27) (THead (Flat Appl) (lift (S O) O 
-x) x0) (THead (Flat Appl) (lift (S O) O x) x4) (Bind b) (pr3_pr2 (CHead c 
-(Bind b) x6) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift 
-(S O) O x) x4) (pr2_thin_dx (CHead c (Bind b) x6) x0 x4 H28 (lift (S O) O x) 
-Appl))))) H32))) x5 H30))) (\lambda (H30: (((eq T x x5) \to (\forall (P: 
-Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x5) x4) (\lambda (H31: (eq T 
-(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) 
-x4))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map 
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
-lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t4))]) in lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
-(lift (S O) O x5) x4) H31) in ((let H33 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat 
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) x4) H31) in 
-(\lambda (H34: (eq T (lift (S O) O x) (lift (S O) O x5))).(let H35 \def 
-(eq_ind_r T x5 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) 
-H30 x (lift_inj x x5 (S O) O H34)) in (let H36 \def (eq_ind_r T x5 (\lambda 
-(t0: T).(pr2 c x t0)) H17 x (lift_inj x x5 (S O) O H34)) in (let H37 \def 
-(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0 
-H33) in (H35 (refl_equal T x) P)))))) H32)))) (pr3_pr3_pr3_t c t1 x6 (pr3_pr2 
-c t1 x6 H27) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift 
-(S O) O x5) x4) (Bind b) (pr3_flat (CHead c (Bind b) x6) (lift (S O) O x) 
-(lift (S O) O x5) (pr3_lift (CHead c (Bind b) x6) c (S O) O (drop_drop (Bind 
-b) O c c (drop_refl c) x6) x x5 (pr3_pr2 c x x5 H17)) x0 x4 (pr3_pr2 (CHead c 
-(Bind b) x6) x0 x4 H28) Appl)))) H29)))) (THead (Bind b) x6 (THead (Flat 
-Appl) (lift (S O) O x5) x4)) (pr3_pr2 c (THead (Bind b) t1 (THead (Flat Appl) 
-(lift (S O) O x5) x4)) (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O 
-x5) x4)) (pr2_head_1 c t1 x6 H27 (Bind b) (THead (Flat Appl) (lift (S O) O 
-x5) x4)))) x1 H25))))))) H22)) H21)) t3 H16)))))))))))))) H13)) 
-H12)))))))))))))) y H4))))) H3))))))) u H0))))).
-
-theorem sn3_appl_beta:
- \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((sn3 c 
-(THead (Flat Appl) u (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) 
-\to (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind Abst) w 
-t))))))))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H: 
-(sn3 c (THead (Flat Appl) u (THead (Bind Abbr) v t)))).(\lambda (w: 
-T).(\lambda (H0: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THead (Bind 
-Abbr) v t) H) in (let H1 \def H_x in (and_ind (sn3 c u) (sn3 c (THead (Bind 
-Abbr) v t)) (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind 
-Abst) w t)))) (\lambda (H2: (sn3 c u)).(\lambda (H3: (sn3 c (THead (Bind 
-Abbr) v t))).(sn3_appl_appl v (THead (Bind Abst) w t) c (sn3_beta c v t H3 w 
-H0) u H2 (\lambda (u2: T).(\lambda (H4: (pr3 c (THead (Flat Appl) v (THead 
-(Bind Abst) w t)) u2)).(\lambda (H5: (((iso (THead (Flat Appl) v (THead (Bind 
-Abst) w t)) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat 
-Appl) u (THead (Bind Abbr) v t)) H (THead (Flat Appl) u u2) (pr3_thin_dx c 
-(THead (Bind Abbr) v t) u2 (pr3_iso_beta v w t c u2 H4 H5) u Appl)))))))) 
-H1))))))))).
-
-theorem sn3_appls_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: 
-T).((sn3 c u) \to (\forall (vs: TList).(\forall (t: T).((sn3 (CHead c (Bind 
-b) u) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to (sn3 c (THeads (Flat 
-Appl) vs (THead (Bind b) u t))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda 
-(u: T).(\lambda (H0: (sn3 c u)).(\lambda (vs: TList).(TList_ind (\lambda (t: 
-TList).(\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) (lifts 
-(S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u t0)))))) 
-(\lambda (t: T).(\lambda (H1: (sn3 (CHead c (Bind b) u) t)).(sn3_bind b c u 
-H0 t H1))) (\lambda (v: T).(\lambda (vs0: TList).(TList_ind (\lambda (t: 
-TList).(((\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) 
-(lifts (S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u 
-t0)))))) \to (\forall (t0: T).((sn3 (CHead c (Bind b) u) (THead (Flat Appl) 
-(lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t) t0))) \to (sn3 c 
-(THead (Flat Appl) v (THeads (Flat Appl) t (THead (Bind b) u t0)))))))) 
-(\lambda (_: ((\forall (t: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) 
-(lifts (S O) O TNil) t)) \to (sn3 c (THeads (Flat Appl) TNil (THead (Bind b) 
-u t))))))).(\lambda (t: T).(\lambda (H2: (sn3 (CHead c (Bind b) u) (THead 
-(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O TNil) 
-t)))).(sn3_appl_bind b H c u H0 t v H2)))) (\lambda (t: T).(\lambda (t0: 
-TList).(\lambda (_: ((((\forall (t1: T).((sn3 (CHead c (Bind b) u) (THeads 
-(Flat Appl) (lifts (S O) O t0) t1)) \to (sn3 c (THeads (Flat Appl) t0 (THead 
-(Bind b) u t1)))))) \to (\forall (t1: T).((sn3 (CHead c (Bind b) u) (THead 
-(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t0) t1))) \to 
-(sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (THead (Bind b) u 
-t1))))))))).(\lambda (H2: ((\forall (t1: T).((sn3 (CHead c (Bind b) u) 
-(THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) \to (sn3 c (THeads 
-(Flat Appl) (TCons t t0) (THead (Bind b) u t1))))))).(\lambda (t1: 
-T).(\lambda (H3: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O 
-v) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))).(let H_x \def 
-(sn3_gen_flat Appl (CHead c (Bind b) u) (lift (S O) O v) (THeads (Flat Appl) 
-(lifts (S O) O (TCons t t0)) t1) H3) in (let H4 \def H_x in (and_ind (sn3 
-(CHead c (Bind b) u) (lift (S O) O v)) (sn3 (CHead c (Bind b) u) (THeads 
-(Flat Appl) (lifts (S O) O (TCons t t0)) t1)) (sn3 c (THead (Flat Appl) v 
-(THeads (Flat Appl) (TCons t t0) (THead (Bind b) u t1)))) (\lambda (H5: (sn3 
-(CHead c (Bind b) u) (lift (S O) O v))).(\lambda (H6: (sn3 (CHead c (Bind b) 
-u) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1))).(let H_y \def 
-(sn3_gen_lift (CHead c (Bind b) u) v (S O) O H5 c) in (sn3_appl_appls t 
-(THead (Bind b) u t1) t0 c (H2 t1 H6) v (H_y (drop_drop (Bind b) O c c 
-(drop_refl c) u)) (\lambda (u2: T).(\lambda (H7: (pr3 c (THeads (Flat Appl) 
-(TCons t t0) (THead (Bind b) u t1)) u2)).(\lambda (H8: (((iso (THeads (Flat 
-Appl) (TCons t t0) (THead (Bind b) u t1)) u2) \to (\forall (P: 
-Prop).P)))).(let H9 \def (pr3_iso_appls_bind b H (TCons t t0) u t1 c u2 H7 
-H8) in (sn3_pr3_trans c (THead (Flat Appl) v (THead (Bind b) u (THeads (Flat 
-Appl) (lifts (S O) O (TCons t t0)) t1))) (sn3_appl_bind b H c u H0 (THeads 
-(Flat Appl) (lifts (S O) O (TCons t t0)) t1) v H3) (THead (Flat Appl) v u2) 
-(pr3_flat c v v (pr3_refl c v) (THead (Bind b) u (THeads (Flat Appl) (lifts 
-(S O) O (TCons t t0)) t1)) u2 H9 Appl)))))))))) H4))))))))) vs0))) vs)))))).
-
-theorem sn3_appls_beta:
- \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (us: TList).((sn3 c 
-(THeads (Flat Appl) us (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c 
-w) \to (sn3 c (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) 
-w t))))))))))
-\def
- \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (us: 
-TList).(TList_ind (\lambda (t0: TList).((sn3 c (THeads (Flat Appl) t0 (THead 
-(Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat 
-Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) (\lambda (H: 
-(sn3 c (THead (Bind Abbr) v t))).(\lambda (w: T).(\lambda (H0: (sn3 c 
-w)).(sn3_beta c v t H w H0)))) (\lambda (u: T).(\lambda (us0: 
-TList).(TList_ind (\lambda (t0: TList).((((sn3 c (THeads (Flat Appl) t0 
-(THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads 
-(Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 
-c (THead (Flat Appl) u (THeads (Flat Appl) t0 (THead (Bind Abbr) v t)))) \to 
-(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat 
-Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))) (\lambda (_: 
-(((sn3 c (THeads (Flat Appl) TNil (THead (Bind Abbr) v t))) \to (\forall (w: 
-T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) TNil (THead (Flat Appl) v (THead 
-(Bind Abst) w t))))))))).(\lambda (H0: (sn3 c (THead (Flat Appl) u (THeads 
-(Flat Appl) TNil (THead (Bind Abbr) v t))))).(\lambda (w: T).(\lambda (H1: 
-(sn3 c w)).(sn3_appl_beta c u v t H0 w H1))))) (\lambda (t0: T).(\lambda (t1: 
-TList).(\lambda (_: (((((sn3 c (THeads (Flat Appl) t1 (THead (Bind Abbr) v 
-t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) t1 (THead 
-(Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 c (THead (Flat Appl) u 
-(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)))) \to (\forall (w: T).((sn3 c 
-w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) t1 (THead (Flat Appl) 
-v (THead (Bind Abst) w t))))))))))).(\lambda (H0: (((sn3 c (THeads (Flat 
-Appl) (TCons t0 t1) (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) 
-\to (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead 
-(Bind Abst) w t))))))))).(\lambda (H1: (sn3 c (THead (Flat Appl) u (THeads 
-(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t))))).(\lambda (w: 
-T).(\lambda (H2: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THeads 
-(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t)) H1) in (let H3 \def H_x in 
-(and_ind (sn3 c u) (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Bind 
-Abbr) v t))) (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) 
-(THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H4: (sn3 c 
-u)).(\lambda (H5: (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) 
-v t)))).(sn3_appl_appls t0 (THead (Flat Appl) v (THead (Bind Abst) w t)) t1 c 
-(H0 H5 w H2) u H4 (\lambda (u2: T).(\lambda (H6: (pr3 c (THeads (Flat Appl) 
-(TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w t))) u2)).(\lambda 
-(H7: (((iso (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead 
-(Bind Abst) w t))) u2) \to (\forall (P: Prop).P)))).(let H8 \def 
-(pr3_iso_appls_beta (TCons t0 t1) v w t c u2 H6 H7) in (sn3_pr3_trans c 
-(THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v 
-t))) H1 (THead (Flat Appl) u u2) (pr3_thin_dx c (THeads (Flat Appl) (TCons t0 
-t1) (THead (Bind Abbr) v t)) u2 H8 u Appl))))))))) H3)))))))))) us0))) us)))).
-
-theorem sn3_appls_abbr:
- \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
-(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).((sn3 c (THeads (Flat Appl) 
-vs (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda 
-(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind 
-(\lambda (t: TList).((sn3 c (THeads (Flat Appl) t (lift (S i) O w))) \to (sn3 
-c (THeads (Flat Appl) t (TLRef i))))) (\lambda (H0: (sn3 c (lift (S i) O 
-w))).(let H_y \def (sn3_gen_lift c w (S i) O H0 d (getl_drop Abbr c d w i H)) 
-in (sn3_abbr c d w i H H_y))) (\lambda (v: T).(\lambda (vs0: 
-TList).(TList_ind (\lambda (t: TList).((((sn3 c (THeads (Flat Appl) t (lift 
-(S i) O w))) \to (sn3 c (THeads (Flat Appl) t (TLRef i))))) \to ((sn3 c 
-(THead (Flat Appl) v (THeads (Flat Appl) t (lift (S i) O w)))) \to (sn3 c 
-(THead (Flat Appl) v (THeads (Flat Appl) t (TLRef i))))))) (\lambda (_: 
-(((sn3 c (THeads (Flat Appl) TNil (lift (S i) O w))) \to (sn3 c (THeads (Flat 
-Appl) TNil (TLRef i)))))).(\lambda (H1: (sn3 c (THead (Flat Appl) v (THeads 
-(Flat Appl) TNil (lift (S i) O w))))).(nf3_appl_abbr c d w i H v H1))) 
-(\lambda (t: T).(\lambda (t0: TList).(\lambda (_: (((((sn3 c (THeads (Flat 
-Appl) t0 (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) t0 (TLRef i))))) 
-\to ((sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (lift (S i) O w)))) 
-\to (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (TLRef 
-i)))))))).(\lambda (H1: (((sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i) 
-O w))) \to (sn3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)))))).(\lambda 
-(H2: (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (lift (S i) 
-O w))))).(let H_x \def (sn3_gen_flat Appl c v (THeads (Flat Appl) (TCons t 
-t0) (lift (S i) O w)) H2) in (let H3 \def H_x in (and_ind (sn3 c v) (sn3 c 
-(THeads (Flat Appl) (TCons t t0) (lift (S i) O w))) (sn3 c (THead (Flat Appl) 
-v (THeads (Flat Appl) (TCons t t0) (TLRef i)))) (\lambda (H4: (sn3 c 
-v)).(\lambda (H5: (sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i) O 
-w)))).(sn3_appl_appls t (TLRef i) t0 c (H1 H5) v H4 (\lambda (u2: T).(\lambda 
-(H6: (pr3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2)).(\lambda (H7: 
-(((iso (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2) \to (\forall (P: 
-Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) v (THeads (Flat Appl) (TCons 
-t t0) (lift (S i) O w))) H2 (THead (Flat Appl) v u2) (pr3_thin_dx c (THeads 
-(Flat Appl) (TCons t t0) (lift (S i) O w)) u2 (pr3_iso_appls_abbr c d w i H 
-(TCons t t0) u2 H6 H7) v Appl)))))))) H3)))))))) vs0))) vs)))))).
-
-theorem sn3_gen_def:
- \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
-(CHead d (Bind Abbr) v)) \to ((sn3 c (TLRef i)) \to (sn3 d v))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda 
-(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 c (TLRef 
-i))).(sn3_gen_lift c v (S i) O (sn3_pr3_trans c (TLRef i) H0 (lift (S i) O v) 
-(pr3_pr2 c (TLRef i) (lift (S i) O v) (pr2_delta c d v i H (TLRef i) (TLRef 
-i) (pr0_refl (TLRef i)) (lift (S i) O v) (subst0_lref v i)))) d (getl_drop 
-Abbr c d v i H))))))).
-
-theorem sn3_cdelta:
- \forall (v: T).(\forall (t: T).(\forall (i: nat).(((\forall (w: T).(ex T 
-(\lambda (u: T).(subst0 i w t u))))) \to (\forall (c: C).(\forall (d: 
-C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to (sn3 d v))))))))
-\def
- \lambda (v: T).(\lambda (t: T).(\lambda (i: nat).(\lambda (H: ((\forall (w: 
-T).(ex T (\lambda (u: T).(subst0 i w t u)))))).(let H_x \def (H v) in (let H0 
-\def H_x in (ex_ind T (\lambda (u: T).(subst0 i v t u)) (\forall (c: 
-C).(\forall (d: C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to 
-(sn3 d v))))) (\lambda (x: T).(\lambda (H1: (subst0 i v t x)).(subst0_ind 
-(\lambda (n: nat).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_: T).(\forall 
-(c: C).(\forall (d: C).((getl n c (CHead d (Bind Abbr) t0)) \to ((sn3 c t1) 
-\to (sn3 d t0))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (c: 
-C).(\lambda (d: C).(\lambda (H2: (getl i0 c (CHead d (Bind Abbr) 
-v0))).(\lambda (H3: (sn3 c (TLRef i0))).(sn3_gen_def c d v0 i0 H2 H3))))))) 
-(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: 
-nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c: 
-C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to 
-(sn3 d v0))))))).(\lambda (t0: T).(\lambda (k: K).(\lambda (c: C).(\lambda 
-(d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) v0))).(\lambda (H5: (sn3 
-c (THead k u1 t0))).(let H_y \def (sn3_gen_head k c u1 t0 H5) in (H3 c d H4 
-H_y)))))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: T).(\lambda 
-(t1: T).(\lambda (i0: nat).(\lambda (H2: (subst0 (s k i0) v0 t1 t2)).(\lambda 
-(H3: ((\forall (c: C).(\forall (d: C).((getl (s k i0) c (CHead d (Bind Abbr) 
-v0)) \to ((sn3 c t1) \to (sn3 d v0))))))).(\lambda (u: T).(\lambda (c: 
-C).(\lambda (d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) 
-v0))).(\lambda (H5: (sn3 c (THead k u t1))).(K_ind (\lambda (k0: K).((subst0 
-(s k0 i0) v0 t1 t2) \to (((\forall (c0: C).(\forall (d0: C).((getl (s k0 i0) 
-c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0)))))) \to ((sn3 
-c (THead k0 u t1)) \to (sn3 d v0))))) (\lambda (b: B).(\lambda (_: (subst0 (s 
-(Bind b) i0) v0 t1 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: 
-C).((getl (s (Bind b) i0) c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to 
-(sn3 d0 v0))))))).(\lambda (H8: (sn3 c (THead (Bind b) u t1))).(let H_x0 \def 
-(sn3_gen_bind b c u t1 H8) in (let H9 \def H_x0 in (and_ind (sn3 c u) (sn3 
-(CHead c (Bind b) u) t1) (sn3 d v0) (\lambda (_: (sn3 c u)).(\lambda (H11: 
-(sn3 (CHead c (Bind b) u) t1)).(H7 (CHead c (Bind b) u) d (getl_clear_bind b 
-(CHead c (Bind b) u) c u (clear_bind b c u) (CHead d (Bind Abbr) v0) i0 H4) 
-H11))) H9))))))) (\lambda (f: F).(\lambda (_: (subst0 (s (Flat f) i0) v0 t1 
-t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: C).((getl (s (Flat f) i0) 
-c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0))))))).(\lambda 
-(H8: (sn3 c (THead (Flat f) u t1))).(let H_x0 \def (sn3_gen_flat f c u t1 H8) 
-in (let H9 \def H_x0 in (and_ind (sn3 c u) (sn3 c t1) (sn3 d v0) (\lambda (_: 
-(sn3 c u)).(\lambda (H11: (sn3 c t1)).(H7 c d H4 H11))) H9))))))) k H2 H3 
-H5))))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda 
-(i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c: 
-C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to 
-(sn3 d v0))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: ((\forall (c: C).(\forall (d: 
-C).((getl (s k i0) c (CHead d (Bind Abbr) v0)) \to ((sn3 c t1) \to (sn3 d 
-v0))))))).(\lambda (c: C).(\lambda (d: C).(\lambda (H6: (getl i0 c (CHead d 
-(Bind Abbr) v0))).(\lambda (H7: (sn3 c (THead k u1 t1))).(let H_y \def 
-(sn3_gen_head k c u1 t1 H7) in (H3 c d H6 H_y))))))))))))))))) i v t x H1))) 
-H0)))))).
-
-inductive csubn: C \to (C \to Prop) \def
-| csubn_sort: \forall (n: nat).(csubn (CSort n) (CSort n))
-| csubn_head: \forall (c1: C).(\forall (c2: C).((csubn c1 c2) \to (\forall 
-(k: K).(\forall (v: T).(csubn (CHead c1 k v) (CHead c2 k v))))))
-| csubn_abst: \forall (c1: C).(\forall (c2: C).((csubn c1 c2) \to (\forall 
-(v: T).(\forall (w: T).((sn3 c2 w) \to (csubn (CHead c1 (Bind Abst) v) (CHead 
-c2 (Bind Abbr) w))))))).
-
-theorem csubc_csuba:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (csuba 
-g c1 c2))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 
-c2)).(csubc_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda 
-(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda 
-(_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda 
-(v: T).(csuba_head g c3 c4 H1 k v))))))) (\lambda (c3: C).(\lambda (c4: 
-C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (v: 
-T).(\lambda (a: A).(\lambda (H2: (sc3 g (asucc g a) c3 v)).(\lambda (w: 
-T).(\lambda (H3: (sc3 g a c4 w)).(csuba_abst g c3 c4 H1 v a (sc3_arity_gen g 
-c3 v (asucc g a) H2) w (sc3_arity_gen g c4 w a H3))))))))))) c1 c2 H)))).
-
-theorem csubc_csubn:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (csubn 
-c1 c2))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 
-c2)).(csubc_ind g (\lambda (c: C).(\lambda (c0: C).(csubn c c0))) (\lambda 
-(n: nat).(csubn_sort n)) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_: 
-(csubc g c3 c4)).(\lambda (H1: (csubn c3 c4)).(\lambda (k: K).(\lambda (v: 
-T).(csubn_head c3 c4 H1 k v))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda 
-(_: (csubc g c3 c4)).(\lambda (H1: (csubn c3 c4)).(\lambda (v: T).(\lambda 
-(a: A).(\lambda (_: (sc3 g (asucc g a) c3 v)).(\lambda (w: T).(\lambda (H3: 
-(sc3 g a c4 w)).(csubn_abst c3 c4 H1 v w (sc3_sn3 g a c4 w H3))))))))))) c1 
-c2 H)))).
-
-theorem ceq_arity_trans:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((ceqc g c2 c1) \to 
-(\forall (t: T).(\forall (a: A).((arity g c1 t a) \to (arity g c2 t a)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (ceqc g c2 
-c1)).(\lambda (t: T).(\lambda (a: A).(\lambda (H0: (arity g c1 t a)).(let H1 
-\def H in (or_ind (csubc g c2 c1) (csubc g c1 c2) (arity g c2 t a) (\lambda 
-(H2: (csubc g c2 c1)).(csuba_arity_rev g c1 t a H0 c2 (csubc_csuba g c2 c1 
-H2))) (\lambda (H2: (csubc g c1 c2)).(csuba_arity g c1 t a H0 c2 (csubc_csuba 
-g c1 c2 H2))) H1)))))))).
-
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/T/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/T/dec.ma
index 8b1a36ba1..7c6fa73da 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/T/dec.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/T/dec.ma
@@ -59,6 +59,16 @@ Abst \Rightarrow False | Void \Rightarrow True])) I Abst H) in (False_ind P
 H0))))) (or_introl (eq B Void Void) ((eq B Void Void) \to (\forall (P: 
 Prop).P)) (refl_equal B Void)) b2)) b1).
 
+theorem bind_dec_not:
+ \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) (not (eq B b1 b2))))
+\def
+ \lambda (b1: B).(\lambda (b2: B).(let H_x \def (terms_props__bind_dec b1 b2) 
+in (let H \def H_x in (or_ind (eq B b1 b2) ((eq B b1 b2) \to (\forall (P: 
+Prop).P)) (or (eq B b1 b2) ((eq B b1 b2) \to False)) (\lambda (H0: (eq B b1 
+b2)).(or_introl (eq B b1 b2) ((eq B b1 b2) \to False) H0)) (\lambda (H0: 
+(((eq B b1 b2) \to (\forall (P: Prop).P)))).(or_intror (eq B b1 b2) ((eq B b1 
+b2) \to False) (\lambda (H1: (eq B b1 b2)).(H0 H1 False)))) H)))).
+
 theorem terms_props__flat_dec:
  \forall (f1: F).(\forall (f2: F).(or (eq F f1 f2) ((eq F f1 f2) \to (\forall 
 (P: Prop).P))))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/T/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/T/defs.ma
index 5db781439..cc5d91731 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/T/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/T/defs.ma
@@ -36,17 +36,6 @@ inductive T: Set \def
 | TLRef: nat \to T
 | THead: K \to (T \to (T \to T)).
 
-inductive TList: Set \def
-| TNil: TList
-| TCons: T \to (TList \to TList).
-
-definition THeads:
- K \to (TList \to (T \to T))
-\def
- let rec THeads (k: K) (us: TList) on us: (T \to T) \def (\lambda (t: 
-T).(match us with [TNil \Rightarrow t | (TCons u ul) \Rightarrow (THead k u 
-(THeads k ul t))])) in THeads.
-
 definition tweight:
  T \to nat
 \def
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/arity/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/arity/defs.ma
index 3ce54e042..287c4446c 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/arity/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/arity/defs.ma
@@ -20,7 +20,7 @@ include "leq/defs.ma".
 
 include "getl/defs.ma".
 
-inductive arity (g:G): C \to (T \to (A \to Prop)) \def
+inductive arity (g: G): C \to (T \to (A \to Prop)) \def
 | arity_sort: \forall (c: C).(\forall (n: nat).(arity g c (TSort n) (ASort O 
 n)))
 | arity_abbr: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ceqc/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ceqc/defs.ma
deleted file mode 100644
index 91ae5de91..000000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ceqc/defs.ma
+++ /dev/null
@@ -1,35 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ceqc/defs".
-
-include "sc3/defs.ma".
-
-inductive csubc (g:G): C \to (C \to Prop) \def
-| csubc_sort: \forall (n: nat).(csubc g (CSort n) (CSort n))
-| csubc_head: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall 
-(k: K).(\forall (v: T).(csubc g (CHead c1 k v) (CHead c2 k v))))))
-| csubc_abst: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall 
-(v: T).(\forall (a: A).((sc3 g (asucc g a) c1 v) \to (\forall (w: T).((sc3 g 
-a c2 w) \to (csubc g (CHead c1 (Bind Abst) v) (CHead c2 (Bind Abbr) 
-w))))))))).
-
-definition ceqc:
- G \to (C \to (C \to Prop))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(or (csubc g c1 c2) (csubc 
-g c2 c1)))).
-
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ceqc/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ceqc/props.ma
deleted file mode 100644
index 7ed7c50d2..000000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ceqc/props.ma
+++ /dev/null
@@ -1,563 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ceqc/props".
-
-include "ceqc/defs.ma".
-
-include "sc3/props.ma".
-
-theorem csubc_refl:
- \forall (g: G).(\forall (c: C).(csubc g c c))
-\def
- \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubc g c0 c0)) 
-(\lambda (n: nat).(csubc_sort g n)) (\lambda (c0: C).(\lambda (H: (csubc g c0 
-c0)).(\lambda (k: K).(\lambda (t: T).(csubc_head g c0 c0 H k t))))) c)).
-
-theorem ceqc_sym:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((ceqc g c1 c2) \to (ceqc g 
-c2 c1))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (ceqc g c1 
-c2)).(let H0 \def H in (or_ind (csubc g c1 c2) (csubc g c2 c1) (ceqc g c2 c1) 
-(\lambda (H1: (csubc g c1 c2)).(or_intror (csubc g c2 c1) (csubc g c1 c2) 
-H1)) (\lambda (H1: (csubc g c2 c1)).(or_introl (csubc g c2 c1) (csubc g c1 
-c2) H1)) H0))))).
-
-theorem drop_csubc_trans:
- \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall 
-(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C 
-(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))))))))
-\def
- \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: 
-C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: 
-C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda 
-(c1: C).(csubc g c c1)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda 
-(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda 
-(e1: C).(\lambda (H0: (csubc g e2 e1)).(and3_ind (eq C e2 (CSort n)) (eq nat 
-h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: 
-C).(csubc g (CSort n) c1))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: 
-(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: 
-nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g 
-(CSort n) c1)))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: 
-C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)))) (let H4 \def 
-(eq_ind C e2 (\lambda (c: C).(csubc g c e1)) H0 (CSort n) H1) in (ex_intro2 C 
-(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)) 
-e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) 
-(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: 
-nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C 
-(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c 
-c1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: 
-nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) 
-e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h 
-n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) (\lambda (h: 
-nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall 
-(e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) 
-(\lambda (c1: C).(csubc g (CHead c k t) c1))))))) (\lambda (H0: (drop O O 
-(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2 
-\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g c0 e1)) H1 (CHead c k t) 
-(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O 
-O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)) e1 (drop_refl e1) 
-H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to 
-(\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 
-e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))).(\lambda (H1: (drop 
-(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 
-e1)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in 
-(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) 
-(\lambda (c1: C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 
-e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))) (\lambda (x: C).(\lambda 
-(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g c x)).(ex_intro2 C 
-(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k 
-t) c1)) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g c x H5 k t))))) 
-H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n 
-(CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda 
-(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) 
-c1))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) 
-e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 e1)).(ex3_2_ind C T (\lambda 
-(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: 
-T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k 
-n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: 
-C).(csubc g (CHead c k t) c1))) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
-(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) 
-x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda 
-(c0: C).(csubc g c0 e1)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 
-(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to 
-(\forall (e3: C).((csubc g c0 e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 
-e3)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) H0 (CHead x0 k x1) 
-H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 
-n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g (CHead x0 k 
-x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: 
-C).(csubc g (CHead c k t0) c1)))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r 
-T (lift h (r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) 
-c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t0) c1)))) (let H9 \def (match 
-H6 in csubc return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csubc ? c0 
-c1)).((eq C c0 (CHead x0 k x1)) \to ((eq C c1 e1) \to (ex2 C (\lambda (c3: 
-C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) 
-x1)) c3)))))))) with [(csubc_sort n0) \Rightarrow (\lambda (H9: (eq C (CSort 
-n0) (CHead x0 k x1))).(\lambda (H10: (eq C (CSort n0) e1)).((let H11 \def 
-(eq_ind C (CSort n0) (\lambda (e: C).(match e in C return (\lambda (_: 
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
-False])) I (CHead x0 k x1) H9) in (False_ind ((eq C (CSort n0) e1) \to (ex2 C 
-(\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g (CHead c k 
-(lift h (r k n) x1)) c1)))) H11)) H10))) | (csubc_head c1 c0 H9 k0 v) 
-\Rightarrow (\lambda (H10: (eq C (CHead c1 k0 v) (CHead x0 k x1))).(\lambda 
-(H11: (eq C (CHead c0 k0 v) e1)).((let H12 \def (f_equal C T (\lambda (e: 
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | 
-(CHead _ _ t0) \Rightarrow t0])) (CHead c1 k0 v) (CHead x0 k x1) H10) in 
-((let H13 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: 
-C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) 
-(CHead c1 k0 v) (CHead x0 k x1) H10) in ((let H14 \def (f_equal C C (\lambda 
-(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 
-| (CHead c3 _ _) \Rightarrow c3])) (CHead c1 k0 v) (CHead x0 k x1) H10) in 
-(eq_ind C x0 (\lambda (c3: C).((eq K k0 k) \to ((eq T v x1) \to ((eq C (CHead 
-c0 k0 v) e1) \to ((csubc g c3 c0) \to (ex2 C (\lambda (c4: C).(drop h (S n) 
-c4 e1)) (\lambda (c4: C).(csubc g (CHead c k (lift h (r k n) x1)) c4)))))))) 
-(\lambda (H15: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T v x1) \to 
-((eq C (CHead c0 k1 v) e1) \to ((csubc g x0 c0) \to (ex2 C (\lambda (c3: 
-C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) 
-x1)) c3))))))) (\lambda (H16: (eq T v x1)).(eq_ind T x1 (\lambda (t0: T).((eq 
-C (CHead c0 k t0) e1) \to ((csubc g x0 c0) \to (ex2 C (\lambda (c3: C).(drop 
-h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) 
-c3)))))) (\lambda (H17: (eq C (CHead c0 k x1) e1)).(eq_ind C (CHead c0 k x1) 
-(\lambda (c3: C).((csubc g x0 c0) \to (ex2 C (\lambda (c4: C).(drop h (S n) 
-c4 c3)) (\lambda (c4: C).(csubc g (CHead c k (lift h (r k n) x1)) c4))))) 
-(\lambda (H18: (csubc g x0 c0)).(let H_x \def (H x0 (r k n) h H5 c0 H18) in 
-(let H19 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r k n) c3 c0)) 
-(\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 
-(CHead c0 k x1))) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) 
-c3))) (\lambda (x: C).(\lambda (H20: (drop h (r k n) x c0)).(\lambda (H21: 
-(csubc g c x)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c0 k 
-x1))) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3)) (CHead x 
-k (lift h (r k n) x1)) (drop_skip k h n x c0 H20 x1) (csubc_head g c x H21 k 
-(lift h (r k n) x1)))))) H19)))) e1 H17)) v (sym_eq T v x1 H16))) k0 (sym_eq 
-K k0 k H15))) c1 (sym_eq C c1 x0 H14))) H13)) H12)) H11 H9))) | (csubc_abst 
-c1 c0 H9 v a H10 w H11) \Rightarrow (\lambda (H12: (eq C (CHead c1 (Bind 
-Abst) v) (CHead x0 k x1))).(\lambda (H13: (eq C (CHead c0 (Bind Abbr) w) 
-e1)).((let H14 \def (f_equal C T (\lambda (e: C).(match e in C return 
-(\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t0) \Rightarrow 
-t0])) (CHead c1 (Bind Abst) v) (CHead x0 k x1) H12) in ((let H15 \def 
-(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with 
-[(CSort _) \Rightarrow (Bind Abst) | (CHead _ k0 _) \Rightarrow k0])) (CHead 
-c1 (Bind Abst) v) (CHead x0 k x1) H12) in ((let H16 \def (f_equal C C 
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow c1 | (CHead c3 _ _) \Rightarrow c3])) (CHead c1 (Bind Abst) v) 
-(CHead x0 k x1) H12) in (eq_ind C x0 (\lambda (c3: C).((eq K (Bind Abst) k) 
-\to ((eq T v x1) \to ((eq C (CHead c0 (Bind Abbr) w) e1) \to ((csubc g c3 c0) 
-\to ((sc3 g (asucc g a) c3 v) \to ((sc3 g a c0 w) \to (ex2 C (\lambda (c4: 
-C).(drop h (S n) c4 e1)) (\lambda (c4: C).(csubc g (CHead c k (lift h (r k n) 
-x1)) c4)))))))))) (\lambda (H17: (eq K (Bind Abst) k)).(eq_ind K (Bind Abst) 
-(\lambda (k0: K).((eq T v x1) \to ((eq C (CHead c0 (Bind Abbr) w) e1) \to 
-((csubc g x0 c0) \to ((sc3 g (asucc g a) x0 v) \to ((sc3 g a c0 w) \to (ex2 C 
-(\lambda (c3: C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k0 
-(lift h (r k0 n) x1)) c3))))))))) (\lambda (H18: (eq T v x1)).(eq_ind T x1 
-(\lambda (t0: T).((eq C (CHead c0 (Bind Abbr) w) e1) \to ((csubc g x0 c0) \to 
-((sc3 g (asucc g a) x0 t0) \to ((sc3 g a c0 w) \to (ex2 C (\lambda (c3: 
-C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c (Bind Abst) (lift 
-h (r (Bind Abst) n) x1)) c3)))))))) (\lambda (H19: (eq C (CHead c0 (Bind 
-Abbr) w) e1)).(eq_ind C (CHead c0 (Bind Abbr) w) (\lambda (c3: C).((csubc g 
-x0 c0) \to ((sc3 g (asucc g a) x0 x1) \to ((sc3 g a c0 w) \to (ex2 C (\lambda 
-(c4: C).(drop h (S n) c4 c3)) (\lambda (c4: C).(csubc g (CHead c (Bind Abst) 
-(lift h (r (Bind Abst) n) x1)) c4))))))) (\lambda (H20: (csubc g x0 
-c0)).(\lambda (H21: (sc3 g (asucc g a) x0 x1)).(\lambda (H22: (sc3 g a c0 
-w)).(let H23 \def (eq_ind_r K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 
-n (CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: 
-C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c3: C).(drop h0 n c3 
-e3)) (\lambda (c3: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c3)))))))) 
-H8 (Bind Abst) H17) in (let H24 \def (eq_ind_r K k (\lambda (k0: K).(drop h 
-(r k0 n) c x0)) H5 (Bind Abst) H17) in (let H_x \def (H x0 (r (Bind Abst) n) 
-h H24 c0 H20) in (let H25 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r 
-(Bind Abst) n) c3 c0)) (\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (c3: 
-C).(drop h (S n) c3 (CHead c0 (Bind Abbr) w))) (\lambda (c3: C).(csubc g 
-(CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c3))) (\lambda (x: 
-C).(\lambda (H26: (drop h (r (Bind Abst) n) x c0)).(\lambda (H27: (csubc g c 
-x)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c0 (Bind Abbr) w))) 
-(\lambda (c3: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) 
-c3)) (CHead x (Bind Abbr) (lift h n w)) (drop_skip_bind h n x c0 H26 Abbr w) 
-(csubc_abst g c x H27 (lift h (r (Bind Abst) n) x1) a (sc3_lift g (asucc g a) 
-x0 x1 H21 c h (r (Bind Abst) n) H24) (lift h n w) (sc3_lift g a c0 w H22 x h 
-n H26)))))) H25)))))))) e1 H19)) v (sym_eq T v x1 H18))) k H17)) c1 (sym_eq C 
-c1 x0 H16))) H15)) H14)) H13 H9 H10 H11)))]) in (H9 (refl_equal C (CHead x0 k 
-x1)) (refl_equal C e1))) t H4))))))))) (drop_gen_skip_l c e2 t h n k 
-H1)))))))) d))))))) c2)).
-
-theorem csubc_drop_conf_rev:
- \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall 
-(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C 
-(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))))))))
-\def
- \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: 
-C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: 
-C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda 
-(c1: C).(csubc g c1 c)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda 
-(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda 
-(e1: C).(\lambda (H0: (csubc g e1 e2)).(and3_ind (eq C e2 (CSort n)) (eq nat 
-h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: 
-C).(csubc g c1 (CSort n)))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: 
-(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: 
-nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g c1 
-(CSort n))))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: 
-C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))))) (let H4 \def 
-(eq_ind C e2 (\lambda (c: C).(csubc g e1 c)) H0 (CSort n) H1) in (ex_intro2 C 
-(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))) 
-e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) 
-(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: 
-nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C 
-(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 
-c))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: 
-nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) 
-e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h 
-n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) (\lambda (h: 
-nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall 
-(e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) 
-(\lambda (c1: C).(csubc g c1 (CHead c k t)))))))) (\lambda (H0: (drop O O 
-(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2 
-\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g e1 c0)) H1 (CHead c k t) 
-(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O 
-O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))) e1 (drop_refl e1) 
-H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to 
-(\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 
-e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))).(\lambda (H1: (drop 
-(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 
-e2)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in 
-(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) 
-(\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 
-e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))) (\lambda (x: C).(\lambda 
-(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g x c)).(ex_intro2 C 
-(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c 
-k t))) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g x c H5 k t))))) 
-H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n 
-(CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda 
-(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k 
-t)))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) 
-e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 e2)).(ex3_2_ind C T (\lambda 
-(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: 
-T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k 
-n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: 
-C).(csubc g c1 (CHead c k t)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
-(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) 
-x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda 
-(c0: C).(csubc g e1 c0)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 
-(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to 
-(\forall (e3: C).((csubc g e3 c0) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 
-e3)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) H0 (CHead x0 k x1) 
-H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 
-n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g e3 (CHead x0 
-k x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc 
-g c1 (CHead c k t0))))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h 
-(r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) 
-(\lambda (c1: C).(csubc g c1 (CHead c k t0))))) (let H9 \def (match H6 in 
-csubc return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csubc ? c0 
-c1)).((eq C c0 e1) \to ((eq C c1 (CHead x0 k x1)) \to (ex2 C (\lambda (c3: 
-C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k 
-n) x1)))))))))) with [(csubc_sort n0) \Rightarrow (\lambda (H9: (eq C (CSort 
-n0) e1)).(\lambda (H10: (eq C (CSort n0) (CHead x0 k x1))).(eq_ind C (CSort 
-n0) (\lambda (c0: C).((eq C (CSort n0) (CHead x0 k x1)) \to (ex2 C (\lambda 
-(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h 
-(r k n) x1))))))) (\lambda (H11: (eq C (CSort n0) (CHead x0 k x1))).(let H12 
-\def (eq_ind C (CSort n0) (\lambda (e: C).(match e in C return (\lambda (_: 
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
-False])) I (CHead x0 k x1) H11) in (False_ind (ex2 C (\lambda (c1: C).(drop h 
-(S n) c1 (CSort n0))) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) 
-x1))))) H12))) e1 H9 H10))) | (csubc_head c1 c0 H9 k0 v) \Rightarrow (\lambda 
-(H10: (eq C (CHead c1 k0 v) e1)).(\lambda (H11: (eq C (CHead c0 k0 v) (CHead 
-x0 k x1))).(eq_ind C (CHead c1 k0 v) (\lambda (c3: C).((eq C (CHead c0 k0 v) 
-(CHead x0 k x1)) \to ((csubc g c1 c0) \to (ex2 C (\lambda (c4: C).(drop h (S 
-n) c4 c3)) (\lambda (c4: C).(csubc g c4 (CHead c k (lift h (r k n) x1)))))))) 
-(\lambda (H12: (eq C (CHead c0 k0 v) (CHead x0 k x1))).(let H13 \def (f_equal 
-C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
-\Rightarrow v | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k0 v) (CHead x0 k 
-x1) H12) in ((let H14 \def (f_equal C K (\lambda (e: C).(match e in C return 
-(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) 
-\Rightarrow k1])) (CHead c0 k0 v) (CHead x0 k x1) H12) in ((let H15 \def 
-(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow c0 | (CHead c3 _ _) \Rightarrow c3])) (CHead c0 k0 v) 
-(CHead x0 k x1) H12) in (eq_ind C x0 (\lambda (c3: C).((eq K k0 k) \to ((eq T 
-v x1) \to ((csubc g c1 c3) \to (ex2 C (\lambda (c4: C).(drop h (S n) c4 
-(CHead c1 k0 v))) (\lambda (c4: C).(csubc g c4 (CHead c k (lift h (r k n) 
-x1))))))))) (\lambda (H16: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T v 
-x1) \to ((csubc g c1 x0) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead 
-c1 k1 v))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))))))) 
-(\lambda (H17: (eq T v x1)).(eq_ind T x1 (\lambda (t0: T).((csubc g c1 x0) 
-\to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k t0))) (\lambda (c3: 
-C).(csubc g c3 (CHead c k (lift h (r k n) x1))))))) (\lambda (H18: (csubc g 
-c1 x0)).(let H19 \def (eq_ind T v (\lambda (t0: T).(eq C (CHead c1 k0 t0) 
-e1)) H10 x1 H17) in (let H20 \def (eq_ind K k0 (\lambda (k1: K).(eq C (CHead 
-c1 k1 x1) e1)) H19 k H16) in (let H_x \def (H x0 (r k n) h H5 c1 H18) in (let 
-H21 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r k n) c3 c1)) (\lambda 
-(c3: C).(csubc g c3 c)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k 
-x1))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1))))) 
-(\lambda (x: C).(\lambda (H22: (drop h (r k n) x c1)).(\lambda (H23: (csubc g 
-x c)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k x1))) 
-(\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))) (CHead x k 
-(lift h (r k n) x1)) (drop_skip k h n x c1 H22 x1) (csubc_head g x c H23 k 
-(lift h (r k n) x1)))))) H21)))))) v (sym_eq T v x1 H17))) k0 (sym_eq K k0 k 
-H16))) c0 (sym_eq C c0 x0 H15))) H14)) H13))) e1 H10 H11 H9))) | (csubc_abst 
-c1 c0 H9 v a H10 w H11) \Rightarrow (\lambda (H12: (eq C (CHead c1 (Bind 
-Abst) v) e1)).(\lambda (H13: (eq C (CHead c0 (Bind Abbr) w) (CHead x0 k 
-x1))).(eq_ind C (CHead c1 (Bind Abst) v) (\lambda (c3: C).((eq C (CHead c0 
-(Bind Abbr) w) (CHead x0 k x1)) \to ((csubc g c1 c0) \to ((sc3 g (asucc g a) 
-c1 v) \to ((sc3 g a c0 w) \to (ex2 C (\lambda (c4: C).(drop h (S n) c4 c3)) 
-(\lambda (c4: C).(csubc g c4 (CHead c k (lift h (r k n) x1)))))))))) (\lambda 
-(H14: (eq C (CHead c0 (Bind Abbr) w) (CHead x0 k x1))).(let H15 \def (f_equal 
-C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
-\Rightarrow w | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 (Bind Abbr) w) 
-(CHead x0 k x1) H14) in ((let H16 \def (f_equal C K (\lambda (e: C).(match e 
-in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow (Bind Abbr) | 
-(CHead _ k0 _) \Rightarrow k0])) (CHead c0 (Bind Abbr) w) (CHead x0 k x1) 
-H14) in ((let H17 \def (f_equal C C (\lambda (e: C).(match e in C return 
-(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c3 _ _) 
-\Rightarrow c3])) (CHead c0 (Bind Abbr) w) (CHead x0 k x1) H14) in (eq_ind C 
-x0 (\lambda (c3: C).((eq K (Bind Abbr) k) \to ((eq T w x1) \to ((csubc g c1 
-c3) \to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a c3 w) \to (ex2 C (\lambda 
-(c4: C).(drop h (S n) c4 (CHead c1 (Bind Abst) v))) (\lambda (c4: C).(csubc g 
-c4 (CHead c k (lift h (r k n) x1))))))))))) (\lambda (H18: (eq K (Bind Abbr) 
-k)).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T w x1) \to ((csubc g c1 x0) 
-\to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a x0 w) \to (ex2 C (\lambda (c3: 
-C).(drop h (S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: C).(csubc g c3 
-(CHead c k0 (lift h (r k0 n) x1)))))))))) (\lambda (H19: (eq T w x1)).(eq_ind 
-T x1 (\lambda (t0: T).((csubc g c1 x0) \to ((sc3 g (asucc g a) c1 v) \to 
-((sc3 g a x0 t0) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 (Bind 
-Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c (Bind Abbr) (lift h (r (Bind 
-Abbr) n) x1))))))))) (\lambda (H20: (csubc g c1 x0)).(\lambda (H21: (sc3 g 
-(asucc g a) c1 v)).(\lambda (H22: (sc3 g a x0 x1)).(let H23 \def (eq_ind_r K 
-k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c k0 (lift h (r k0 
-n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3 (CHead x0 k0 x1)) 
-\to (ex2 C (\lambda (c3: C).(drop h0 n c3 e3)) (\lambda (c3: C).(csubc g c3 
-(CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind Abbr) H18) in (let H24 
-\def (eq_ind_r K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5 (Bind Abbr) 
-H18) in (let H_x \def (H x0 (r (Bind Abbr) n) h H24 c1 H20) in (let H25 \def 
-H_x in (ex2_ind C (\lambda (c3: C).(drop h (r (Bind Abbr) n) c3 c1)) (\lambda 
-(c3: C).(csubc g c3 c)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 
-(Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c (Bind Abbr) (lift h (r 
-(Bind Abbr) n) x1))))) (\lambda (x: C).(\lambda (H26: (drop h (r (Bind Abbr) 
-n) x c1)).(\lambda (H27: (csubc g x c)).(ex_intro2 C (\lambda (c3: C).(drop h 
-(S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c 
-(Bind Abbr) (lift h (r (Bind Abbr) n) x1)))) (CHead x (Bind Abst) (lift h n 
-v)) (drop_skip_bind h n x c1 H26 Abst v) (csubc_abst g x c H27 (lift h n v) a 
-(sc3_lift g (asucc g a) c1 v H21 x h n H26) (lift h (r (Bind Abbr) n) x1) 
-(sc3_lift g a x0 x1 H22 c h (r (Bind Abbr) n) H24)))))) H25)))))))) w (sym_eq 
-T w x1 H19))) k H18)) c0 (sym_eq C c0 x0 H17))) H16)) H15))) e1 H12 H13 H9 
-H10 H11)))]) in (H9 (refl_equal C e1) (refl_equal C (CHead x0 k x1)))) t 
-H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
-
-theorem drop1_csubc_trans:
- \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: 
-C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C 
-(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))
-\def
- \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
-(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 
-e1) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 
-c1))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 
-e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e2 e1)).(let H1 \def (match H in 
-drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda 
-(_: (drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 e2) \to 
-(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c2 
-c1)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil 
-PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c e2)).(eq_ind C c2 
-(\lambda (c0: C).((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1 
-e1)) (\lambda (c1: C).(csubc g c2 c1))))) (\lambda (H4: (eq C c2 e2)).(eq_ind 
-C e2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda 
-(c1: C).(csubc g c0 c1)))) (let H5 \def (eq_ind_r C e2 (\lambda (c0: 
-C).(csubc g c0 e1)) H0 c2 H4) in (eq_ind C c2 (\lambda (c0: C).(ex2 C 
-(\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c0 c1)))) 
-(ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g 
-c2 c1)) e1 (drop1_nil e1) H5) e2 H4)) c2 (sym_eq C c2 e2 H4))) c (sym_eq C c 
-c2 H2) H3)))) | (drop1_cons c1 c0 h d H1 c3 hds0 H2) \Rightarrow (\lambda 
-(H3: (eq PList (PCons h d hds0) PNil)).(\lambda (H4: (eq C c1 c2)).(\lambda 
-(H5: (eq C c3 e2)).((let H6 \def (eq_ind PList (PCons h d hds0) (\lambda (e: 
-PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil 
-\Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in 
-(False_ind ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop h d c1 c0) \to ((drop1 
-hds0 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 PNil c4 e1)) (\lambda (c4: 
-C).(csubc g c2 c4))))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList 
-PNil) (refl_equal C c2) (refl_equal C e2)))))))) (\lambda (n: nat).(\lambda 
-(n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: C).(\forall (e2: 
-C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda 
-(c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))).(\lambda 
-(c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n n0 p) c2 e2)).(\lambda 
-(e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2 \def (match H0 in drop1 return 
-(\lambda (p0: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p0 
-c c0)).((eq PList p0 (PCons n n0 p)) \to ((eq C c c2) \to ((eq C c0 e2) \to 
-(ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc 
-g c2 c1)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList 
-PNil (PCons n n0 p))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c 
-e2)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e in PList 
-return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) 
-\Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c2) \to ((eq 
-C c e2) \to (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda 
-(c1: C).(csubc g c2 c1))))) H5)) H3 H4)))) | (drop1_cons c1 c0 h d H2 c3 hds0 
-H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds0) (PCons n n0 
-p))).(\lambda (H5: (eq C c1 c2)).(\lambda (H6: (eq C c3 e2)).((let H7 \def 
-(f_equal PList PList (\lambda (e: PList).(match e in PList return (\lambda 
-(_: PList).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p0) \Rightarrow 
-p0])) (PCons h d hds0) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat 
-(\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with 
-[PNil \Rightarrow d | (PCons _ n1 _) \Rightarrow n1])) (PCons h d hds0) 
-(PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e: 
-PList).(match e in PList return (\lambda (_: PList).nat) with [PNil 
-\Rightarrow h | (PCons n1 _ _) \Rightarrow n1])) (PCons h d hds0) (PCons n n0 
-p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds0 
-p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n1 d c1 c0) \to ((drop1 
-hds0 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) 
-(\lambda (c4: C).(csubc g c2 c4)))))))))) (\lambda (H10: (eq nat d 
-n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds0 p) \to ((eq C c1 c2) 
-\to ((eq C c3 e2) \to ((drop n n1 c1 c0) \to ((drop1 hds0 c0 c3) \to (ex2 C 
-(\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c2 
-c4))))))))) (\lambda (H11: (eq PList hds0 p)).(eq_ind PList p (\lambda (p0: 
-PList).((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n n0 c1 c0) \to ((drop1 p0 
-c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda 
-(c4: C).(csubc g c2 c4)))))))) (\lambda (H12: (eq C c1 c2)).(eq_ind C c2 
-(\lambda (c: C).((eq C c3 e2) \to ((drop n n0 c c0) \to ((drop1 p c0 c3) \to 
-(ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc 
-g c2 c4))))))) (\lambda (H13: (eq C c3 e2)).(eq_ind C e2 (\lambda (c: 
-C).((drop n n0 c2 c0) \to ((drop1 p c0 c) \to (ex2 C (\lambda (c4: C).(drop1 
-(PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c2 c4)))))) (\lambda (H14: 
-(drop n n0 c2 c0)).(\lambda (H15: (drop1 p c0 e2)).(let H_x \def (H c0 e2 H15 
-e1 H1) in (let H16 \def H_x in (ex2_ind C (\lambda (c4: C).(drop1 p c4 e1)) 
-(\lambda (c4: C).(csubc g c0 c4)) (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 
-p) c4 e1)) (\lambda (c4: C).(csubc g c2 c4))) (\lambda (x: C).(\lambda (H17: 
-(drop1 p x e1)).(\lambda (H18: (csubc g c0 x)).(let H_x0 \def 
-(drop_csubc_trans g c2 c0 n0 n H14 x H18) in (let H19 \def H_x0 in (ex2_ind C 
-(\lambda (c4: C).(drop n n0 c4 x)) (\lambda (c4: C).(csubc g c2 c4)) (ex2 C 
-(\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c2 
-c4))) (\lambda (x0: C).(\lambda (H20: (drop n n0 x0 x)).(\lambda (H21: (csubc 
-g c2 x0)).(ex_intro2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) 
-(\lambda (c4: C).(csubc g c2 c4)) x0 (drop1_cons x0 x n n0 H20 e1 p H17) 
-H21)))) H19)))))) H16))))) c3 (sym_eq C c3 e2 H13))) c1 (sym_eq C c1 c2 
-H12))) hds0 (sym_eq PList hds0 p H11))) d (sym_eq nat d n0 H10))) h (sym_eq 
-nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n 
-n0 p)) (refl_equal C c2) (refl_equal C e2)))))))))))) hds)).
-
-theorem csubc_drop1_conf_rev:
- \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: 
-C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C 
-(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))
-\def
- \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
-(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 
-e2) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 
-c2))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 
-e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e1 e2)).(let H1 \def (match H in 
-drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda 
-(_: (drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 e2) \to 
-(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 
-c2)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil 
-PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c e2)).(eq_ind C c2 
-(\lambda (c0: C).((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1 
-e1)) (\lambda (c1: C).(csubc g c1 c2))))) (\lambda (H4: (eq C c2 e2)).(eq_ind 
-C e2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda 
-(c1: C).(csubc g c1 c0)))) (let H5 \def (eq_ind_r C e2 (\lambda (c0: 
-C).(csubc g e1 c0)) H0 c2 H4) in (eq_ind C c2 (\lambda (c0: C).(ex2 C 
-(\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 c0)))) 
-(ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g 
-c1 c2)) e1 (drop1_nil e1) H5) e2 H4)) c2 (sym_eq C c2 e2 H4))) c (sym_eq C c 
-c2 H2) H3)))) | (drop1_cons c1 c0 h d H1 c3 hds0 H2) \Rightarrow (\lambda 
-(H3: (eq PList (PCons h d hds0) PNil)).(\lambda (H4: (eq C c1 c2)).(\lambda 
-(H5: (eq C c3 e2)).((let H6 \def (eq_ind PList (PCons h d hds0) (\lambda (e: 
-PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil 
-\Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in 
-(False_ind ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop h d c1 c0) \to ((drop1 
-hds0 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 PNil c4 e1)) (\lambda (c4: 
-C).(csubc g c4 c2))))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList 
-PNil) (refl_equal C c2) (refl_equal C e2)))))))) (\lambda (n: nat).(\lambda 
-(n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: C).(\forall (e2: 
-C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda 
-(c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))).(\lambda 
-(c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n n0 p) c2 e2)).(\lambda 
-(e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2 \def (match H0 in drop1 return 
-(\lambda (p0: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p0 
-c c0)).((eq PList p0 (PCons n n0 p)) \to ((eq C c c2) \to ((eq C c0 e2) \to 
-(ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc 
-g c1 c2)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList 
-PNil (PCons n n0 p))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c 
-e2)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e in PList 
-return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) 
-\Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c2) \to ((eq 
-C c e2) \to (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda 
-(c1: C).(csubc g c1 c2))))) H5)) H3 H4)))) | (drop1_cons c1 c0 h d H2 c3 hds0 
-H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds0) (PCons n n0 
-p))).(\lambda (H5: (eq C c1 c2)).(\lambda (H6: (eq C c3 e2)).((let H7 \def 
-(f_equal PList PList (\lambda (e: PList).(match e in PList return (\lambda 
-(_: PList).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p0) \Rightarrow 
-p0])) (PCons h d hds0) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat 
-(\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with 
-[PNil \Rightarrow d | (PCons _ n1 _) \Rightarrow n1])) (PCons h d hds0) 
-(PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e: 
-PList).(match e in PList return (\lambda (_: PList).nat) with [PNil 
-\Rightarrow h | (PCons n1 _ _) \Rightarrow n1])) (PCons h d hds0) (PCons n n0 
-p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds0 
-p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n1 d c1 c0) \to ((drop1 
-hds0 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) 
-(\lambda (c4: C).(csubc g c4 c2)))))))))) (\lambda (H10: (eq nat d 
-n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds0 p) \to ((eq C c1 c2) 
-\to ((eq C c3 e2) \to ((drop n n1 c1 c0) \to ((drop1 hds0 c0 c3) \to (ex2 C 
-(\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c4 
-c2))))))))) (\lambda (H11: (eq PList hds0 p)).(eq_ind PList p (\lambda (p0: 
-PList).((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n n0 c1 c0) \to ((drop1 p0 
-c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda 
-(c4: C).(csubc g c4 c2)))))))) (\lambda (H12: (eq C c1 c2)).(eq_ind C c2 
-(\lambda (c: C).((eq C c3 e2) \to ((drop n n0 c c0) \to ((drop1 p c0 c3) \to 
-(ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc 
-g c4 c2))))))) (\lambda (H13: (eq C c3 e2)).(eq_ind C e2 (\lambda (c: 
-C).((drop n n0 c2 c0) \to ((drop1 p c0 c) \to (ex2 C (\lambda (c4: C).(drop1 
-(PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c4 c2)))))) (\lambda (H14: 
-(drop n n0 c2 c0)).(\lambda (H15: (drop1 p c0 e2)).(let H_x \def (H c0 e2 H15 
-e1 H1) in (let H16 \def H_x in (ex2_ind C (\lambda (c4: C).(drop1 p c4 e1)) 
-(\lambda (c4: C).(csubc g c4 c0)) (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 
-p) c4 e1)) (\lambda (c4: C).(csubc g c4 c2))) (\lambda (x: C).(\lambda (H17: 
-(drop1 p x e1)).(\lambda (H18: (csubc g x c0)).(let H_x0 \def 
-(csubc_drop_conf_rev g c2 c0 n0 n H14 x H18) in (let H19 \def H_x0 in 
-(ex2_ind C (\lambda (c4: C).(drop n n0 c4 x)) (\lambda (c4: C).(csubc g c4 
-c2)) (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: 
-C).(csubc g c4 c2))) (\lambda (x0: C).(\lambda (H20: (drop n n0 x0 
-x)).(\lambda (H21: (csubc g x0 c2)).(ex_intro2 C (\lambda (c4: C).(drop1 
-(PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c4 c2)) x0 (drop1_cons x0 x 
-n n0 H20 e1 p H17) H21)))) H19)))))) H16))))) c3 (sym_eq C c3 e2 H13))) c1 
-(sym_eq C c1 c2 H12))) hds0 (sym_eq PList hds0 p H11))) d (sym_eq nat d n0 
-H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal 
-PList (PCons n n0 p)) (refl_equal C c2) (refl_equal C e2)))))))))))) hds)).
-
-theorem drop1_ceqc_trans:
- \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: 
-C).((drop1 hds c2 e2) \to (\forall (e1: C).((ceqc g e2 e1) \to (ex2 C 
-(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(ceqc g c2 c1)))))))))
-\def
- \lambda (g: G).(\lambda (hds: PList).(\lambda (c2: C).(\lambda (e2: 
-C).(\lambda (H: (drop1 hds c2 e2)).(\lambda (e1: C).(\lambda (H0: (ceqc g e2 
-e1)).(let H1 \def H0 in (or_ind (csubc g e2 e1) (csubc g e1 e2) (ex2 C 
-(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(ceqc g c2 c1))) 
-(\lambda (H2: (csubc g e2 e1)).(let H_x \def (drop1_csubc_trans g hds c2 e2 H 
-e1 H2) in (let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop1 hds c1 e1)) 
-(\lambda (c1: C).(csubc g c2 c1)) (ex2 C (\lambda (c1: C).(drop1 hds c1 e1)) 
-(\lambda (c1: C).(ceqc g c2 c1))) (\lambda (x: C).(\lambda (H4: (drop1 hds x 
-e1)).(\lambda (H5: (csubc g c2 x)).(ex_intro2 C (\lambda (c1: C).(drop1 hds 
-c1 e1)) (\lambda (c1: C).(ceqc g c2 c1)) x H4 (or_introl (csubc g c2 x) 
-(csubc g x c2) H5))))) H3)))) (\lambda (H2: (csubc g e1 e2)).(let H_x \def 
-(csubc_drop1_conf_rev g hds c2 e2 H e1 H2) in (let H3 \def H_x in (ex2_ind C 
-(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c1 c2)) (ex2 C 
-(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(ceqc g c2 c1))) 
-(\lambda (x: C).(\lambda (H4: (drop1 hds x e1)).(\lambda (H5: (csubc g x 
-c2)).(ex_intro2 C (\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(ceqc 
-g c2 c1)) x H4 (or_intror (csubc g c2 x) (csubc g x c2) H5))))) H3)))) 
-H1)))))))).
-
-axiom sc3_ceqc_trans:
- \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c1: 
-C).(\forall (t: T).((sc3 g a c1 (THeads (Flat Appl) vs t)) \to (\forall (c2: 
-C).((ceqc g c2 c1) \to (sc3 g a c2 (THeads (Flat Appl) vs t)))))))))
-.
-
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/clear.ma
deleted file mode 100644
index 86b43ab7b..000000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/clear.ma
+++ /dev/null
@@ -1,72 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/clear".
-
-include "csub3/defs.ma".
-
-include "clear/fwd.ma".
-
-theorem csub3_clear_conf:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to 
-(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) 
-(\lambda (e2: C).(clear c2 e2))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 g c1 
-c2)).(csub3_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c 
-e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c0 
-e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) 
-e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) 
-(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: 
-C).(\lambda (H0: (csub3 g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 
-e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c4 
-e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: 
-(clear (CHead c3 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c3 k0 u) 
-e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear 
-(CHead c4 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c3 (Bind 
-b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda 
-(e2: C).(csub3 g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)))) 
-(ex_intro2 C (\lambda (e2: C).(csub3 g (CHead c3 (Bind b) u) e2)) (\lambda 
-(e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csub3_head g 
-c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3)))) 
-(\lambda (f: F).(\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def 
-(H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csub3 g 
-e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csub3 g e1 
-e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: 
-C).(\lambda (H5: (csub3 g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C 
-(\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) 
-u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: 
-C).(\lambda (c4: C).(\lambda (H0: (csub3 g c3 c4)).(\lambda (_: ((\forall 
-(e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda 
-(e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b 
-Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: 
-(clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1) 
-(\lambda (c: C).(ex2 C (\lambda (e2: C).(csub3 g c e2)) (\lambda (e2: 
-C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csub3 
-g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) 
-u2) e2)) (CHead c4 (Bind b) u2) (csub3_void g c3 c4 H0 b H2 u1 u2) 
-(clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3)))))))))))) 
-(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csub3 g c3 c4)).(\lambda (_: 
-((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) 
-(\lambda (e2: C).(clear c4 e2))))))).(\lambda (u: T).(\lambda (t: T).(\lambda 
-(H2: (ty3 g c4 u t)).(\lambda (e1: C).(\lambda (H3: (clear (CHead c3 (Bind 
-Abst) t) e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda (c: C).(ex2 C 
-(\lambda (e2: C).(csub3 g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind 
-Abbr) u) e2)))) (ex_intro2 C (\lambda (e2: C).(csub3 g (CHead c3 (Bind Abst) 
-t) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)) (CHead c4 (Bind 
-Abbr) u) (csub3_abst g c3 c4 H0 u t H2) (clear_bind Abbr c4 u)) e1 
-(clear_gen_bind Abst c3 e1 t H3))))))))))) c1 c2 H)))).
-
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/defs.ma
deleted file mode 100644
index df8b56ada..000000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/defs.ma
+++ /dev/null
@@ -1,31 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/defs".
-
-include "ty3/defs.ma".
-
-inductive csub3 (g:G): C \to (C \to Prop) \def
-| csub3_sort: \forall (n: nat).(csub3 g (CSort n) (CSort n))
-| csub3_head: \forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall 
-(k: K).(\forall (u: T).(csub3 g (CHead c1 k u) (CHead c2 k u))))))
-| csub3_void: \forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall 
-(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csub3 g 
-(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2))))))))
-| csub3_abst: \forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall 
-(u: T).(\forall (t: T).((ty3 g c2 u t) \to (csub3 g (CHead c1 (Bind Abst) t) 
-(CHead c2 (Bind Abbr) u))))))).
-
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/drop.ma
deleted file mode 100644
index 59141ad9c..000000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/drop.ma
+++ /dev/null
@@ -1,805 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/drop".
-
-include "csub3/defs.ma".
-
-include "drop/fwd.ma".
-
-theorem csub3_drop_flat:
- \forall (g: G).(\forall (f: F).(\forall (n: nat).(\forall (c1: C).(\forall 
-(c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 
-(CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop n O c2 (CHead d2 (Flat f) u))))))))))))
-\def
- \lambda (g: G).(\lambda (f: F).(\lambda (n: nat).(nat_ind (\lambda (n0: 
-nat).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: 
-C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda 
-(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f) 
-u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 g c1 
-c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 
-(Flat f) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csub3 g c c2)) H 
-(CHead d1 (Flat f) u) (drop_gen_refl c1 (CHead d1 (Flat f) u) H0)) in (let H2 
-\def (match H1 in csub3 return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: 
-(csub3 ? c c0)).((eq C c (CHead d1 (Flat f) u)) \to ((eq C c0 c2) \to (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 
-(Flat f) u))))))))) with [(csub3_sort n0) \Rightarrow (\lambda (H2: (eq C 
-(CSort n0) (CHead d1 (Flat f) u))).(\lambda (H3: (eq C (CSort n0) c2)).((let 
-H4 \def (eq_ind C (CSort n0) (\lambda (e: C).(match e in C return (\lambda 
-(_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
-False])) I (CHead d1 (Flat f) u) H2) in (False_ind ((eq C (CSort n0) c2) \to 
-(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead 
-d2 (Flat f) u))))) H4)) H3))) | (csub3_head c0 c3 H2 k u0) \Rightarrow 
-(\lambda (H3: (eq C (CHead c0 k u0) (CHead d1 (Flat f) u))).(\lambda (H4: (eq 
-C (CHead c3 k u0) c2)).((let H5 \def (f_equal C T (\lambda (e: C).(match e in 
-C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) 
-\Rightarrow t])) (CHead c0 k u0) (CHead d1 (Flat f) u) H3) in ((let H6 \def 
-(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with 
-[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u0) 
-(CHead d1 (Flat f) u) H3) in ((let H7 \def (f_equal C C (\lambda (e: 
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | 
-(CHead c _ _) \Rightarrow c])) (CHead c0 k u0) (CHead d1 (Flat f) u) H3) in 
-(eq_ind C d1 (\lambda (c: C).((eq K k (Flat f)) \to ((eq T u0 u) \to ((eq C 
-(CHead c3 k u0) c2) \to ((csub3 g c c3) \to (ex2 C (\lambda (d2: C).(csub3 g 
-d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))))) (\lambda 
-(H8: (eq K k (Flat f))).(eq_ind K (Flat f) (\lambda (k0: K).((eq T u0 u) \to 
-((eq C (CHead c3 k0 u0) c2) \to ((csub3 g d1 c3) \to (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) 
-u)))))))) (\lambda (H9: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C 
-(CHead c3 (Flat f) t) c2) \to ((csub3 g d1 c3) \to (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))) 
-(\lambda (H10: (eq C (CHead c3 (Flat f) u) c2)).(eq_ind C (CHead c3 (Flat f) 
-u) (\lambda (c: C).((csub3 g d1 c3) \to (ex2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(drop O O c (CHead d2 (Flat f) u)))))) (\lambda (H11: 
-(csub3 g d1 c3)).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop O O (CHead c3 (Flat f) u) (CHead d2 (Flat f) u))) c3 H11 (drop_refl 
-(CHead c3 (Flat f) u)))) c2 H10)) u0 (sym_eq T u0 u H9))) k (sym_eq K k (Flat 
-f) H8))) c0 (sym_eq C c0 d1 H7))) H6)) H5)) H4 H2))) | (csub3_void c0 c3 H2 b 
-H3 u1 u2) \Rightarrow (\lambda (H4: (eq C (CHead c0 (Bind Void) u1) (CHead d1 
-(Flat f) u))).(\lambda (H5: (eq C (CHead c3 (Bind b) u2) c2)).((let H6 \def 
-(eq_ind C (CHead c0 (Bind Void) u1) (\lambda (e: C).(match e in C return 
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 (Flat f) u) 
-H4) in (False_ind ((eq C (CHead c3 (Bind b) u2) c2) \to ((csub3 g c0 c3) \to 
-((not (eq B b Void)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))) H6)) H5 H2 H3))) | 
-(csub3_abst c0 c3 H2 u0 t H3) \Rightarrow (\lambda (H4: (eq C (CHead c0 (Bind 
-Abst) t) (CHead d1 (Flat f) u))).(\lambda (H5: (eq C (CHead c3 (Bind Abbr) 
-u0) c2)).((let H6 \def (eq_ind C (CHead c0 (Bind Abst) t) (\lambda (e: 
-C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow 
-False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) 
-with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 
-(Flat f) u) H4) in (False_ind ((eq C (CHead c3 (Bind Abbr) u0) c2) \to 
-((csub3 g c0 c3) \to ((ty3 g c3 u0 t) \to (ex2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))) H6)) H5 H2 
-H3)))]) in (H2 (refl_equal C (CHead d1 (Flat f) u)) (refl_equal C 
-c2)))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: 
-C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 
-(CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop n0 O c2 (CHead d2 (Flat f) u)))))))))))).(\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (H0: (csub3 g c1 c2)).(csub3_ind g (\lambda (c: 
-C).(\lambda (c0: C).(\forall (d1: C).(\forall (u: T).((drop (S n0) O c (CHead 
-d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop (S n0) O c0 (CHead d2 (Flat f) u))))))))) (\lambda (n1: 
-nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n0) O (CSort n1) 
-(CHead d1 (Flat f) u))).(let H2 \def (match H1 in drop return (\lambda (n2: 
-nat).(\lambda (n3: nat).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop 
-n2 n3 c c0)).((eq nat n2 (S n0)) \to ((eq nat n3 O) \to ((eq C c (CSort n1)) 
-\to ((eq C c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) 
-u))))))))))))) with [(drop_refl c) \Rightarrow (\lambda (H2: (eq nat O (S 
-n0))).(\lambda (H3: (eq nat O O)).(\lambda (H4: (eq C c (CSort n1))).(\lambda 
-(H5: (eq C c (CHead d1 (Flat f) u))).((let H6 \def (eq_ind nat O (\lambda (e: 
-nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True 
-| (S _) \Rightarrow False])) I (S n0) H2) in (False_ind ((eq nat O O) \to 
-((eq C c (CSort n1)) \to ((eq C c (CHead d1 (Flat f) u)) \to (ex2 C (\lambda 
-(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 
-(Flat f) u))))))) H6)) H3 H4 H5))))) | (drop_drop k h c e H2 u0) \Rightarrow 
-(\lambda (H3: (eq nat (S h) (S n0))).(\lambda (H4: (eq nat O O)).(\lambda 
-(H5: (eq C (CHead c k u0) (CSort n1))).(\lambda (H6: (eq C e (CHead d1 (Flat 
-f) u))).((let H7 \def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat 
-return (\lambda (_: nat).nat) with [O \Rightarrow h | (S n2) \Rightarrow 
-n2])) (S h) (S n0) H3) in (eq_ind nat n0 (\lambda (n2: nat).((eq nat O O) \to 
-((eq C (CHead c k u0) (CSort n1)) \to ((eq C e (CHead d1 (Flat f) u)) \to 
-((drop (r k n2) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u))))))))) (\lambda (_: 
-(eq nat O O)).(\lambda (H9: (eq C (CHead c k u0) (CSort n1))).(let H10 \def 
-(eq_ind C (CHead c k u0) (\lambda (e0: C).(match e0 in C return (\lambda (_: 
-C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow 
-True])) I (CSort n1) H9) in (False_ind ((eq C e (CHead d1 (Flat f) u)) \to 
-((drop (r k n0) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u)))))) H10)))) h 
-(sym_eq nat h n0 H7))) H4 H5 H6 H2))))) | (drop_skip k h d c e H2 u0) 
-\Rightarrow (\lambda (H3: (eq nat h (S n0))).(\lambda (H4: (eq nat (S d) 
-O)).(\lambda (H5: (eq C (CHead c k (lift h (r k d) u0)) (CSort n1))).(\lambda 
-(H6: (eq C (CHead e k u0) (CHead d1 (Flat f) u))).(eq_ind nat (S n0) (\lambda 
-(n2: nat).((eq nat (S d) O) \to ((eq C (CHead c k (lift n2 (r k d) u0)) 
-(CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 (Flat f) u)) \to ((drop n2 (r 
-k d) c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
-(S n0) O (CSort n1) (CHead d2 (Flat f) u))))))))) (\lambda (H7: (eq nat (S d) 
-O)).(let H8 \def (eq_ind nat (S d) (\lambda (e0: nat).(match e0 in nat return 
-(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) 
-I O H7) in (False_ind ((eq C (CHead c k (lift (S n0) (r k d) u0)) (CSort n1)) 
-\to ((eq C (CHead e k u0) (CHead d1 (Flat f) u)) \to ((drop (S n0) (r k d) c 
-e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) 
-O (CSort n1) (CHead d2 (Flat f) u))))))) H8))) h (sym_eq nat h (S n0) H3) H4 
-H5 H6 H2)))))]) in (H2 (refl_equal nat (S n0)) (refl_equal nat O) (refl_equal 
-C (CSort n1)) (refl_equal C (CHead d1 (Flat f) u)))))))) (\lambda (c0: 
-C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (H2: ((\forall 
-(d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead 
-d2 (Flat f) u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: 
-T).(\forall (d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead 
-d1 (Flat f) u0)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Flat f) u0))))))))) (\lambda (b: 
-B).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S 
-n0) O (CHead c0 (Bind b) u) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) 
-(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
-(CHead c3 (Bind b) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: 
-(csub3 g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x (Flat f) 
-u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
-(S n0) O (CHead c3 (Bind b) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Bind 
-b) n0 c3 (CHead x (Flat f) u0) H5 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop 
-(Bind b) c0 (CHead d1 (Flat f) u0) u n0 H3)))))))) (\lambda (f0: F).(\lambda 
-(u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead 
-c0 (Flat f0) u) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g 
-d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u0))) (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Flat f0) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csub3 g 
-d1 x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Flat f) u0))).(ex_intro2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Flat f0) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Flat f0) n0 c3 (CHead 
-x (Flat f) u0) H5 u))))) (H2 d1 u0 (drop_gen_drop (Flat f0) c0 (CHead d1 
-(Flat f) u0) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: 
-C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: 
-T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) 
-u))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: 
-T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S 
-n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Flat f) u))).(ex2_ind C (\lambda 
-(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) 
-u))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
-(CHead c3 (Bind b) u2) (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H5: 
-(csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u))).(ex_intro2 
-C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Bind b) u2) (CHead d2 (Flat f) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead x 
-(Flat f) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 (CHead 
-d1 (Flat f) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: 
-C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: 
-T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) 
-u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c3 u 
-t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H4: (drop (S n0) O (CHead c0 
-(Bind Abst) t) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g 
-d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Bind Abbr) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H5: (csub3 
-g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Bind Abbr) u) (CHead d2 (Flat f) u0))) x H5 (drop_drop (Bind Abbr) n0 c3 
-(CHead x (Flat f) u0) H6 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) 
-c0 (CHead d1 (Flat f) u0) t n0 H4))))))))))))) c1 c2 H0)))))) n))).
-
-theorem csub3_drop_abbr:
- \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csub3 g 
-c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind 
-Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
-n O c2 (CHead d2 (Bind Abbr) u)))))))))))
-\def
- \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: 
-C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u: 
-T).((drop n0 O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) 
-u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 g c1 
-c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 
-(Bind Abbr) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csub3 g c c2)) H 
-(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H0)) in 
-(let H2 \def (match H1 in csub3 return (\lambda (c: C).(\lambda (c0: 
-C).(\lambda (_: (csub3 ? c c0)).((eq C c (CHead d1 (Bind Abbr) u)) \to ((eq C 
-c0 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O 
-O c2 (CHead d2 (Bind Abbr) u))))))))) with [(csub3_sort n0) \Rightarrow 
-(\lambda (H2: (eq C (CSort n0) (CHead d1 (Bind Abbr) u))).(\lambda (H3: (eq C 
-(CSort n0) c2)).((let H4 \def (eq_ind C (CSort n0) (\lambda (e: C).(match e 
-in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ 
-_ _) \Rightarrow False])) I (CHead d1 (Bind Abbr) u) H2) in (False_ind ((eq C 
-(CSort n0) c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop O O c2 (CHead d2 (Bind Abbr) u))))) H4)) H3))) | (csub3_head c0 c3 
-H2 k u0) \Rightarrow (\lambda (H3: (eq C (CHead c0 k u0) (CHead d1 (Bind 
-Abbr) u))).(\lambda (H4: (eq C (CHead c3 k u0) c2)).((let H5 \def (f_equal C 
-T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
-\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u0) (CHead d1 
-(Bind Abbr) u) H3) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in 
-C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) 
-\Rightarrow k0])) (CHead c0 k u0) (CHead d1 (Bind Abbr) u) H3) in ((let H7 
-\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) 
-with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k 
-u0) (CHead d1 (Bind Abbr) u) H3) in (eq_ind C d1 (\lambda (c: C).((eq K k 
-(Bind Abbr)) \to ((eq T u0 u) \to ((eq C (CHead c3 k u0) c2) \to ((csub3 g c 
-c3) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O 
-c2 (CHead d2 (Bind Abbr) u))))))))) (\lambda (H8: (eq K k (Bind 
-Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead 
-c3 k0 u0) c2) \to ((csub3 g d1 c3) \to (ex2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u)))))))) (\lambda 
-(H9: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c3 (Bind Abbr) t) 
-c2) \to ((csub3 g d1 c3) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))))))) (\lambda (H10: 
-(eq C (CHead c3 (Bind Abbr) u) c2)).(eq_ind C (CHead c3 (Bind Abbr) u) 
-(\lambda (c: C).((csub3 g d1 c3) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u)))))) (\lambda (H11: 
-(csub3 g d1 c3)).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) c3 H11 
-(drop_refl (CHead c3 (Bind Abbr) u)))) c2 H10)) u0 (sym_eq T u0 u H9))) k 
-(sym_eq K k (Bind Abbr) H8))) c0 (sym_eq C c0 d1 H7))) H6)) H5)) H4 H2))) | 
-(csub3_void c0 c3 H2 b H3 u1 u2) \Rightarrow (\lambda (H4: (eq C (CHead c0 
-(Bind Void) u1) (CHead d1 (Bind Abbr) u))).(\lambda (H5: (eq C (CHead c3 
-(Bind b) u2) c2)).((let H6 \def (eq_ind C (CHead c0 (Bind Void) u1) (\lambda 
-(e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow 
-False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) 
-with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: B).Prop) with 
-[Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | 
-(Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H4) in (False_ind 
-((eq C (CHead c3 (Bind b) u2) c2) \to ((csub3 g c0 c3) \to ((not (eq B b 
-Void)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O 
-O c2 (CHead d2 (Bind Abbr) u))))))) H6)) H5 H2 H3))) | (csub3_abst c0 c3 H2 
-u0 t H3) \Rightarrow (\lambda (H4: (eq C (CHead c0 (Bind Abst) t) (CHead d1 
-(Bind Abbr) u))).(\lambda (H5: (eq C (CHead c3 (Bind Abbr) u0) c2)).((let H6 
-\def (eq_ind C (CHead c0 (Bind Abst) t) (\lambda (e: C).(match e in C return 
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) 
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow 
-False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) 
-\Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H4) in (False_ind ((eq C 
-(CHead c3 (Bind Abbr) u0) c2) \to ((csub3 g c0 c3) \to ((ty3 g c3 u0 t) \to 
-(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead 
-d2 (Bind Abbr) u))))))) H6)) H5 H2 H3)))]) in (H2 (refl_equal C (CHead d1 
-(Bind Abbr) u)) (refl_equal C c2)))))))))) (\lambda (n0: nat).(\lambda (H: 
-((\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: 
-C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 
-(Bind Abbr) u)))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H0: 
-(csub3 g c1 c2)).(csub3_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (d1: 
-C).(\forall (u: T).((drop (S n0) O c (CHead d1 (Bind Abbr) u)) \to (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c0 (CHead 
-d2 (Bind Abbr) u))))))))) (\lambda (n1: nat).(\lambda (d1: C).(\lambda (u: 
-T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Bind Abbr) u))).(let H2 
-\def (match H1 in drop return (\lambda (n2: nat).(\lambda (n3: nat).(\lambda 
-(c: C).(\lambda (c0: C).(\lambda (_: (drop n2 n3 c c0)).((eq nat n2 (S n0)) 
-\to ((eq nat n3 O) \to ((eq C c (CSort n1)) \to ((eq C c0 (CHead d1 (Bind 
-Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
-(S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))))))))) with [(drop_refl c) 
-\Rightarrow (\lambda (H2: (eq nat O (S n0))).(\lambda (H3: (eq nat O 
-O)).(\lambda (H4: (eq C c (CSort n1))).(\lambda (H5: (eq C c (CHead d1 (Bind 
-Abbr) u))).((let H6 \def (eq_ind nat O (\lambda (e: nat).(match e in nat 
-return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow 
-False])) I (S n0) H2) in (False_ind ((eq nat O O) \to ((eq C c (CSort n1)) 
-\to ((eq C c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g 
-d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) 
-u))))))) H6)) H3 H4 H5))))) | (drop_drop k h c e H2 u0) \Rightarrow (\lambda 
-(H3: (eq nat (S h) (S n0))).(\lambda (H4: (eq nat O O)).(\lambda (H5: (eq C 
-(CHead c k u0) (CSort n1))).(\lambda (H6: (eq C e (CHead d1 (Bind Abbr) 
-u))).((let H7 \def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat 
-return (\lambda (_: nat).nat) with [O \Rightarrow h | (S n2) \Rightarrow 
-n2])) (S h) (S n0) H3) in (eq_ind nat n0 (\lambda (n2: nat).((eq nat O O) \to 
-((eq C (CHead c k u0) (CSort n1)) \to ((eq C e (CHead d1 (Bind Abbr) u)) \to 
-((drop (r k n2) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))))) (\lambda 
-(_: (eq nat O O)).(\lambda (H9: (eq C (CHead c k u0) (CSort n1))).(let H10 
-\def (eq_ind C (CHead c k u0) (\lambda (e0: C).(match e0 in C return (\lambda 
-(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow 
-True])) I (CSort n1) H9) in (False_ind ((eq C e (CHead d1 (Bind Abbr) u)) \to 
-((drop (r k n0) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))))) H10)))) h 
-(sym_eq nat h n0 H7))) H4 H5 H6 H2))))) | (drop_skip k h d c e H2 u0) 
-\Rightarrow (\lambda (H3: (eq nat h (S n0))).(\lambda (H4: (eq nat (S d) 
-O)).(\lambda (H5: (eq C (CHead c k (lift h (r k d) u0)) (CSort n1))).(\lambda 
-(H6: (eq C (CHead e k u0) (CHead d1 (Bind Abbr) u))).(eq_ind nat (S n0) 
-(\lambda (n2: nat).((eq nat (S d) O) \to ((eq C (CHead c k (lift n2 (r k d) 
-u0)) (CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 (Bind Abbr) u)) \to 
-((drop n2 (r k d) c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))))) (\lambda 
-(H7: (eq nat (S d) O)).(let H8 \def (eq_ind nat (S d) (\lambda (e0: 
-nat).(match e0 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow 
-False | (S _) \Rightarrow True])) I O H7) in (False_ind ((eq C (CHead c k 
-(lift (S n0) (r k d) u0)) (CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 
-(Bind Abbr) u)) \to ((drop (S n0) (r k d) c e) \to (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 
-(Bind Abbr) u))))))) H8))) h (sym_eq nat h (S n0) H3) H4 H5 H6 H2)))))]) in 
-(H2 (refl_equal nat (S n0)) (refl_equal nat O) (refl_equal C (CSort n1)) 
-(refl_equal C (CHead d1 (Bind Abbr) u)))))))) (\lambda (c0: C).(\lambda (c3: 
-C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall 
-(u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) 
-u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall 
-(d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Bind 
-Abbr) u0)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abbr) u0))))))))) (\lambda 
-(b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop 
-(S n0) O (CHead c0 (Bind b) u) (CHead d1 (Bind Abbr) u0))).(ex2_ind C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 
-(Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda 
-(x: C).(\lambda (H4: (csub3 g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x 
-(Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0))) x H4 
-(drop_drop (Bind b) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) (H c0 c3 H1 d1 
-u0 (drop_gen_drop (Bind b) c0 (CHead d1 (Bind Abbr) u0) u n0 H3)))))))) 
-(\lambda (f: F).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda 
-(H3: (drop (S n0) O (CHead c0 (Flat f) u) (CHead d1 (Bind Abbr) 
-u0))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S 
-n0) O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) 
-u0)))) (\lambda (x: C).(\lambda (H4: (csub3 g d1 x)).(\lambda (H5: (drop (S 
-n0) O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind 
-Abbr) u0))) x H4 (drop_drop (Flat f) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) 
-(H2 d1 u0 (drop_gen_drop (Flat f) c0 (CHead d1 (Bind Abbr) u0) u n0 
-H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g 
-c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 
-(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda 
-(b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S n0) O (CHead c0 
-(Bind Void) u1) (CHead d1 (Bind Abbr) u))).(ex2_ind C (\lambda (d2: C).(csub3 
-g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u))) (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (x: C).(\lambda (H5: (csub3 
-g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind Abbr) u))).(ex_intro2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Bind b) u2) (CHead d2 (Bind Abbr) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead 
-x (Bind Abbr) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 
-(CHead d1 (Bind Abbr) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda 
-(c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: 
-C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead 
-d2 (Bind Abbr) u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g 
-c3 u t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H4: (drop (S n0) O 
-(CHead c0 (Bind Abst) t) (CHead d1 (Bind Abbr) u0))).(ex2_ind C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) 
-u0))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) 
-O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (x: 
-C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind 
-Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0))) x H5 
-(drop_drop (Bind Abbr) n0 c3 (CHead x (Bind Abbr) u0) H6 u))))) (H c0 c3 H1 
-d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Bind Abbr) u0) t n0 
-H4))))))))))))) c1 c2 H0)))))) n)).
-
-theorem csub3_drop_abst:
- \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csub3 g 
-c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n O c1 (CHead d1 (Bind 
-Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n 
-O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t))))))))))))
-\def
- \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: 
-C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (t: 
-T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst) 
-t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
-(d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda 
-(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) (\lambda (c1: C).(\lambda 
-(c2: C).(\lambda (H: (csub3 g c1 c2)).(\lambda (d1: C).(\lambda (t: 
-T).(\lambda (H0: (drop O O c1 (CHead d1 (Bind Abst) t))).(let H1 \def (eq_ind 
-C c1 (\lambda (c: C).(csub3 g c c2)) H (CHead d1 (Bind Abst) t) 
-(drop_gen_refl c1 (CHead d1 (Bind Abst) t) H0)) in (let H2 \def (match H1 in 
-csub3 return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 ? c 
-c0)).((eq C c (CHead d1 (Bind Abst) t)) \to ((eq C c0 c2) \to (or (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 
-(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) 
-u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) with 
-[(csub3_sort n0) \Rightarrow (\lambda (H2: (eq C (CSort n0) (CHead d1 (Bind 
-Abst) t))).(\lambda (H3: (eq C (CSort n0) c2)).((let H4 \def (eq_ind C (CSort 
-n0) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort 
-_) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind 
-Abst) t) H2) in (False_ind ((eq C (CSort n0) c2) \to (or (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) 
-(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
-C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
-C).(\lambda (u: T).(ty3 g d2 u t)))))) H4)) H3))) | (csub3_head c0 c3 H2 k u) 
-\Rightarrow (\lambda (H3: (eq C (CHead c0 k u) (CHead d1 (Bind Abst) 
-t))).(\lambda (H4: (eq C (CHead c3 k u) c2)).((let H5 \def (f_equal C T 
-(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
-\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k u) (CHead d1 
-(Bind Abst) t) H3) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in 
-C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) 
-\Rightarrow k0])) (CHead c0 k u) (CHead d1 (Bind Abst) t) H3) in ((let H7 
-\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) 
-with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k 
-u) (CHead d1 (Bind Abst) t) H3) in (eq_ind C d1 (\lambda (c: C).((eq K k 
-(Bind Abst)) \to ((eq T u t) \to ((eq C (CHead c3 k u) c2) \to ((csub3 g c 
-c3) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O 
-O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 
-(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) 
-(\lambda (H8: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda (k0: 
-K).((eq T u t) \to ((eq C (CHead c3 k0 u) c2) \to ((csub3 g d1 c3) \to (or 
-(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead 
-d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) 
-u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))))) (\lambda 
-(H9: (eq T u t)).(eq_ind T t (\lambda (t0: T).((eq C (CHead c3 (Bind Abst) 
-t0) c2) \to ((csub3 g d1 c3) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda 
-(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: 
-T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
-T).(ty3 g d2 u0 t)))))))) (\lambda (H10: (eq C (CHead c3 (Bind Abst) t) 
-c2)).(eq_ind C (CHead c3 (Bind Abst) t) (\lambda (c: C).((csub3 g d1 c3) \to 
-(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c 
-(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c (CHead d2 
-(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))) 
-(\lambda (H11: (csub3 g d1 c3)).(or_introl (ex2 C (\lambda (d2: C).(csub3 g 
-d1 d2)) (\lambda (d2: C).(drop O O (CHead c3 (Bind Abst) t) (CHead d2 (Bind 
-Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) 
-(\lambda (d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abst) t) (CHead 
-d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) 
-(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O 
-(CHead c3 (Bind Abst) t) (CHead d2 (Bind Abst) t))) c3 H11 (drop_refl (CHead 
-c3 (Bind Abst) t))))) c2 H10)) u (sym_eq T u t H9))) k (sym_eq K k (Bind 
-Abst) H8))) c0 (sym_eq C c0 d1 H7))) H6)) H5)) H4 H2))) | (csub3_void c0 c3 
-H2 b H3 u1 u2) \Rightarrow (\lambda (H4: (eq C (CHead c0 (Bind Void) u1) 
-(CHead d1 (Bind Abst) t))).(\lambda (H5: (eq C (CHead c3 (Bind b) u2) 
-c2)).((let H6 \def (eq_ind C (CHead c0 (Bind Void) u1) (\lambda (e: C).(match 
-e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | 
-(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
-[(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr 
-\Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat 
-_) \Rightarrow False])])) I (CHead d1 (Bind Abst) t) H4) in (False_ind ((eq C 
-(CHead c3 (Bind b) u2) c2) \to ((csub3 g c0 c3) \to ((not (eq B b Void)) \to 
-(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 
-(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c2 (CHead d2 
-(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))) H6)) 
-H5 H2 H3))) | (csub3_abst c0 c3 H2 u t0 H3) \Rightarrow (\lambda (H4: (eq C 
-(CHead c0 (Bind Abst) t0) (CHead d1 (Bind Abst) t))).(\lambda (H5: (eq C 
-(CHead c3 (Bind Abbr) u) c2)).((let H6 \def (f_equal C T (\lambda (e: 
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t0 | 
-(CHead _ _ t1) \Rightarrow t1])) (CHead c0 (Bind Abst) t0) (CHead d1 (Bind 
-Abst) t) H4) in ((let H7 \def (f_equal C C (\lambda (e: C).(match e in C 
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) 
-\Rightarrow c])) (CHead c0 (Bind Abst) t0) (CHead d1 (Bind Abst) t) H4) in 
-(eq_ind C d1 (\lambda (c: C).((eq T t0 t) \to ((eq C (CHead c3 (Bind Abbr) u) 
-c2) \to ((csub3 g c c3) \to ((ty3 g c3 u t0) \to (or (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) 
-(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
-C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
-C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) (\lambda (H8: (eq T t0 
-t)).(eq_ind T t (\lambda (t1: T).((eq C (CHead c3 (Bind Abbr) u) c2) \to 
-((csub3 g d1 c3) \to ((ty3 g c3 u t1) \to (or (ex2 C (\lambda (d2: C).(csub3 
-g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C 
-T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
-C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
-C).(\lambda (u0: T).(ty3 g d2 u0 t))))))))) (\lambda (H9: (eq C (CHead c3 
-(Bind Abbr) u) c2)).(eq_ind C (CHead c3 (Bind Abbr) u) (\lambda (c: 
-C).((csub3 g d1 c3) \to ((ty3 g c3 u t) \to (or (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) t)))) 
-(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
-C).(\lambda (u0: T).(drop O O c (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
-C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))) (\lambda (H10: (csub3 g d1 
-c3)).(\lambda (H11: (ty3 g c3 u t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 
-g d1 d2)) (\lambda (d2: C).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind 
-Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) 
-(\lambda (d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abbr) u) (CHead 
-d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) 
-(ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
-(d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind 
-Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) c3 u H10 
-(drop_refl (CHead c3 (Bind Abbr) u)) H11)))) c2 H9)) t0 (sym_eq T t0 t H8))) 
-c0 (sym_eq C c0 d1 H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal C (CHead d1 
-(Bind Abst) t)) (refl_equal C c2)))))))))) (\lambda (n0: nat).(\lambda (H: 
-((\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: 
-C).(\forall (t: T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 
-(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) 
-u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))))).(\lambda 
-(c1: C).(\lambda (c2: C).(\lambda (H0: (csub3 g c1 c2)).(csub3_ind g (\lambda 
-(c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (t: T).((drop (S n0) O c 
-(CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(drop (S n0) O c0 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
-C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (n1: nat).(\lambda (d1: 
-C).(\lambda (t: T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Bind 
-Abst) t))).(let H2 \def (match H1 in drop return (\lambda (n2: nat).(\lambda 
-(n3: nat).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop n2 n3 c 
-c0)).((eq nat n2 (S n0)) \to ((eq nat n3 O) \to ((eq C c (CSort n1)) \to ((eq 
-C c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) 
-(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
-C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) 
-(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))))) with [(drop_refl 
-c) \Rightarrow (\lambda (H2: (eq nat O (S n0))).(\lambda (H3: (eq nat O 
-O)).(\lambda (H4: (eq C c (CSort n1))).(\lambda (H5: (eq C c (CHead d1 (Bind 
-Abst) t))).((let H6 \def (eq_ind nat O (\lambda (e: nat).(match e in nat 
-return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow 
-False])) I (S n0) H2) in (False_ind ((eq nat O O) \to ((eq C c (CSort n1)) 
-\to ((eq C c (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 
-g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) 
-t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
-(d2: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) 
-(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))) H6)) H3 H4 H5))))) | 
-(drop_drop k h c e H2 u) \Rightarrow (\lambda (H3: (eq nat (S h) (S 
-n0))).(\lambda (H4: (eq nat O O)).(\lambda (H5: (eq C (CHead c k u) (CSort 
-n1))).(\lambda (H6: (eq C e (CHead d1 (Bind Abst) t))).((let H7 \def (f_equal 
-nat nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) 
-with [O \Rightarrow h | (S n2) \Rightarrow n2])) (S h) (S n0) H3) in (eq_ind 
-nat n0 (\lambda (n2: nat).((eq nat O O) \to ((eq C (CHead c k u) (CSort n1)) 
-\to ((eq C e (CHead d1 (Bind Abst) t)) \to ((drop (r k n2) O c e) \to (or 
-(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
-(CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda 
-(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O 
-(CSort n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
-T).(ty3 g d2 u0 t)))))))))) (\lambda (_: (eq nat O O)).(\lambda (H9: (eq C 
-(CHead c k u) (CSort n1))).(let H10 \def (eq_ind C (CHead c k u) (\lambda 
-(e0: C).(match e0 in C return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n1) H9) in 
-(False_ind ((eq C e (CHead d1 (Bind Abst) t)) \to ((drop (r k n0) O c e) \to 
-(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
-(CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda 
-(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O 
-(CSort n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
-T).(ty3 g d2 u0 t))))))) H10)))) h (sym_eq nat h n0 H7))) H4 H5 H6 H2))))) | 
-(drop_skip k h d c e H2 u) \Rightarrow (\lambda (H3: (eq nat h (S 
-n0))).(\lambda (H4: (eq nat (S d) O)).(\lambda (H5: (eq C (CHead c k (lift h 
-(r k d) u)) (CSort n1))).(\lambda (H6: (eq C (CHead e k u) (CHead d1 (Bind 
-Abst) t))).(eq_ind nat (S n0) (\lambda (n2: nat).((eq nat (S d) O) \to ((eq C 
-(CHead c k (lift n2 (r k d) u)) (CSort n1)) \to ((eq C (CHead e k u) (CHead 
-d1 (Bind Abst) t)) \to ((drop n2 (r k d) c e) \to (or (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 
-(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CSort n1) (CHead d2 
-(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) 
-(\lambda (H7: (eq nat (S d) O)).(let H8 \def (eq_ind nat (S d) (\lambda (e0: 
-nat).(match e0 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow 
-False | (S _) \Rightarrow True])) I O H7) in (False_ind ((eq C (CHead c k 
-(lift (S n0) (r k d) u)) (CSort n1)) \to ((eq C (CHead e k u) (CHead d1 (Bind 
-Abst) t)) \to ((drop (S n0) (r k d) c e) \to (or (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 
-(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CSort n1) (CHead d2 
-(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))) 
-H8))) h (sym_eq nat h (S n0) H3) H4 H5 H6 H2)))))]) in (H2 (refl_equal nat (S 
-n0)) (refl_equal nat O) (refl_equal C (CSort n1)) (refl_equal C (CHead d1 
-(Bind Abst) t)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 
-g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (t: T).((drop (S n0) O c0 
-(CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
-C).(\lambda (u: T).(ty3 g d2 u t)))))))))).(\lambda (k: K).(K_ind (\lambda 
-(k0: K).(\forall (u: T).(\forall (d1: C).(\forall (t: T).((drop (S n0) O 
-(CHead c0 k0 u) (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 
-(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 k0 u) (CHead 
-d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 
-t)))))))))) (\lambda (b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda (t: 
-T).(\lambda (H3: (drop (S n0) O (CHead c0 (Bind b) u) (CHead d1 (Bind Abst) 
-t))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
-n0 O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead 
-d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) 
-(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
-(CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
-(S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
-C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (H4: (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) 
-t))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 
-O c3 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) 
-t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
-(d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind 
-Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda 
-(x: C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x 
-(Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) 
-(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
-C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind 
-Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex_intro2 
-C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Bind b) u) (CHead d2 (Bind Abst) t))) x H5 (drop_drop (Bind b) n0 c3 (CHead 
-x (Bind Abst) t) H6 u)))))) H4)) (\lambda (H4: (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
-n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g 
-d2 u0 t))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) 
-u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) (or (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda 
-(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O 
-(CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda 
-(u0: T).(ty3 g d2 u0 t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: 
-(csub3 g d1 x0)).(\lambda (H6: (drop n0 O c3 (CHead x0 (Bind Abbr) 
-x1))).(\lambda (H7: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) 
-(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
-c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
-T).(ty3 g d2 u0 t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
-c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
-T).(ty3 g d2 u0 t))) x0 x1 H5 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) 
-x1) H6 u) H7))))))) H4)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind b) c0 (CHead 
-d1 (Bind Abst) t) u n0 H3)))))))) (\lambda (f: F).(\lambda (u: T).(\lambda 
-(d1: C).(\lambda (t: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f) u) 
-(CHead d1 (Bind Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u0: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
-C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (or (ex2 C (\lambda (d2: C).(csub3 g 
-d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind 
-Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) 
-(\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead 
-d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) 
-(\lambda (H4: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
-(S n0) O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csub3 g 
-d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t))) (or 
-(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
-(CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
-(S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
-C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (x: C).(\lambda (H5: (csub3 
-g d1 x)).(\lambda (H6: (drop (S n0) O c3 (CHead x (Bind Abst) t))).(or_introl 
-(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
-(CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
-(S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
-C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g 
-d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind 
-Abst) t))) x H5 (drop_drop (Flat f) n0 c3 (CHead x (Bind Abst) t) H6 u)))))) 
-H4)) (\lambda (H4: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O c3 (CHead d2 (Bind 
-Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))).(ex3_2_ind 
-C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
-C).(\lambda (u0: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda 
-(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) (or (ex2 C (\lambda (d2: C).(csub3 
-g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 
-(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) 
-(CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 
-t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (csub3 g d1 
-x0)).(\lambda (H6: (drop (S n0) O c3 (CHead x0 (Bind Abbr) x1))).(\lambda 
-(H7: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) 
-t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
-(d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind 
-Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) 
-(ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
-(d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind 
-Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) x0 x1 H5 
-(drop_drop (Flat f) n0 c3 (CHead x0 (Bind Abbr) x1) H6 u) H7))))))) H4)) (H2 
-d1 t (drop_gen_drop (Flat f) c0 (CHead d1 (Bind Abst) t) u n0 H3)))))))) 
-k)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 
-c3)).(\lambda (_: ((\forall (d1: C).(\forall (t: T).((drop (S n0) O c0 (CHead 
-d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda 
-(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: 
-T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda 
-(u: T).(ty3 g d2 u t)))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b 
-Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (t: 
-T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Bind 
-Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop n0 O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop 
-n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
-d2 u t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C 
-T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind 
-Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H5: 
-(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 
-(CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda 
-(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) 
-u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead 
-c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t))))) (\lambda (x: C).(\lambda (H6: (csub3 g d1 x)).(\lambda 
-(H7: (drop n0 O c3 (CHead x (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) 
-(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead 
-c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t))) x H6 
-(drop_drop (Bind b) n0 c3 (CHead x (Bind Abst) t) H7 u2)))))) H5)) (\lambda 
-(H5: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
-(d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda 
-(d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop 
-n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
-d2 u t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
-(S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) 
-(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x0: C).(\lambda 
-(x1: T).(\lambda (H6: (csub3 g d1 x0)).(\lambda (H7: (drop n0 O c3 (CHead x0 
-(Bind Abbr) x1))).(\lambda (H8: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda 
-(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) 
-u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead 
-c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 
-g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) 
-u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t))) x0 x1 H6 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) x1) H7 u2) 
-H8))))))) H5)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind Void) c0 (CHead d1 (Bind 
-Abst) t) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda 
-(H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (t: T).((drop 
-(S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 
-g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) 
-(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
-C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda 
-(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))).(\lambda (u: T).(\lambda (t: 
-T).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (t0: T).(\lambda 
-(H4: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind Abst) 
-t0))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop 
-n0 O c3 (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead 
-d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) 
-(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
-(CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda 
-(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: 
-T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) 
-(\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))) (\lambda (H5: (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 
-(Bind Abst) t0))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t0))) (or (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) 
-(CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
-c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
-T).(ty3 g d2 u0 t0))))) (\lambda (x: C).(\lambda (H6: (csub3 g d1 
-x)).(\lambda (H7: (drop n0 O c3 (CHead x (Bind Abst) t0))).(or_introl (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
-(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
-C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex_intro2 C (\lambda (d2: C).(csub3 
-g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 
-(Bind Abst) t0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind Abst) t0) 
-H7 u)))))) H5)) (\lambda (H5: (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead 
-d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 
-t0))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) 
-(\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) 
-(\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))) (or (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) 
-(CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
-c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
-T).(ty3 g d2 u0 t0))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: 
-(csub3 g d1 x0)).(\lambda (H7: (drop n0 O c3 (CHead x0 (Bind Abbr) 
-x1))).(\lambda (H8: (ty3 g x0 x1 t0)).(or_intror (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) 
-(CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
-c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
-T).(ty3 g d2 u0 t0)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
-c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
-T).(ty3 g d2 u0 t0))) x0 x1 H6 (drop_drop (Bind Abbr) n0 c3 (CHead x0 (Bind 
-Abbr) x1) H7 u) H8))))))) H5)) (H c0 c3 H1 d1 t0 (drop_gen_drop (Bind Abst) 
-c0 (CHead d1 (Bind Abst) t0) t n0 H4))))))))))))) c1 c2 H0)))))) n)).
-
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/fwd.ma
deleted file mode 100644
index 99b9dae96..000000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/fwd.ma
+++ /dev/null
@@ -1,336 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/fwd".
-
-include "csub3/defs.ma".
-
-theorem csub3_gen_abbr:
- \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csub3 g 
-(CHead e1 (Bind Abbr) v) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 
-(Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))))
-\def
- \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda 
-(H: (csub3 g (CHead e1 (Bind Abbr) v) c2)).(let H0 \def (match H in csub3 
-return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 ? c c0)).((eq C c 
-(CHead e1 (Bind Abbr) v)) \to ((eq C c0 c2) \to (ex2 C (\lambda (e2: C).(eq C 
-c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))))) with 
-[(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead e1 (Bind 
-Abbr) v))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def (eq_ind C (CSort 
-n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abbr) 
-v) H0) in (False_ind ((eq C (CSort n) c2) \to (ex2 C (\lambda (e2: C).(eq C 
-c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))) H2)) H1))) 
-| (csub3_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) 
-(CHead e1 (Bind Abbr) v))).(\lambda (H2: (eq C (CHead c0 k u) c2)).((let H3 
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) 
-with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) 
-(CHead e1 (Bind Abbr) v) H1) in ((let H4 \def (f_equal C K (\lambda (e: 
-C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | 
-(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind Abbr) v) H1) 
-in ((let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda 
-(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) 
-(CHead c1 k u) (CHead e1 (Bind Abbr) v) H1) in (eq_ind C e1 (\lambda (c: 
-C).((eq K k (Bind Abbr)) \to ((eq T u v) \to ((eq C (CHead c0 k u) c2) \to 
-((csub3 g c c0) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) 
-v))) (\lambda (e2: C).(csub3 g e1 e2)))))))) (\lambda (H6: (eq K k (Bind 
-Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u v) \to ((eq C (CHead 
-c0 k0 u) c2) \to ((csub3 g e1 c0) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead 
-e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2))))))) (\lambda (H7: (eq 
-T u v)).(eq_ind T v (\lambda (t: T).((eq C (CHead c0 (Bind Abbr) t) c2) \to 
-((csub3 g e1 c0) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) 
-v))) (\lambda (e2: C).(csub3 g e1 e2)))))) (\lambda (H8: (eq C (CHead c0 
-(Bind Abbr) v) c2)).(eq_ind C (CHead c0 (Bind Abbr) v) (\lambda (c: 
-C).((csub3 g e1 c0) \to (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abbr) 
-v))) (\lambda (e2: C).(csub3 g e1 e2))))) (\lambda (H9: (csub3 g e1 c0)).(let 
-H10 \def (eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind Abbr) v) c)) 
-H (CHead c0 (Bind Abbr) v) H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead 
-c0 (Bind Abbr) v) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 
-e2)) c0 (refl_equal C (CHead c0 (Bind Abbr) v)) H9))) c2 H8)) u (sym_eq T u v 
-H7))) k (sym_eq K k (Bind Abbr) H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 
-H0))) | (csub3_void c1 c0 H0 b H1 u1 u2) \Rightarrow (\lambda (H2: (eq C 
-(CHead c1 (Bind Void) u1) (CHead e1 (Bind Abbr) v))).(\lambda (H3: (eq C 
-(CHead c0 (Bind b) u2) c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Void) u1) 
-(\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: 
-B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void 
-\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr) 
-v) H2) in (False_ind ((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c1 c0) 
-\to ((not (eq B b Void)) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind 
-Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))) H4)) H3 H0 H1))) | 
-(csub3_abst c1 c0 H0 u t H1) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind 
-Abst) t) (CHead e1 (Bind Abbr) v))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) 
-u) c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: 
-C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow 
-False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) 
-with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with 
-[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | 
-(Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H2) in (False_ind 
-((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g c1 c0) \to ((ty3 g c0 u t) 
-\to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: 
-C).(csub3 g e1 e2)))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead e1 
-(Bind Abbr) v)) (refl_equal C c2))))))).
-
-theorem csub3_gen_abst:
- \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csub3 g 
-(CHead e1 (Bind Abst) v1) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead 
-e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda 
-(e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
-C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g 
-e2 v2 v1)))))))))
-\def
- \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda 
-(H: (csub3 g (CHead e1 (Bind Abst) v1) c2)).(let H0 \def (match H in csub3 
-return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 ? c c0)).((eq C c 
-(CHead e1 (Bind Abst) v1)) \to ((eq C c0 c2) \to (or (ex2 C (\lambda (e2: 
-C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) 
-(ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) 
-v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: 
-C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))))) with [(csub3_sort n) 
-\Rightarrow (\lambda (H0: (eq C (CSort n) (CHead e1 (Bind Abst) 
-v1))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def (eq_ind C (CSort n) 
-(\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abst) 
-v1) H0) in (False_ind ((eq C (CSort n) c2) \to (or (ex2 C (\lambda (e2: 
-C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) 
-(ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) 
-v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: 
-C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) H2)) H1))) | (csub3_head c1 c0 H0 k 
-u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) (CHead e1 (Bind Abst) 
-v1))).(\lambda (H2: (eq C (CHead c0 k u) c2)).((let H3 \def (f_equal C T 
-(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
-\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind 
-Abst) v1) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e in C 
-return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) 
-\Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H1) in ((let H5 
-\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) 
-with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k 
-u) (CHead e1 (Bind Abst) v1) H1) in (eq_ind C e1 (\lambda (c: C).((eq K k 
-(Bind Abst)) \to ((eq T u v1) \to ((eq C (CHead c0 k u) c2) \to ((csub3 g c 
-c0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) 
-(\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: 
-T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
-T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
-v1)))))))))) (\lambda (H6: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) 
-(\lambda (k0: K).((eq T u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csub3 g e1 
-c0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) 
-(\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: 
-T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
-T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
-v1))))))))) (\lambda (H7: (eq T u v1)).(eq_ind T v1 (\lambda (t: T).((eq C 
-(CHead c0 (Bind Abst) t) c2) \to ((csub3 g e1 c0) \to (or (ex2 C (\lambda 
-(e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 
-e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind 
-Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: 
-C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (H8: (eq C (CHead c0 
-(Bind Abst) v1) c2)).(eq_ind C (CHead c0 (Bind Abst) v1) (\lambda (c: 
-C).((csub3 g e1 c0) \to (or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind 
-Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: 
-C).(\lambda (v2: T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
-C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g 
-e2 v2 v1))))))) (\lambda (H9: (csub3 g e1 c0)).(let H10 \def (eq_ind_r C c2 
-(\lambda (c: C).(csub3 g (CHead e1 (Bind Abst) v1) c)) H (CHead c0 (Bind 
-Abst) v1) H8) in (or_introl (ex2 C (\lambda (e2: C).(eq C (CHead c0 (Bind 
-Abst) v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) 
-(ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind Abst) v1) 
-(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 
-e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) (ex_intro2 C 
-(\lambda (e2: C).(eq C (CHead c0 (Bind Abst) v1) (CHead e2 (Bind Abst) v1))) 
-(\lambda (e2: C).(csub3 g e1 e2)) c0 (refl_equal C (CHead c0 (Bind Abst) v1)) 
-H9)))) c2 H8)) u (sym_eq T u v1 H7))) k (sym_eq K k (Bind Abst) H6))) c1 
-(sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 H0 b H1 u1 u2) 
-\Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind 
-Abst) v1))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) c2)).((let H4 \def 
-(eq_ind C (CHead c1 (Bind Void) u1) (\lambda (e: C).(match e in C return 
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b0) 
-\Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr 
-\Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat 
-_) \Rightarrow False])])) I (CHead e1 (Bind Abst) v1) H2) in (False_ind ((eq 
-C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c1 c0) \to ((not (eq B b Void)) 
-\to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda 
-(e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C 
-c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 
-e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) H4)) H3 H0 
-H1))) | (csub3_abst c1 c0 H0 u t H1) \Rightarrow (\lambda (H2: (eq C (CHead 
-c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1))).(\lambda (H3: (eq C (CHead c0 
-(Bind Abbr) u) c2)).((let H4 \def (f_equal C T (\lambda (e: C).(match e in C 
-return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t0) 
-\Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H2) in 
-((let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: 
-C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead 
-c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H2) in (eq_ind C e1 (\lambda (c: 
-C).((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g c c0) 
-\to ((ty3 g c0 u t) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind 
-Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: 
-C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
-C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g 
-e2 v2 v1)))))))))) (\lambda (H6: (eq T t v1)).(eq_ind T v1 (\lambda (t0: 
-T).((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g e1 c0) \to ((ty3 g c0 u 
-t0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) 
-(\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: 
-T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
-T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
-v1))))))))) (\lambda (H7: (eq C (CHead c0 (Bind Abbr) u) c2)).(eq_ind C 
-(CHead c0 (Bind Abbr) u) (\lambda (c: C).((csub3 g e1 c0) \to ((ty3 g c0 u 
-v1) \to (or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abst) v1))) 
-(\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: 
-T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
-T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
-v1)))))))) (\lambda (H8: (csub3 g e1 c0)).(\lambda (H9: (ty3 g c0 u v1)).(let 
-H10 \def (eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind Abst) v1) 
-c)) H (CHead c0 (Bind Abbr) u) H7) in (or_intror (ex2 C (\lambda (e2: C).(eq 
-C (CHead c0 (Bind Abbr) u) (CHead e2 (Bind Abst) v1))) (\lambda (e2: 
-C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C 
-(CHead c0 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
-C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g 
-e2 v2 v1)))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead 
-c0 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
-T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))) c0 
-u (refl_equal C (CHead c0 (Bind Abbr) u)) H8 H9))))) c2 H7)) t (sym_eq T t v1 
-H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead 
-e1 (Bind Abst) v1)) (refl_equal C c2))))))).
-
-theorem csub3_gen_bind:
- \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall 
-(v1: T).((csub3 g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))))))
-\def
- \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda 
-(v1: T).(\lambda (H: (csub3 g (CHead e1 (Bind b1) v1) c2)).(let H0 \def 
-(match H in csub3 return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 
-? c c0)).((eq C c (CHead e1 (Bind b1) v1)) \to ((eq C c0 c2) \to (ex2_3 B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind 
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 
-e2)))))))))) with [(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) 
-(CHead e1 (Bind b1) v1))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def 
-(eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_: 
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
-False])) I (CHead e1 (Bind b1) v1) H0) in (False_ind ((eq C (CSort n) c2) \to 
-(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
-T).(csub3 g e1 e2)))))) H2)) H1))) | (csub3_head c1 c0 H0 k u) \Rightarrow 
-(\lambda (H1: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(\lambda (H2: 
-(eq C (CHead c0 k u) c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e 
-in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) 
-\Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in ((let H4 \def 
-(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with 
-[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) 
-(CHead e1 (Bind b1) v1) H1) in ((let H5 \def (f_equal C C (\lambda (e: 
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | 
-(CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in 
-(eq_ind C e1 (\lambda (c: C).((eq K k (Bind b1)) \to ((eq T u v1) \to ((eq C 
-(CHead c0 k u) c2) \to ((csub3 g c c0) \to (ex2_3 B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))))) 
-(\lambda (H6: (eq K k (Bind b1))).(eq_ind K (Bind b1) (\lambda (k0: K).((eq T 
-u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csub3 g e1 c0) \to (ex2_3 B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind 
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 
-e2))))))))) (\lambda (H7: (eq T u v1)).(eq_ind T v1 (\lambda (t: T).((eq C 
-(CHead c0 (Bind b1) t) c2) \to ((csub3 g e1 c0) \to (ex2_3 B C T (\lambda 
-(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) 
-v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 
-e2)))))))) (\lambda (H8: (eq C (CHead c0 (Bind b1) v1) c2)).(eq_ind C (CHead 
-c0 (Bind b1) v1) (\lambda (c: C).((csub3 g e1 c0) \to (ex2_3 B C T (\lambda 
-(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))) 
-(\lambda (H9: (csub3 g e1 c0)).(let H10 \def (eq_ind_r C c2 (\lambda (c: 
-C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind b1) v1) H8) in 
-(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
-(CHead c0 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda 
-(e2: C).(\lambda (_: T).(csub3 g e1 e2)))) b1 c0 v1 (refl_equal C (CHead c0 
-(Bind b1) v1)) H9))) c2 H8)) u (sym_eq T u v1 H7))) k (sym_eq K k (Bind b1) 
-H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 H0 b H1 
-u1 u2) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) u1) (CHead e1 
-(Bind b1) v1))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) c2)).((let H4 \def 
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind 
-Void) u1) (CHead e1 (Bind b1) v1) H2) in ((let H5 \def (f_equal C B (\lambda 
-(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow 
-Void | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with 
-[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Void])])) (CHead c1 (Bind 
-Void) u1) (CHead e1 (Bind b1) v1) H2) in ((let H6 \def (f_equal C C (\lambda 
-(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 
-| (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind 
-b1) v1) H2) in (eq_ind C e1 (\lambda (c: C).((eq B Void b1) \to ((eq T u1 v1) 
-\to ((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c c0) \to ((not (eq B b 
-Void)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
-T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csub3 g e1 e2))))))))))) (\lambda (H7: (eq B Void 
-b1)).(eq_ind B Void (\lambda (_: B).((eq T u1 v1) \to ((eq C (CHead c0 (Bind 
-b) u2) c2) \to ((csub3 g e1 c0) \to ((not (eq B b Void)) \to (ex2_3 B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind 
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 
-e2)))))))))) (\lambda (H8: (eq T u1 v1)).(eq_ind T v1 (\lambda (_: T).((eq C 
-(CHead c0 (Bind b) u2) c2) \to ((csub3 g e1 c0) \to ((not (eq B b Void)) \to 
-(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
-T).(csub3 g e1 e2))))))))) (\lambda (H9: (eq C (CHead c0 (Bind b) u2) 
-c2)).(eq_ind C (CHead c0 (Bind b) u2) (\lambda (c: C).((csub3 g e1 c0) \to 
-((not (eq B b Void)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
-C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: 
-B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))) (\lambda (H10: 
-(csub3 g e1 c0)).(\lambda (_: (not (eq B b Void))).(let H12 \def (eq_ind_r C 
-c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind b) 
-u2) H9) in (let H13 \def (eq_ind_r B b1 (\lambda (b0: B).(csub3 g (CHead e1 
-(Bind b0) v1) (CHead c0 (Bind b) u2))) H12 Void H7) in (ex2_3_intro B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind b) 
-u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
-T).(csub3 g e1 e2)))) b c0 u2 (refl_equal C (CHead c0 (Bind b) u2)) H10))))) 
-c2 H9)) u1 (sym_eq T u1 v1 H8))) b1 H7)) c1 (sym_eq C c1 e1 H6))) H5)) H4)) 
-H3 H0 H1))) | (csub3_abst c1 c0 H0 u t H1) \Rightarrow (\lambda (H2: (eq C 
-(CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1))).(\lambda (H3: (eq C (CHead 
-c0 (Bind Abbr) u) c2)).((let H4 \def (f_equal C T (\lambda (e: C).(match e in 
-C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t0) 
-\Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H2) in 
-((let H5 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: 
-C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k 
-in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) 
-\Rightarrow Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H2) in 
-((let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: 
-C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead 
-c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H2) in (eq_ind C e1 (\lambda (c: 
-C).((eq B Abst b1) \to ((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) 
-\to ((csub3 g c c0) \to ((ty3 g c0 u t) \to (ex2_3 B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))))))) 
-(\lambda (H7: (eq B Abst b1)).(eq_ind B Abst (\lambda (_: B).((eq T t v1) \to 
-((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g e1 c0) \to ((ty3 g c0 u t) 
-\to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
-T).(csub3 g e1 e2)))))))))) (\lambda (H8: (eq T t v1)).(eq_ind T v1 (\lambda 
-(t0: T).((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g e1 c0) \to ((ty3 g 
-c0 u t0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
-T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csub3 g e1 e2))))))))) (\lambda (H9: (eq C (CHead c0 
-(Bind Abbr) u) c2)).(eq_ind C (CHead c0 (Bind Abbr) u) (\lambda (c: 
-C).((csub3 g e1 c0) \to ((ty3 g c0 u v1) \to (ex2_3 B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))) 
-(\lambda (H10: (csub3 g e1 c0)).(\lambda (_: (ty3 g c0 u v1)).(let H12 \def 
-(eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead 
-c0 (Bind Abbr) u) H9) in (let H13 \def (eq_ind_r B b1 (\lambda (b: B).(csub3 
-g (CHead e1 (Bind b) v1) (CHead c0 (Bind Abbr) u))) H12 Abst H7) in 
-(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
-(CHead c0 (Bind Abbr) u) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda 
-(e2: C).(\lambda (_: T).(csub3 g e1 e2)))) Abbr c0 u (refl_equal C (CHead c0 
-(Bind Abbr) u)) H10))))) c2 H9)) t (sym_eq T t v1 H8))) b1 H7)) c1 (sym_eq C 
-c1 e1 H6))) H5)) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead e1 (Bind b1) 
-v1)) (refl_equal C c2)))))))).
-
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/getl.ma
deleted file mode 100644
index acb352ea3..000000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/getl.ma
+++ /dev/null
@@ -1,398 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/getl".
-
-include "csub3/fwd.ma".
-
-include "csub3/clear.ma".
-
-include "csub3/drop.ma".
-
-include "getl/clear.ma".
-
-theorem csub3_getl_abbr:
- \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall 
-(n: nat).((getl n c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csub3 g 
-c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n 
-c2 (CHead d2 (Bind Abbr) u)))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda 
-(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abbr) u))).(let H0 \def 
-(getl_gen_all c1 (CHead d1 (Bind Abbr) u) n H) in (ex2_ind C (\lambda (e: 
-C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u))) 
-(\forall (c2: C).((csub3 g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))) (\lambda (x: 
-C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind 
-Abbr) u))).(C_ind (\lambda (c: C).((drop n O c1 c) \to ((clear c (CHead d1 
-(Bind Abbr) u)) \to (\forall (c2: C).((csub3 g c1 c2) \to (ex2 C (\lambda 
-(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) 
-u))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 (CSort n0))).(\lambda 
-(H4: (clear (CSort n0) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1 
-(Bind Abbr) u) n0 H4 (\forall (c2: C).((csub3 g c1 c2) \to (ex2 C (\lambda 
-(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) 
-u)))))))))) (\lambda (x0: C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0 
-(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csub3 g c1 c2) \to (ex2 C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind 
-Abbr) u)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: (drop n O c1 
-(CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 (Bind Abbr) 
-u))).(K_ind (\lambda (k0: K).((drop n O c1 (CHead x0 k0 t)) \to ((clear 
-(CHead x0 k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csub3 g c1 
-c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 
-(CHead d2 (Bind Abbr) u))))))))) (\lambda (b: B).(\lambda (H5: (drop n O c1 
-(CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 
-(Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C 
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) 
-\Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) 
-(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def 
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with 
-[(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K 
-return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
-\Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) 
-(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def 
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind 
-Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) 
-t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda 
-(c2: C).(\lambda (H12: (csub3 g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda 
-(t0: T).(drop n O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def 
-(eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) u))) H13 Abbr 
-H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1 (CHead c 
-(Bind Abbr) u))) H14 d1 H11) in (ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))) 
-(\lambda (x1: C).(\lambda (H16: (csub3 g d1 x1)).(\lambda (H17: (drop n O c2 
-(CHead x1 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) x1 H16 (getl_intro n 
-c2 (CHead x1 (Bind Abbr) u) (CHead x1 (Bind Abbr) u) H17 (clear_bind Abbr x1 
-u)))))) (csub3_drop_abbr g n c1 c2 H12 d1 u H15)))))))))) H8)) H7))))) 
-(\lambda (f: F).(\lambda (H5: (drop n O c1 (CHead x0 (Flat f) t))).(\lambda 
-(H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr) u))).(let H7 \def H5 
-in (unintro C c1 (\lambda (c: C).((drop n O c (CHead x0 (Flat f) t)) \to 
-(\forall (c2: C).((csub3 g c c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))))) (nat_ind (\lambda 
-(n0: nat).(\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall 
-(c2: C).((csub3 g x1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u))))))))) (\lambda (x1: 
-C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: 
-C).(\lambda (H9: (csub3 g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: 
-C).(csub3 g c c2)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat 
-f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abbr) u) 
-(clear_gen_flat f x0 (CHead d1 (Bind Abbr) u) t H6) f t) in (let H11 \def 
-(csub3_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u) 
-H_y) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead d1 (Bind Abbr) u) e2)) 
-(\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x2: 
-C).(\lambda (H12: (csub3 g (CHead d1 (Bind Abbr) u) x2)).(\lambda (H13: 
-(clear c2 x2)).(let H14 \def (csub3_gen_abbr g d1 x2 u H12) in (ex2_ind C 
-(\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csub3 
-g d1 e2)) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x3: C).(\lambda (H15: (eq C x2 
-(CHead x3 (Bind Abbr) u))).(\lambda (H16: (csub3 g d1 x3)).(let H17 \def 
-(eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) u) H15) 
-in (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 
-(CHead d2 (Bind Abbr) u))) x3 H16 (getl_intro O c2 (CHead x3 (Bind Abbr) u) 
-c2 (drop_refl c2) H17)))))) H14))))) H11)))))))) (\lambda (n0: nat).(\lambda 
-(H8: ((\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall 
-(c2: C).((csub3 g x1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u)))))))))).(\lambda (x1: 
-C).(\lambda (H9: (drop (S n0) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: 
-C).(\lambda (H10: (csub3 g x1 c2)).(let H11 \def (drop_clear x1 (CHead x0 
-(Flat f) t) n0 H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: 
-C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) (\lambda (_: 
-B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead x0 (Flat f) t))))) 
-(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 
-(CHead d2 (Bind Abbr) u)))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: 
-T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0 
-O x3 (CHead x0 (Flat f) t))).(let H14 \def (csub3_clear_conf g x1 c2 H10 
-(CHead x3 (Bind x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead 
-x3 (Bind x2) x4) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) 
-u)))) (\lambda (x5: C).(\lambda (H15: (csub3 g (CHead x3 (Bind x2) x4) 
-x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def (csub3_gen_bind g x2 x3 x5 
-x4 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
-T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csub3 g x3 e2)))) (ex2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda 
-(x6: B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 
-(Bind x6) x8))).(\lambda (H19: (csub3 g x3 x7)).(let H20 \def (eq_ind C x5 
-(\lambda (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 
-\def (H8 x3 H13 x7 H19) in (ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) 
-u)))) (\lambda (x9: C).(\lambda (H22: (csub3 g d1 x9)).(\lambda (H23: (getl 
-n0 x7 (CHead x9 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u))) x9 H22 
-(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) u) n0 H23))))) 
-H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))) k H3 H4))))))) x H1 
-H2)))) H0))))))).
-
-theorem csub3_getl_abst:
- \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (t: T).(\forall 
-(n: nat).((getl n c1 (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csub3 g 
-c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t))))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (t: T).(\lambda 
-(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abst) t))).(let H0 \def 
-(getl_gen_all c1 (CHead d1 (Bind Abst) t) n H) in (ex2_ind C (\lambda (e: 
-C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) t))) 
-(\forall (c2: C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t))))))) (\lambda (x: C).(\lambda (H1: (drop n O c1 
-x)).(\lambda (H2: (clear x (CHead d1 (Bind Abst) t))).(C_ind (\lambda (c: 
-C).((drop n O c1 c) \to ((clear c (CHead d1 (Bind Abst) t)) \to (\forall (c2: 
-C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t)))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 (CSort 
-n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abst) 
-t))).(clear_gen_sort (CHead d1 (Bind Abst) t) n0 H4 (\forall (c2: C).((csub3 
-g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t))))))))))) (\lambda (x0: C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0 
-(CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csub3 g c1 c2) \to (or (ex2 
-C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 
-(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) 
-(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))).(\lambda (k: 
-K).(\lambda (t0: T).(\lambda (H3: (drop n O c1 (CHead x0 k t0))).(\lambda 
-(H4: (clear (CHead x0 k t0) (CHead d1 (Bind Abst) t))).(K_ind (\lambda (k0: 
-K).((drop n O c1 (CHead x0 k0 t0)) \to ((clear (CHead x0 k0 t0) (CHead d1 
-(Bind Abst) t)) \to (\forall (c2: C).((csub3 g c1 c2) \to (or (ex2 C (\lambda 
-(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) 
-t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
-(d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
-C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (b: B).(\lambda (H5: 
-(drop n O c1 (CHead x0 (Bind b) t0))).(\lambda (H6: (clear (CHead x0 (Bind b) 
-t0) (CHead d1 (Bind Abst) t))).(let H7 \def (f_equal C C (\lambda (e: 
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | 
-(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) t) (CHead x0 (Bind b) 
-t0) (clear_gen_bind b x0 (CHead d1 (Bind Abst) t) t0 H6)) in ((let H8 \def 
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with 
-[(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 in K 
-return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
-\Rightarrow Abst])])) (CHead d1 (Bind Abst) t) (CHead x0 (Bind b) t0) 
-(clear_gen_bind b x0 (CHead d1 (Bind Abst) t) t0 H6)) in ((let H9 \def 
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow t | (CHead _ _ t1) \Rightarrow t1])) (CHead d1 (Bind 
-Abst) t) (CHead x0 (Bind b) t0) (clear_gen_bind b x0 (CHead d1 (Bind Abst) t) 
-t0 H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 
-x0)).(\lambda (c2: C).(\lambda (H12: (csub3 g c1 c2)).(let H13 \def (eq_ind_r 
-T t0 (\lambda (t1: T).(drop n O c1 (CHead x0 (Bind b) t1))) H5 t H9) in (let 
-H14 \def (eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) t))) 
-H13 Abst H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1 
-(CHead c (Bind Abst) t))) H14 d1 H11) in (or_ind (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t)))) 
-(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
-C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
-C).(\lambda (u: T).(ty3 g d2 u t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t))))) (\lambda (H16: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t))))).(ex2_ind C 
-(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 
-(Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t))))) (\lambda (x1: C).(\lambda (H17: (csub3 g d1 x1)).(\lambda (H18: (drop 
-n O c2 (CHead x1 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g 
-d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(getl n c2 (CHead d2 (Bind Abst) t))) x1 H17 (getl_intro n c2 (CHead 
-x1 (Bind Abst) t) (CHead x1 (Bind Abst) t) H18 (clear_bind Abst x1 t))))))) 
-H16)) (\lambda (H16: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) 
-u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t))))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (H17: (csub3 g d1 
-x1)).(\lambda (H18: (drop n O c2 (CHead x1 (Bind Abbr) x2))).(\lambda (H19: 
-(ty3 g x1 x2 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) 
-(\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) 
-(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x1 x2 H17 (getl_intro n c2 
-(CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2) H18 (clear_bind Abbr x1 
-x2)) H19))))))) H16)) (csub3_drop_abst g n c1 c2 H12 d1 t H15)))))))))) H8)) 
-H7))))) (\lambda (f: F).(\lambda (H5: (drop n O c1 (CHead x0 (Flat f) 
-t0))).(\lambda (H6: (clear (CHead x0 (Flat f) t0) (CHead d1 (Bind Abst) 
-t))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop n O c (CHead x0 
-(Flat f) t0)) \to (\forall (c2: C).((csub3 g c c2) \to (or (ex2 C (\lambda 
-(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) 
-t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
-(d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
-C).(\lambda (u: T).(ty3 g d2 u t))))))))) (nat_ind (\lambda (n0: 
-nat).(\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t0)) \to (\forall 
-(c2: C).((csub3 g x1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda 
-(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: 
-T).(getl n0 c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t)))))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead 
-x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H9: (csub3 g x1 c2)).(let H10 
-\def (eq_ind C x1 (\lambda (c: C).(csub3 g c c2)) H9 (CHead x0 (Flat f) t0) 
-(drop_gen_refl x1 (CHead x0 (Flat f) t0) H8)) in (let H_y \def (clear_flat x0 
-(CHead d1 (Bind Abst) t) (clear_gen_flat f x0 (CHead d1 (Bind Abst) t) t0 H6) 
-f t0) in (let H11 \def (csub3_clear_conf g (CHead x0 (Flat f) t0) c2 H10 
-(CHead d1 (Bind Abst) t) H_y) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead 
-d1 (Bind Abst) t) e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda 
-(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) 
-t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda 
-(d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
-C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x2: C).(\lambda (H12: (csub3 
-g (CHead d1 (Bind Abst) t) x2)).(\lambda (H13: (clear c2 x2)).(let H14 \def 
-(csub3_gen_abst g d1 x2 t H12) in (or_ind (ex2 C (\lambda (e2: C).(eq C x2 
-(CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2))) (ex3_2 C T 
-(\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 (Bind Abbr) v2)))) 
-(\lambda (e2: C).(\lambda (_: T).(csub3 g d1 e2))) (\lambda (e2: C).(\lambda 
-(v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda 
-(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: 
-T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t))))) (\lambda (H15: (ex2 C (\lambda (e2: C).(eq C x2 (CHead 
-e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2)))).(ex2_ind C (\lambda 
-(e2: C).(eq C x2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2)) 
-(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 
-(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 
-(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
-(\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst) t))).(\lambda 
-(H17: (csub3 g d1 x3)).(let H18 \def (eq_ind C x2 (\lambda (c: C).(clear c2 
-c)) H13 (CHead x3 (Bind Abst) t) H16) in (or_introl (ex2 C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) 
-(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: 
-C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
-C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g 
-d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))) x3 H17 
-(getl_intro O c2 (CHead x3 (Bind Abst) t) c2 (drop_refl c2) H18))))))) H15)) 
-(\lambda (H15: (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead 
-e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g d1 e2))) 
-(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))))).(ex3_2_ind C T (\lambda 
-(e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
-C).(\lambda (_: T).(csub3 g d1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g 
-e2 v2 t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 
-(Bind Abbr) x4))).(\lambda (H17: (csub3 g d1 x3)).(\lambda (H18: (ty3 g x3 x4 
-t)).(let H19 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 
-(Bind Abbr) x4) H16) in (or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda 
-(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: 
-T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 
-g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) 
-u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x3 x4 H17 (getl_intro 
-O c2 (CHead x3 (Bind Abbr) x4) c2 (drop_refl c2) H19) H18)))))))) H15)) 
-H14))))) H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x1: 
-C).((drop n0 O x1 (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csub3 g x1 
-c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl 
-n0 c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 
-(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n0) O x1 (CHead x0 (Flat 
-f) t0))).(\lambda (c2: C).(\lambda (H10: (csub3 g x1 c2)).(let H11 \def 
-(drop_clear x1 (CHead x0 (Flat f) t0) n0 H9) in (ex2_3_ind B C T (\lambda (b: 
-B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) 
-(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead x0 (Flat 
-f) t0))))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl 
-(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
-d2 u t))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: 
-(clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0 O x3 (CHead x0 
-(Flat f) t0))).(let H14 \def (csub3_clear_conf g x1 c2 H10 (CHead x3 (Bind 
-x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead x3 (Bind x2) x4) 
-e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda 
-(u: T).(ty3 g d2 u t))))) (\lambda (x5: C).(\lambda (H15: (csub3 g (CHead x3 
-(Bind x2) x4) x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def 
-(csub3_gen_bind g x2 x3 x5 x4 H15) in (ex2_3_ind B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g x3 e2)))) (or (ex2 
-C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead 
-d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 
-d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) 
-u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x6: 
-B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind 
-x6) x8))).(\lambda (H19: (csub3 g x3 x7)).(let H20 \def (eq_ind C x5 (\lambda 
-(c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 
-H13 x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(getl n0 x7 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl 
-n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 
-u t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl 
-(S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda 
-(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 
-(CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: 
-C).(getl n0 x7 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: 
-C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abst) t))) 
-(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 
-(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead 
-d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
-(\lambda (x9: C).(\lambda (H23: (csub3 g d1 x9)).(\lambda (H24: (getl n0 x7 
-(CHead x9 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 
-d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda 
-(u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t))) x9 H23 
-(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abst) t) n0 H24)))))) H22)) 
-(\lambda (H22: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) 
-(\lambda (d2: C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) 
-(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda 
-(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: 
-T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
-T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda 
-(d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl 
-(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
-d2 u t))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (H23: (csub3 g d1 
-x9)).(\lambda (H24: (getl n0 x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H25: 
-(ty3 g x9 x10 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) 
-(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda 
-(u: T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: 
-T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead 
-d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x9 x10 
-H23 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) x10) n0 H24) 
-H25))))))) H22)) H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))) k H3 
-H4))))))) x H1 H2)))) H0))))))).
-
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/pc3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/pc3.ma
deleted file mode 100644
index 980672598..000000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/pc3.ma
+++ /dev/null
@@ -1,58 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/pc3".
-
-include "csub3/getl.ma".
-
-include "pc3/left.ma".
-
-theorem csub3_pr2:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 
-t1 t2) \to (\forall (c2: C).((csub3 g c1 c2) \to (pr2 c2 t1 t2)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: 
-T).(\forall (c2: C).((csub3 g c c2) \to (pr2 c2 t t0)))))) (\lambda (c: 
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c2: 
-C).(\lambda (_: (csub3 g c c2)).(pr2_free c2 t3 t4 H0))))))) (\lambda (c: 
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c 
-(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
-(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2: 
-C).(\lambda (H3: (csub3 g c c2)).(let H4 \def (csub3_getl_abbr g c d u i H0 
-c2 H3) in (ex2_ind C (\lambda (d2: C).(csub3 g d d2)) (\lambda (d2: C).(getl 
-i c2 (CHead d2 (Bind Abbr) u))) (pr2 c2 t3 t) (\lambda (x: C).(\lambda (_: 
-(csub3 g d x)).(\lambda (H6: (getl i c2 (CHead x (Bind Abbr) u))).(pr2_delta 
-c2 x u i H6 t3 t4 H1 t H2)))) H4)))))))))))))) c1 t1 t2 H))))).
-
-theorem csub3_pc3:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1 
-t1 t2) \to (\forall (c2: C).((csub3 g c1 c2) \to (pc3 c2 t1 t2)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (pc3 c1 t1 t2)).(pc3_ind_left c1 (\lambda (t: T).(\lambda (t0: 
-T).(\forall (c2: C).((csub3 g c1 c2) \to (pc3 c2 t t0))))) (\lambda (t: 
-T).(\lambda (c2: C).(\lambda (_: (csub3 g c1 c2)).(pc3_refl c2 t)))) (\lambda 
-(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 t0 t3)).(\lambda (t4: 
-T).(\lambda (_: (pc3 c1 t3 t4)).(\lambda (H2: ((\forall (c2: C).((csub3 g c1 
-c2) \to (pc3 c2 t3 t4))))).(\lambda (c2: C).(\lambda (H3: (csub3 g c1 
-c2)).(pc3_pr2_u c2 t3 t0 (csub3_pr2 g c1 t0 t3 H0 c2 H3) t4 (H2 c2 
-H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 t0 
-t3)).(\lambda (t4: T).(\lambda (_: (pc3 c1 t0 t4)).(\lambda (H2: ((\forall 
-(c2: C).((csub3 g c1 c2) \to (pc3 c2 t0 t4))))).(\lambda (c2: C).(\lambda 
-(H3: (csub3 g c1 c2)).(pc3_t t0 c2 t3 (pc3_pr2_x c2 t3 t0 (csub3_pr2 g c1 t0 
-t3 H0 c2 H3)) t4 (H2 c2 H3)))))))))) t1 t2 H))))).
-
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/props.ma
deleted file mode 100644
index 3258cda57..000000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/props.ma
+++ /dev/null
@@ -1,27 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/props".
-
-include "csub3/defs.ma".
-
-theorem csub3_refl:
- \forall (g: G).(\forall (c: C).(csub3 g c c))
-\def
- \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csub3 g c0 c0)) 
-(\lambda (n: nat).(csub3_sort g n)) (\lambda (c0: C).(\lambda (H: (csub3 g c0 
-c0)).(\lambda (k: K).(\lambda (t: T).(csub3_head g c0 c0 H k t))))) c)).
-
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/ty3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/ty3.ma
deleted file mode 100644
index 26ad14f35..000000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/ty3.ma
+++ /dev/null
@@ -1,99 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/ty3".
-
-include "csub3/pc3.ma".
-
-include "csub3/props.ma".
-
-theorem csub3_ty3:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 
-t1 t2) \to (\forall (c2: C).((csub3 g c1 c2) \to (ty3 g c2 t1 t2)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda 
-(t0: T).(\forall (c2: C).((csub3 g c c2) \to (ty3 g c2 t t0)))))) (\lambda 
-(c: C).(\lambda (t0: T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda 
-(H1: ((\forall (c2: C).((csub3 g c c2) \to (ty3 g c2 t0 t))))).(\lambda (u: 
-T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2: 
-C).((csub3 g c c2) \to (ty3 g c2 u t3))))).(\lambda (H4: (pc3 c t3 
-t0)).(\lambda (c2: C).(\lambda (H5: (csub3 g c c2)).(ty3_conv g c2 t0 t (H1 
-c2 H5) u t3 (H3 c2 H5) (csub3_pc3 g c t3 t0 H4 c2 H5)))))))))))))) (\lambda 
-(c: C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (csub3 g c 
-c2)).(ty3_sort g c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda 
-(t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((csub3 g 
-d c2) \to (ty3 g c2 u t))))).(\lambda (c2: C).(\lambda (H3: (csub3 g c 
-c2)).(let H4 \def (csub3_getl_abbr g c d u n H0 c2 H3) in (ex2_ind C (\lambda 
-(d2: C).(csub3 g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) 
-u))) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x: C).(\lambda (H5: 
-(csub3 g d x)).(\lambda (H6: (getl n c2 (CHead x (Bind Abbr) u))).(ty3_abbr g 
-n c2 x u H6 t (H2 x H5))))) H4)))))))))))) (\lambda (n: nat).(\lambda (c: 
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind 
-Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: 
-((\forall (c2: C).((csub3 g d c2) \to (ty3 g c2 u t))))).(\lambda (c2: 
-C).(\lambda (H3: (csub3 g c c2)).(let H4 \def (csub3_getl_abst g c d u n H0 
-c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(csub3 g d d2)) (\lambda (d2: 
-C).(getl n c2 (CHead d2 (Bind Abst) u)))) (ex3_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 
-u0 u)))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (H5: (ex2 C (\lambda 
-(d2: C).(csub3 g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) 
-u))))).(ex2_ind C (\lambda (d2: C).(csub3 g d d2)) (\lambda (d2: C).(getl n 
-c2 (CHead d2 (Bind Abst) u))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda 
-(x: C).(\lambda (H6: (csub3 g d x)).(\lambda (H7: (getl n c2 (CHead x (Bind 
-Abst) u))).(ty3_abst g n c2 x u H7 t (H2 x H6))))) H5)) (\lambda (H5: (ex3_2 
-C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d d2))) (\lambda (d2: 
-C).(\lambda (u0: T).(getl n c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
-C).(\lambda (u0: T).(ty3 g d2 u0 u))))).(ex3_2_ind C T (\lambda (d2: 
-C).(\lambda (_: T).(csub3 g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 
-u0 u))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x0: C).(\lambda (x1: 
-T).(\lambda (_: (csub3 g d x0)).(\lambda (H7: (getl n c2 (CHead x0 (Bind 
-Abbr) x1))).(\lambda (H8: (ty3 g x0 x1 u)).(ty3_abbr g n c2 x0 x1 H7 u 
-H8)))))) H5)) H4)))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t: 
-T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c2: C).((csub3 g c 
-c2) \to (ty3 g c2 u t))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t0 t3)).(\lambda (H3: ((\forall 
-(c2: C).((csub3 g (CHead c (Bind b) u) c2) \to (ty3 g c2 t0 t3))))).(\lambda 
-(t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t3 t4)).(\lambda (H5: 
-((\forall (c2: C).((csub3 g (CHead c (Bind b) u) c2) \to (ty3 g c2 t3 
-t4))))).(\lambda (c2: C).(\lambda (H6: (csub3 g c c2)).(ty3_bind g c2 u t (H1 
-c2 H6) b t0 t3 (H3 (CHead c2 (Bind b) u) (csub3_head g c c2 H6 (Bind b) u)) 
-t4 (H5 (CHead c2 (Bind b) u) (csub3_head g c c2 H6 (Bind b) 
-u)))))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda 
-(_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((csub3 g c c2) \to (ty3 g 
-c2 w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead 
-(Bind Abst) u t))).(\lambda (H3: ((\forall (c2: C).((csub3 g c c2) \to (ty3 g 
-c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (csub3 g c 
-c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c: 
-C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t0 t3)).(\lambda 
-(H1: ((\forall (c2: C).((csub3 g c c2) \to (ty3 g c2 t0 t3))))).(\lambda (t4: 
-T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3: ((\forall (c2: C).((csub3 g c 
-c2) \to (ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (csub3 g c 
-c2)).(ty3_cast g c2 t0 t3 (H1 c2 H4) t4 (H3 c2 H4)))))))))))) c1 t1 t2 H))))).
-
-theorem csub3_ty3_ld:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (v: T).((ty3 g c u 
-v) \to (\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind Abst) v) t1 
-t2) \to (ty3 g (CHead c (Bind Abbr) u) t1 t2))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (H: 
-(ty3 g c u v)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead 
-c (Bind Abst) v) t1 t2)).(csub3_ty3 g (CHead c (Bind Abst) v) t1 t2 H0 (CHead 
-c (Bind Abbr) u) (csub3_abst g c c (csub3_refl g c) u v H))))))))).
-
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/arity.ma
index acd564afb..fe9436497 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/arity.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/arity.ma
@@ -103,10 +103,83 @@ g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2:
 C).(\lambda (H3: (csuba g c c2)).(arity_repl g c2 t0 a1 (H1 c2 H3) a2 
 H2)))))))))) c1 t a H))))).
 
-axiom csuba_arity_rev:
+theorem csuba_arity_rev:
  \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 
 t a) \to (\forall (c2: C).((csuba g c2 c1) \to (arity g c2 t a)))))))
-.
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
+(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: 
+A).(\forall (c2: C).((csuba g c2 c) \to (arity g c2 t0 a0)))))) (\lambda (c: 
+C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (csuba g c2 
+c)).(arity_sort g c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) 
+u))).(\lambda (a0: A).(\lambda (H1: (arity g d u a0)).(\lambda (H2: ((\forall 
+(c2: C).((csuba g c2 d) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda 
+(H3: (csuba g c2 c)).(let H4 \def (csuba_getl_abbr_rev g c d u i H0 c2 H3) in 
+(or_ind (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) 
+(\lambda (d2: C).(csuba g d2 d))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 (asucc g a1))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u a1))))) (arity g c2 
+(TLRef i) a0) (\lambda (H5: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2: 
+C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)) 
+(arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H6: (getl i c2 (CHead x 
+(Bind Abbr) u))).(\lambda (H7: (csuba g x d)).(arity_abbr g c2 x u i H6 a0 
+(H2 x H7))))) H5)) (\lambda (H5: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 (asucc g a1))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u a1)))))).(ex4_3_ind C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 
+(asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d 
+u a1)))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
+(x2: A).(\lambda (H6: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (_: 
+(csuba g x0 d)).(\lambda (H8: (arity g x0 x1 (asucc g x2))).(\lambda (H9: 
+(arity g d u x2)).(arity_repl g c2 (TLRef i) x2 (arity_abst g c2 x0 x1 i H6 
+x2 H8) a0 (arity_mono g d u x2 H9 a0 H1))))))))) H5)) H4)))))))))))) (\lambda 
+(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl 
+i c (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u 
+(asucc g a0))).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to (arity g 
+c2 u (asucc g a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2 c)).(let H4 
+\def (csuba_getl_abst_rev g c d u i H0 c2 H3) in (ex2_ind C (\lambda (d2: 
+C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d)) 
+(arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x 
+(Bind Abst) u))).(\lambda (H6: (csuba g x d)).(arity_abst g c2 x u i H5 a0 
+(H2 x H6))))) H4)))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b 
+Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity 
+g c u a1)).(\lambda (H2: ((\forall (c2: C).((csuba g c2 c) \to (arity g c2 u 
+a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c 
+(Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c2: C).((csuba g c2 (CHead c 
+(Bind b) u)) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H5: (csuba 
+g c2 c)).(arity_bind g b H0 c2 u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) 
+u) (csuba_head g c2 c H5 (Bind b) u)))))))))))))))) (\lambda (c: C).(\lambda 
+(u: T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda 
+(H1: ((\forall (c2: C).((csuba g c2 c) \to (arity g c2 u (asucc g 
+a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c 
+(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c2: C).((csuba g c2 (CHead c 
+(Bind Abst) u)) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: 
+(csuba g c2 c)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind 
+Abst) u) (csuba_head g c2 c H4 (Bind Abst) u)))))))))))))) (\lambda (c: 
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda 
+(H1: ((\forall (c2: C).((csuba g c2 c) \to (arity g c2 u a1))))).(\lambda 
+(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda 
+(H3: ((\forall (c2: C).((csuba g c2 c) \to (arity g c2 t0 (AHead a1 
+a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2 c)).(arity_appl g c2 u a1 
+(H1 c2 H4) t0 a2 (H3 c2 H4))))))))))))) (\lambda (c: C).(\lambda (u: 
+T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: 
+((\forall (c2: C).((csuba g c2 c) \to (arity g c2 u (asucc g 
+a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3: 
+((\forall (c2: C).((csuba g c2 c) \to (arity g c2 t0 a0))))).(\lambda (c2: 
+C).(\lambda (H4: (csuba g c2 c)).(arity_cast g c2 u a0 (H1 c2 H4) t0 (H3 c2 
+H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: 
+(arity g c t0 a1)).(\lambda (H1: ((\forall (c2: C).((csuba g c2 c) \to (arity 
+g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: 
+C).(\lambda (H3: (csuba g c2 c)).(arity_repl g c2 t0 a1 (H1 c2 H3) a2 
+H2)))))))))) c1 t a H))))).
 
 theorem arity_appls_appl:
  \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (a1: A).((arity g c 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/clear.ma
index a7eb560d7..f5a06a36f 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/clear.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/clear.ma
@@ -61,3 +61,44 @@ e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind Abst) t) e2))
 u) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abbr c4 u)) e1 
 (clear_gen_bind Abst c3 e1 t H4))))))))))))) c1 c2 H)))).
 
+theorem csuba_clear_trans:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c2 c1) \to 
+(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) 
+(\lambda (e2: C).(clear c2 e2))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c2 
+c1)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear 
+c0 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c 
+e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) 
+e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e2 e1)) 
+(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: 
+C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c4 
+e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c3 
+e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: 
+(clear (CHead c4 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c4 k0 u) 
+e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear 
+(CHead c3 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c4 (Bind 
+b) u) e1)).(eq_ind_r C (CHead c4 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda 
+(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind b) u) e2)))) 
+(ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind b) u))) (\lambda 
+(e2: C).(clear (CHead c3 (Bind b) u) e2)) (CHead c3 (Bind b) u) (csuba_head g 
+c3 c4 H0 (Bind b) u) (clear_bind b c3 u)) e1 (clear_gen_bind b c4 e1 u H3)))) 
+(\lambda (f: F).(\lambda (H3: (clear (CHead c4 (Flat f) u) e1)).(let H4 \def 
+(H1 e1 (clear_gen_flat f c4 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g 
+e2 e1)) (\lambda (e2: C).(clear c3 e2)) (ex2 C (\lambda (e2: C).(csuba g e2 
+e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f) u) e2))) (\lambda (x: 
+C).(\lambda (H5: (csuba g x e1)).(\lambda (H6: (clear c3 x)).(ex_intro2 C 
+(\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f) 
+u) e2)) x H5 (clear_flat c3 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: 
+C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall 
+(e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda 
+(e2: C).(clear c3 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H2: 
+(arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u 
+a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c4 (Bind Abbr) u) 
+e1)).(eq_ind_r C (CHead c4 (Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda 
+(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) 
+e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind Abbr) u))) 
+(\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) e2)) (CHead c3 (Bind Abst) 
+t) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abst c3 t)) e1 
+(clear_gen_bind Abbr c4 e1 u H4))))))))))))) c2 c1 H)))).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/defs.ma
index d108b9bba..857c2ab89 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/defs.ma
@@ -18,7 +18,7 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/defs".
 
 include "arity/defs.ma".
 
-inductive csuba (g:G): C \to (C \to Prop) \def
+inductive csuba (g: G): C \to (C \to Prop) \def
 | csuba_sort: \forall (n: nat).(csuba g (CSort n) (CSort n))
 | csuba_head: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall 
 (k: K).(\forall (u: T).(csuba g (CHead c1 k u) (CHead c2 k u))))))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/drop.ma
index ac8e3fae8..e145c6e11 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/drop.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/drop.ma
@@ -813,3 +813,796 @@ A).(arity g d2 u2 a)))) x2 x3 x4 (drop_drop (Flat f) n x0 (CHead x2 (Bind
 Abbr) x3) H10 x1) H11 H12 H13))))))))) H9)) H8)) c2 H6))))) H5)))))) k H2 
 (drop_gen_drop k c (CHead d1 (Bind Abst) u1) t n H1)))))))))))) c1)))) i).
 
+theorem csuba_drop_abst_rev:
+ \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i 
+O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g 
+c2 c1) \to (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1))))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: 
+C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: 
+G).(\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda (d2: C).(drop n O c2 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))))))))) 
+(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 
+(CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: 
+(csuba g c2 c1)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c2 c)) H0 
+(CHead d1 (Bind Abst) u) (drop_gen_refl c1 (CHead d1 (Bind Abst) u) H)) in 
+(let H_x \def (csuba_gen_abst_rev g d1 c2 u H1) in (let H2 \def H_x in 
+(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x: C).(\lambda (H3: 
+(eq C c2 (CHead x (Bind Abst) u))).(\lambda (H4: (csuba g x d1)).(eq_ind_r C 
+(CHead x (Bind Abst) u) (\lambda (c: C).(ex2 C (\lambda (d2: C).(drop O O c 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (ex_intro2 C 
+(\lambda (d2: C).(drop O O (CHead x (Bind Abst) u) (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1)) x (drop_refl (CHead x (Bind Abst) u)) H4) 
+c2 H3)))) H2))))))))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1: 
+C).(\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abst) u)) 
+\to (\forall (g: G).(\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda 
+(d2: C).(drop n O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
+d1)))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: 
+C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall 
+(g: G).(\forall (c2: C).((csuba g c2 c) \to (ex2 C (\lambda (d2: C).(drop (S 
+n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))))) 
+(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n) 
+O (CSort n0) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2: 
+C).(\lambda (_: (csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind 
+Abst) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (ex2 C (\lambda (d2: 
+C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
+d1))) (\lambda (H2: (eq C (CHead d1 (Bind Abst) u) (CSort n0))).(\lambda (_: 
+(eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (match H2 in eq 
+return (\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c (CSort n0)) \to (ex2 
+C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1)))))) with [refl_equal \Rightarrow (\lambda (H5: (eq C 
+(CHead d1 (Bind Abst) u) (CSort n0))).(let H6 \def (eq_ind C (CHead d1 (Bind 
+Abst) u) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with 
+[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n0) 
+H5) in (False_ind (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) H6)))]) in (H5 (refl_equal C 
+(CSort n0))))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u) 
+H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u: 
+T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall 
+(c2: C).((csuba g c2 c) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead 
+d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))))))))).(\lambda (k: 
+K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n) 
+O (CHead c k t) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2: 
+C).(\lambda (H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: K).((csuba 
+g c2 (CHead c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abst) u)) \to 
+(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda 
+(d2: C).(csuba g d2 d1)))))) (\lambda (b: B).(\lambda (H3: (csuba g c2 (CHead 
+c (Bind b) t))).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abst) 
+u))).(B_ind (\lambda (b0: B).((csuba g c2 (CHead c (Bind b0) t)) \to ((drop 
+(r (Bind b0) n) O c (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: 
+C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
+d1)))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) t))).(\lambda (H6: 
+(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abst) u))).(let H_x \def 
+(csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C 
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba 
+g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq 
+C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g c t a))))) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (H8: (ex2 C 
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba 
+g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) 
+(\lambda (d2: C).(csuba g d2 c)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x: 
+C).(\lambda (H9: (eq C c2 (CHead x (Bind Abbr) t))).(\lambda (H10: (csuba g x 
+c)).(eq_ind_r C (CHead x (Bind Abbr) t) (\lambda (c0: C).(ex2 C (\lambda (d2: 
+C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
+d1)))) (let H11 \def (H c d1 u H6 g x H10) in (ex2_ind C (\lambda (d2: 
+C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) 
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x0: C).(\lambda (H12: 
+(drop n O x (CHead x0 (Bind Abst) u))).(\lambda (H13: (csuba g x0 d1)).(let 
+H14 \def (refl_equal nat (r (Bind Abst) n)) in (let H15 \def (eq_ind nat n 
+(\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abst) u))) H12 (r (Bind Abst) 
+n) H14) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) 
+t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop 
+(Bind Abbr) n x (CHead x0 (Bind Abst) u) H15 t) H13)))))) H11)) c2 H9)))) 
+H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) (ex2 C 
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: 
+A).(\lambda (H9: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H10: (csuba g 
+x0 c)).(\lambda (_: (arity g x0 x1 (asucc g x2))).(\lambda (_: (arity g c t 
+x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c0: C).(ex2 C (\lambda 
+(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g 
+d2 d1)))) (let H13 \def (H c d1 u H6 g x0 H10) in (ex2_ind C (\lambda (d2: 
+C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) 
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x: C).(\lambda 
+(H14: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H15: (csuba g x 
+d1)).(let H16 \def (refl_equal nat (r (Bind Abst) n)) in (let H17 \def 
+(eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x (Bind Abst) u))) H14 
+(r (Bind Abst) n) H16) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead 
+x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
+d1)) x (drop_drop (Bind Abst) n x0 (CHead x (Bind Abst) u) H17 x1) H15)))))) 
+H13)) c2 H9)))))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind 
+Abst) t))).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) 
+u))).(let H_x \def (csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in 
+(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: 
+C).(csuba g d2 c)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x: C).(\lambda (H8: 
+(eq C c2 (CHead x (Bind Abst) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C 
+(CHead x (Bind Abst) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) 
+O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H10 
+\def (H c d1 u H6 g x H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead 
+d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda 
+(d2: C).(csuba g d2 d1))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead 
+x0 (Bind Abst) u))).(\lambda (H12: (csuba g x0 d1)).(let H13 \def (refl_equal 
+nat (r (Bind Abst) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: 
+nat).(drop n0 O x (CHead x0 (Bind Abst) u))) H11 (r (Bind Abst) n) H13) in 
+(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop (Bind Abst) 
+n x (CHead x0 (Bind Abst) u) H14 t) H12)))))) H10)) c2 H8)))) H7))))) 
+(\lambda (H5: (csuba g c2 (CHead c (Bind Void) t))).(\lambda (H6: (drop (r 
+(Bind Void) n) O c (CHead d1 (Bind Abst) u))).(let H_x \def 
+(csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda 
+(d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: C).(csuba g d2 c)) 
+(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda 
+(d2: C).(csuba g d2 d1))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x 
+(Bind Void) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C (CHead x (Bind 
+Void) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H10 \def (H c d1 u 
+H6 g x H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) 
+u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(drop (S n) O 
+(CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g 
+d2 d1))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead x0 (Bind Abst) 
+u))).(\lambda (H12: (csuba g x0 d1)).(let H13 \def (refl_equal nat (r (Bind 
+Abst) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x 
+(CHead x0 (Bind Abst) u))) H11 (r (Bind Abst) n) H13) in (ex_intro2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) 
+u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop (Bind Void) n x (CHead 
+x0 (Bind Abst) u) H14 t) H12)))))) H10)) c2 H8)))) H7))))) b H3 H4)))) 
+(\lambda (f: F).(\lambda (H3: (csuba g c2 (CHead c (Flat f) t))).(\lambda 
+(H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abst) u))).(let H_x \def 
+(csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T 
+(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 c))) (ex2 C (\lambda (d2: C).(drop (S n) 
+O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda 
+(x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) 
+x1))).(\lambda (H7: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Flat f) x1) 
+(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H8 \def (H0 d1 u H4 g x0 
+H7) in (ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) 
+u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(drop (S n) O 
+(CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g 
+d2 d1))) (\lambda (x: C).(\lambda (H9: (drop (S n) O x0 (CHead x (Bind Abst) 
+u))).(\lambda (H10: (csuba g x d1)).(ex_intro2 C (\lambda (d2: C).(drop (S n) 
+O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g 
+d2 d1)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u) H9 x1) H10)))) 
+H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abst) u) t n 
+H1)))))))))))) c1)))) i).
+
+theorem csuba_drop_abbr_rev:
+ \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i 
+O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba 
+g c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+a)))))))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: 
+C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: 
+G).(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop n 
+O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: 
+T).(\lambda (H: (drop O O c1 (CHead d1 (Bind Abbr) u1))).(\lambda (g: 
+G).(\lambda (c2: C).(\lambda (H0: (csuba g c2 c1)).(let H1 \def (eq_ind C c1 
+(\lambda (c: C).(csuba g c2 c)) H0 (CHead d1 (Bind Abbr) u1) (drop_gen_refl 
+c1 (CHead d1 (Bind Abbr) u1) H)) in (let H_x \def (csuba_gen_abbr_rev g d1 c2 
+u1 H1) in (let H2 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead 
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H3: 
+(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: 
+C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O 
+O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x 
+(Bind Abbr) u1))).(\lambda (H5: (csuba g x d1)).(eq_ind_r C (CHead x (Bind 
+Abbr) u1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+a))))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) 
+u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x (Bind 
+Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: 
+C).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1)) x (drop_refl (CHead x (Bind Abbr) u1)) H5)) c2 
+H4)))) H3)) (\lambda (H3: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H4: (eq C c2 (CHead x0 (Bind 
+Abst) x1))).(\lambda (H5: (csuba g x0 d1)).(\lambda (H6: (arity g x0 x1 
+(asucc g x2))).(\lambda (H7: (arity g d1 u1 x2)).(eq_ind_r C (CHead x0 (Bind 
+Abst) x1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+a))))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Abst) 
+x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind 
+Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abst) x1) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_refl (CHead x0 (Bind Abst) x1)) H5 
+H6 H7)) c2 H4)))))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H: 
+((\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop n O c1 (CHead d1 
+(Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c1) \to 
+(or (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+a))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: 
+C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to (\forall 
+(g: G).(\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop 
+(S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
+c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))))))))))) (\lambda (n0: nat).(\lambda (d1: 
+C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) (CHead d1 (Bind 
+Abbr) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (csuba g c2 (CSort 
+n0))).(and3_ind (eq C (CHead d1 (Bind Abbr) u1) (CSort n0)) (eq nat (S n) O) 
+(eq nat O O) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) 
+(\lambda (H2: (eq C (CHead d1 (Bind Abbr) u1) (CSort n0))).(\lambda (_: (eq 
+nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (match H2 in eq return 
+(\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c (CSort n0)) \to (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))) with 
+[refl_equal \Rightarrow (\lambda (H5: (eq C (CHead d1 (Bind Abbr) u1) (CSort 
+n0))).(let H6 \def (eq_ind C (CHead d1 (Bind Abbr) u1) (\lambda (e: C).(match 
+e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | 
+(CHead _ _ _) \Rightarrow True])) I (CSort n0) H5) in (False_ind (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) H6)))]) in (H5 
+(refl_equal C (CSort n0))))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind 
+Abbr) u1) H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: 
+C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to (\forall 
+(g: G).(\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop 
+(S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
+c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))))))))))))).(\lambda (k: K).(\lambda (t: 
+T).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H1: (drop (S n) O (CHead c k 
+t) (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: 
+(csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: K).((csuba g c2 (CHead c k0 
+t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))) (\lambda (b: 
+B).(\lambda (H3: (csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop (r 
+(Bind b) n) O c (CHead d1 (Bind Abbr) u1))).(B_ind (\lambda (b0: B).((csuba g 
+c2 (CHead c (Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind 
+Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+a))))))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) t))).(\lambda (H6: 
+(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def 
+(csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C 
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba 
+g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq 
+C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g c t a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind 
+Abbr) t))) (\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq 
+C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x: 
+C).(\lambda (H9: (eq C c2 (CHead x (Bind Abbr) t))).(\lambda (H10: (csuba g x 
+c)).(eq_ind_r C (CHead x (Bind Abbr) t) (\lambda (c0: C).(or (ex2 C (\lambda 
+(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))))) (let H11 \def (H c d1 u1 H6 g 
+x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or 
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n 
+O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind 
+C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind 
+Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
+(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x0: 
+C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14: 
+(csuba g x0 d1)).(let H15 \def (refl_equal nat (r (Bind Abst) n)) in (let H16 
+\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) 
+u1))) H13 (r (Bind Abst) n) H15) in (or_introl (ex2 C (\lambda (d2: C).(drop 
+(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop (Bind Abbr) 
+n x (CHead x0 (Bind Abbr) u1) H16 t) H14))))))) H12)) (\lambda (H12: (ex4_3 C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or (ex2 C (\lambda (d2: C).(drop 
+(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x 
+(CHead x0 (Bind Abst) x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15: 
+(arity g x0 x1 (asucc g x2))).(\lambda (H16: (arity g d1 u1 x2)).(let H17 
+\def (refl_equal nat (r (Bind Abst) n)) in (let H18 \def (eq_ind nat n 
+(\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abst) x1))) H13 (r (Bind 
+Abst) n) H17) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x 
+(Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
+(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex4_3_intro C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
+(Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Abbr) n 
+x (CHead x0 (Bind Abst) x1) H18 t) H14 H15 H16))))))))))) H12)) H11)) c2 
+H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc 
+g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) 
+(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 
+(CHead x0 (Bind Abst) x1))).(\lambda (H10: (csuba g x0 c)).(\lambda (_: 
+(arity g x0 x1 (asucc g x2))).(\lambda (_: (arity g c t x2)).(eq_ind_r C 
+(CHead x0 (Bind Abst) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop 
+(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
+c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in 
+(or_ind (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))))) (\lambda (H14: (ex2 C (\lambda (d2: C).(drop n 
+O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind 
+C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind 
+Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
+(CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x: 
+C).(\lambda (H15: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H16: 
+(csuba g x d1)).(let H17 \def (refl_equal nat (r (Bind Abst) n)) in (let H18 
+\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x (Bind Abbr) 
+u1))) H15 (r (Bind Abst) n) H17) in (or_introl (ex2 C (\lambda (d2: C).(drop 
+(S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead 
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind 
+Abst) n x0 (CHead x (Bind Abbr) u1) H18 x1) H16))))))) H14)) (\lambda (H14: 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: 
+A).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Abst) x4))).(\lambda (H16: 
+(csuba g x3 d1)).(\lambda (H17: (arity g x3 x4 (asucc g x5))).(\lambda (H18: 
+(arity g d1 u1 x5)).(let H19 \def (refl_equal nat (r (Bind Abst) n)) in (let 
+H20 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x3 (Bind Abst) 
+x4))) H15 (r (Bind Abst) n) H19) in (or_intror (ex2 C (\lambda (d2: C).(drop 
+(S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
+n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x3 x4 x5 
+(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Abst) x4) H20 x1) H16 H17 
+H18))))))))))) H14)) H13)) c2 H9)))))))) H8)) H7))))) (\lambda (H5: (csuba g 
+c2 (CHead c (Bind Abst) t))).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead 
+d1 (Bind Abbr) u1))).(let H_x \def (csuba_gen_abst_rev g c c2 t H5) in (let 
+H7 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) 
+t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) 
+O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead 
+d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x 
+(Bind Abst) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C (CHead x (Bind 
+Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead 
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda 
+(d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n 
+O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead 
+x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
+n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H11: 
+(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr) 
+u1))).(\lambda (H13: (csuba g x0 d1)).(let H14 \def (refl_equal nat (r (Bind 
+Abst) n)) in (let H15 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x 
+(CHead x0 (Bind Abbr) u1))) H12 (r (Bind Abst) n) H14) in (or_introl (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O 
+(CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g 
+d2 d1)) x0 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abbr) u1) H15 t) 
+H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
+(Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst) 
+x1))).(\lambda (H13: (csuba g x0 d1)).(\lambda (H14: (arity g x0 x1 (asucc g 
+x2))).(\lambda (H15: (arity g d1 u1 x2)).(let H16 \def (refl_equal nat (r 
+(Bind Abst) n)) in (let H17 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O 
+x (CHead x0 (Bind Abst) x1))) H12 (r (Bind Abst) n) H16) in (or_intror (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))) x0 x1 x2 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abst) x1) H17 t) 
+H13 H14 H15))))))))))) H11)) H10)) c2 H8)))) H7))))) (\lambda (H5: (csuba g 
+c2 (CHead c (Bind Void) t))).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead 
+d1 (Bind Abbr) u1))).(let H_x \def (csuba_gen_void_rev g c c2 t H5) in (let 
+H7 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) 
+t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) 
+O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead 
+d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x 
+(Bind Void) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C (CHead x (Bind 
+Void) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead 
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda 
+(d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n 
+O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead 
+x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
+n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H11: 
+(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr) 
+u1))).(\lambda (H13: (csuba g x0 d1)).(let H14 \def (refl_equal nat (r (Bind 
+Abst) n)) in (let H15 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x 
+(CHead x0 (Bind Abbr) u1))) H12 (r (Bind Abst) n) H14) in (or_introl (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O 
+(CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g 
+d2 d1)) x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abbr) u1) H15 t) 
+H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
+(Bind Void) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst) 
+x1))).(\lambda (H13: (csuba g x0 d1)).(\lambda (H14: (arity g x0 x1 (asucc g 
+x2))).(\lambda (H15: (arity g d1 u1 x2)).(let H16 \def (refl_equal nat (r 
+(Bind Abst) n)) in (let H17 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O 
+x (CHead x0 (Bind Abst) x1))) H12 (r (Bind Abst) n) H16) in (or_intror (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))) x0 x1 x2 (drop_drop (Bind Void) n x (CHead x0 (Bind Abst) x1) H17 t) 
+H13 H14 H15))))))))))) H11)) H10)) c2 H8)))) H7))))) b H3 H4)))) (\lambda (f: 
+F).(\lambda (H3: (csuba g c2 (CHead c (Flat f) t))).(\lambda (H4: (drop (r 
+(Flat f) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def (csuba_gen_flat_rev 
+g c c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T (\lambda (d2: 
+C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O 
+c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead 
+d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: 
+(eq C c2 (CHead x0 (Flat f) x1))).(\lambda (H7: (csuba g x0 c)).(eq_ind_r C 
+(CHead x0 (Flat f) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S 
+n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 
+C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))))) (let H8 \def (H0 d1 u1 H4 g x0 H7) in (or_ind 
+(ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S 
+n) O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1)))).(ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S 
+n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) 
+(\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abbr) 
+u1))).(\lambda (H11: (csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop 
+(S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Flat f) n 
+x0 (CHead x (Bind Abbr) u1) H10 x1) H11))))) H9)) (\lambda (H9: (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop (S n) O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or (ex2 C (\lambda (d2: C).(drop 
+(S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) 
+(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H10: (drop (S n) 
+O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H11: (csuba g x2 d1)).(\lambda 
+(H12: (arity g x2 x3 (asucc g x4))).(\lambda (H13: (arity g d1 u1 
+x4)).(or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
+(Flat f) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))) x2 x3 x4 (drop_drop (Flat f) n x0 (CHead x2 (Bind 
+Abst) x3) H10 x1) H11 H12 H13))))))))) H9)) H8)) c2 H6))))) H5)))))) k H2 
+(drop_gen_drop k c (CHead d1 (Bind Abbr) u1) t n H1)))))))))))) c1)))) i).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/fwd.ma
index c5c65696a..1bb036cca 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/fwd.ma
@@ -409,3 +409,435 @@ H14 Abst H8) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
 H9))) b1 H8)) c1 (sym_eq C c1 e1 H7))) H6)) H5)) H4 H0 H1 H2)))]) in (H0 
 (refl_equal C (CHead e1 (Bind b1) v1)) (refl_equal C c2)))))))).
 
+theorem csuba_gen_abst_rev:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c 
+(CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: 
+(csuba g c (CHead d1 (Bind Abst) u))).(let H0 \def (match H in csuba return 
+(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 c) 
+\to ((eq C c1 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq C c 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) with 
+[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1: 
+(eq C (CSort n) (CHead d1 (Bind Abst) u))).(eq_ind C (CSort n) (\lambda (c0: 
+C).((eq C (CSort n) (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq 
+C c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda 
+(H2: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H3 \def (eq_ind C (CSort 
+n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) 
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abst) 
+u) H2) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) H3))) c H0 H1))) | (csuba_head 
+c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) c)).(\lambda 
+(H2: (eq C (CHead c2 k u0) (CHead d1 (Bind Abst) u))).(eq_ind C (CHead c1 k 
+u0) (\lambda (c0: C).((eq C (CHead c2 k u0) (CHead d1 (Bind Abst) u)) \to 
+((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abst) 
+u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda (H3: (eq C (CHead c2 k 
+u0) (CHead d1 (Bind Abst) u))).(let H4 \def (f_equal C T (\lambda (e: 
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | 
+(CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) 
+in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda 
+(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) 
+(CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) in ((let H6 \def (f_equal C C 
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
+\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 
+(Bind Abst) u) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind Abst)) \to 
+((eq T u0 u) \to ((csuba g c1 c0) \to (ex2 C (\lambda (d2: C).(eq C (CHead c1 
+k u0) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))))) 
+(\lambda (H7: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda (k0: 
+K).((eq T u0 u) \to ((csuba g c1 d1) \to (ex2 C (\lambda (d2: C).(eq C (CHead 
+c1 k0 u0) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))) 
+(\lambda (H8: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((csuba g c1 d1) \to 
+(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst) 
+u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda (H9: (csuba g c1 
+d1)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) u) (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 
+(Bind Abst) u)) H9)) u0 (sym_eq T u0 u H8))) k (sym_eq K k (Bind Abst) H7))) 
+c2 (sym_eq C c2 d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a 
+H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) 
+c)).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst) 
+u))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2 
+(Bind Abbr) u0) (CHead d1 (Bind Abst) u)) \to ((csuba g c1 c2) \to ((arity g 
+c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c0 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) (\lambda 
+(H5: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst) u))).(let H6 \def 
+(eq_ind C (CHead c2 (Bind Abbr) u0) (\lambda (e: C).(match e in C return 
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) 
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow 
+True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) 
+\Rightarrow False])])) I (CHead d1 (Bind Abst) u) H5) in (False_ind ((csuba g 
+c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C 
+(\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1)))))) H6))) c H3 H4 H0 H1 H2)))]) in (H0 
+(refl_equal C c) (refl_equal C (CHead d1 (Bind Abst) u)))))))).
+
+theorem csuba_gen_void_rev:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c 
+(CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind 
+Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: 
+(csuba g c (CHead d1 (Bind Void) u))).(let H0 \def (match H in csuba return 
+(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 c) 
+\to ((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c 
+(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) with 
+[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1: 
+(eq C (CSort n) (CHead d1 (Bind Void) u))).(eq_ind C (CSort n) (\lambda (c0: 
+C).((eq C (CSort n) (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq 
+C c0 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda 
+(H2: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H3 \def (eq_ind C (CSort 
+n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) 
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Void) 
+u) H2) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind 
+Void) u))) (\lambda (d2: C).(csuba g d2 d1))) H3))) c H0 H1))) | (csuba_head 
+c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) c)).(\lambda 
+(H2: (eq C (CHead c2 k u0) (CHead d1 (Bind Void) u))).(eq_ind C (CHead c1 k 
+u0) (\lambda (c0: C).((eq C (CHead c2 k u0) (CHead d1 (Bind Void) u)) \to 
+((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Void) 
+u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda (H3: (eq C (CHead c2 k 
+u0) (CHead d1 (Bind Void) u))).(let H4 \def (f_equal C T (\lambda (e: 
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | 
+(CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3) 
+in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda 
+(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) 
+(CHead c2 k u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \def (f_equal C C 
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
+\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 
+(Bind Void) u) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind Void)) \to 
+((eq T u0 u) \to ((csuba g c1 c0) \to (ex2 C (\lambda (d2: C).(eq C (CHead c1 
+k u0) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))))) 
+(\lambda (H7: (eq K k (Bind Void))).(eq_ind K (Bind Void) (\lambda (k0: 
+K).((eq T u0 u) \to ((csuba g c1 d1) \to (ex2 C (\lambda (d2: C).(eq C (CHead 
+c1 k0 u0) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))) 
+(\lambda (H8: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((csuba g c1 d1) \to 
+(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) t) (CHead d2 (Bind Void) 
+u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda (H9: (csuba g c1 
+d1)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u) (CHead d2 
+(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 
+(Bind Void) u)) H9)) u0 (sym_eq T u0 u H8))) k (sym_eq K k (Bind Void) H7))) 
+c2 (sym_eq C c2 d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a 
+H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) 
+c)).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void) 
+u))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2 
+(Bind Abbr) u0) (CHead d1 (Bind Void) u)) \to ((csuba g c1 c2) \to ((arity g 
+c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c0 
+(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) (\lambda 
+(H5: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void) u))).(let H6 \def 
+(eq_ind C (CHead c2 (Bind Abbr) u0) (\lambda (e: C).(match e in C return 
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) 
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow 
+True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) 
+\Rightarrow False])])) I (CHead d1 (Bind Void) u) H5) in (False_ind ((csuba g 
+c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C 
+(\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u))) 
+(\lambda (d2: C).(csuba g d2 d1)))))) H6))) c H3 H4 H0 H1 H2)))]) in (H0 
+(refl_equal C c) (refl_equal C (CHead d1 (Bind Void) u)))))))).
+
+theorem csuba_gen_abbr_rev:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g c 
+(CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+a))))))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda 
+(H: (csuba g c (CHead d1 (Bind Abbr) u1))).(let H0 \def (match H in csuba 
+return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C 
+c0 c) \to ((eq C c1 (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2: 
+C).(eq C c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead 
+d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))))))))) with [(csuba_sort n) \Rightarrow (\lambda 
+(H0: (eq C (CSort n) c)).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr) 
+u1))).(eq_ind C (CSort n) (\lambda (c0: C).((eq C (CSort n) (CHead d1 (Bind 
+Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))))) 
+(\lambda (H2: (eq C (CSort n) (CHead d1 (Bind Abbr) u1))).(let H3 \def 
+(eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_: 
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
+False])) I (CHead d1 (Bind Abbr) u1) H2) in (False_ind (or (ex2 C (\lambda 
+(d2: C).(eq C (CSort n) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g 
+d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C 
+(CSort n) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) H3))) c H0 H1))) | (csuba_head 
+c1 c2 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) c)).(\lambda 
+(H2: (eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1))).(eq_ind C (CHead c1 k 
+u) (\lambda (c0: C).((eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1)) \to 
+((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))) 
+(\lambda (H3: (eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1))).(let H4 \def 
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u) 
+(CHead d1 (Bind Abbr) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: 
+C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | 
+(CHead _ k0 _) \Rightarrow k0])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) 
+in ((let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda 
+(_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) 
+(CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in (eq_ind C d1 (\lambda (c0: 
+C).((eq K k (Bind Abbr)) \to ((eq T u u1) \to ((csuba g c1 c0) \to (or (ex2 C 
+(\lambda (d2: C).(eq C (CHead c1 k u) (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(eq C (CHead c1 k u) (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))))))) 
+(\lambda (H7: (eq K k (Bind Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0: 
+K).((eq T u u1) \to ((csuba g c1 d1) \to (or (ex2 C (\lambda (d2: C).(eq C 
+(CHead c1 k0 u) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C 
+(CHead c1 k0 u) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))))))) (\lambda (H8: (eq T u 
+u1)).(eq_ind T u1 (\lambda (t: T).((csuba g c1 d1) \to (or (ex2 C (\lambda 
+(d2: C).(eq C (CHead c1 (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(eq C (CHead c1 (Bind Abbr) t) (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+a)))))))) (\lambda (H9: (csuba g c1 d1)).(or_introl (ex2 C (\lambda (d2: 
+C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C 
+(\lambda (d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind Abbr) u1)) 
+H9))) u (sym_eq T u u1 H8))) k (sym_eq K k (Bind Abbr) H7))) c2 (sym_eq C c2 
+d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u H2) 
+\Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) c)).(\lambda (H4: 
+(eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1))).(eq_ind C (CHead 
+c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2 (Bind Abbr) u) (CHead d1 
+(Bind Abbr) u1)) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to 
+((arity g c2 u a) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g a0))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 a0))))))))))) 
+(\lambda (H5: (eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1))).(let 
+H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) 
+with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 
+(Bind Abbr) u) (CHead d1 (Bind Abbr) u1) H5) in ((let H7 \def (f_equal C C 
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
+\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind Abbr) u) 
+(CHead d1 (Bind Abbr) u1) H5) in (eq_ind C d1 (\lambda (c0: C).((eq T u u1) 
+\to ((csuba g c1 c0) \to ((arity g c1 t (asucc g a)) \to ((arity g c0 u a) 
+\to (or (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: 
+A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a0: A).(arity g d1 u1 a0))))))))))) (\lambda (H8: (eq T u u1)).(eq_ind T u1 
+(\lambda (t0: T).((csuba g c1 d1) \to ((arity g c1 t (asucc g a)) \to ((arity 
+g d1 t0 a) \to (or (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) 
+t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: 
+A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a0: A).(arity g d1 u1 a0)))))))))) (\lambda (H9: (csuba g c1 d1)).(\lambda 
+(H10: (arity g c1 t (asucc g a))).(\lambda (H11: (arity g d1 u1 
+a)).(or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead 
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: 
+A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a0: A).(arity g d1 u1 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g 
+a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 
+a0)))) c1 t a (refl_equal C (CHead c1 (Bind Abst) t)) H9 H10 H11))))) u 
+(sym_eq T u u1 H8))) c2 (sym_eq C c2 d1 H7))) H6))) c H3 H4 H0 H1 H2)))]) in 
+(H0 (refl_equal C c) (refl_equal C (CHead d1 (Bind Abbr) u1)))))))).
+
+theorem csuba_gen_flat_rev:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall 
+(f: F).((csuba g c (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda 
+(f: F).(\lambda (H: (csuba g c (CHead d1 (Flat f) u1))).(let H0 \def (match H 
+in csuba return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 
+c1)).((eq C c0 c) \to ((eq C c1 (CHead d1 (Flat f) u1)) \to (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) with [(csuba_sort n) 
+\Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1: (eq C (CSort n) 
+(CHead d1 (Flat f) u1))).(eq_ind C (CSort n) (\lambda (c0: C).((eq C (CSort 
+n) (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(eq C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba 
+g d2 d1)))))) (\lambda (H2: (eq C (CSort n) (CHead d1 (Flat f) u1))).(let H3 
+\def (eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_: 
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
+False])) I (CHead d1 (Flat f) u1) H2) in (False_ind (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))) H3))) c H0 H1))) | (csuba_head c1 c2 H0 
+k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) c)).(\lambda (H2: (eq C 
+(CHead c2 k u) (CHead d1 (Flat f) u1))).(eq_ind C (CHead c1 k u) (\lambda 
+(c0: C).((eq C (CHead c2 k u) (CHead d1 (Flat f) u1)) \to ((csuba g c1 c2) 
+\to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Flat f) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (H3: 
+(eq C (CHead c2 k u) (CHead d1 (Flat f) u1))).(let H4 \def (f_equal C T 
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
+\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Flat 
+f) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return 
+(\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow 
+k0])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C C 
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
+\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u) (CHead d1 
+(Flat f) u1) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Flat f)) \to ((eq 
+T u u1) \to ((csuba g c1 c0) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(eq C (CHead c1 k u) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1)))))))) (\lambda (H7: (eq K k (Flat f))).(eq_ind K 
+(Flat f) (\lambda (k0: K).((eq T u u1) \to ((csuba g c1 d1) \to (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k0 u) (CHead d2 (Flat f) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (H8: 
+(eq T u u1)).(eq_ind T u1 (\lambda (t: T).((csuba g c1 d1) \to (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Flat f) t) (CHead d2 (Flat 
+f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (\lambda (H9: 
+(csuba g c1 d1)).(ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(eq C 
+(CHead c1 (Flat f) u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1))) c1 u1 (refl_equal C (CHead c1 (Flat f) u1)) H9)) u 
+(sym_eq T u u1 H8))) k (sym_eq K k (Flat f) H7))) c2 (sym_eq C c2 d1 H6))) 
+H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u H2) \Rightarrow 
+(\lambda (H3: (eq C (CHead c1 (Bind Abst) t) c)).(\lambda (H4: (eq C (CHead 
+c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(eq_ind C (CHead c1 (Bind Abst) t) 
+(\lambda (c0: C).((eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1)) \to 
+((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u a) \to 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Flat f) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) (\lambda (H5: 
+(eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind 
+C (CHead c2 (Bind Abbr) u) (\lambda (e: C).(match e in C return (\lambda (_: 
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match 
+k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat 
+_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind ((csuba 
+g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u a) \to (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 
+(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H6))) 
+c H3 H4 H0 H1 H2)))]) in (H0 (refl_equal C c) (refl_equal C (CHead d1 (Flat 
+f) u1))))))))).
+
+theorem csuba_gen_bind_rev:
+ \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall 
+(v1: T).((csuba g c2 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))))))
+\def
+ \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda 
+(v1: T).(\lambda (H: (csuba g c2 (CHead e1 (Bind b1) v1))).(let H0 \def 
+(match H in csuba return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csuba 
+? c c0)).((eq C c c2) \to ((eq C c0 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind 
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 
+e1)))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) 
+c2)).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind b1) v1))).(eq_ind C (CSort 
+n) (\lambda (c: C).((eq C (CSort n) (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind 
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 
+e1))))))) (\lambda (H2: (eq C (CSort n) (CHead e1 (Bind b1) v1))).(let H3 
+\def (eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_: 
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
+False])) I (CHead e1 (Bind b1) v1) H2) in (False_ind (ex2_3 B C T (\lambda 
+(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 (Bind b2) 
+v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))) 
+H3))) c2 H0 H1))) | (csuba_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C 
+(CHead c1 k u) c2)).(\lambda (H2: (eq C (CHead c0 k u) (CHead e1 (Bind b1) 
+v1))).(eq_ind C (CHead c1 k u) (\lambda (c: C).((eq C (CHead c0 k u) (CHead 
+e1 (Bind b1) v1)) \to ((csuba g c1 c0) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))) 
+(\lambda (H3: (eq C (CHead c0 k u) (CHead e1 (Bind b1) v1))).(let H4 \def 
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u) 
+(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: 
+C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | 
+(CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u) (CHead e1 (Bind b1) v1) H3) 
+in ((let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda 
+(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) 
+(CHead c0 k u) (CHead e1 (Bind b1) v1) H3) in (eq_ind C e1 (\lambda (c: 
+C).((eq K k (Bind b1)) \to ((eq T u v1) \to ((csuba g c1 c) \to (ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k u) 
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
+T).(csuba g e2 e1))))))))) (\lambda (H7: (eq K k (Bind b1))).(eq_ind K (Bind 
+b1) (\lambda (k0: K).((eq T u v1) \to ((csuba g c1 e1) \to (ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k0 u) 
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
+T).(csuba g e2 e1)))))))) (\lambda (H8: (eq T u v1)).(eq_ind T v1 (\lambda 
+(t: T).((csuba g c1 e1) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
+C).(\lambda (v2: T).(eq C (CHead c1 (Bind b1) t) (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) 
+(\lambda (H9: (csuba g c1 e1)).(let H10 \def (eq_ind T u (\lambda (t: T).(eq 
+C (CHead c1 k t) c2)) H1 v1 H8) in (let H11 \def (eq_ind K k (\lambda (k0: 
+K).(eq C (CHead c1 k0 v1) c2)) H10 (Bind b1) H7) in (let H12 \def (eq_ind_r C 
+c2 (\lambda (c: C).(csuba g c (CHead e1 (Bind b1) v1))) H (CHead c1 (Bind b1) 
+v1) H11) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda 
+(v2: T).(eq C (CHead c1 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda 
+(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))) b1 c1 v1 
+(refl_equal C (CHead c1 (Bind b1) v1)) H9))))) u (sym_eq T u v1 H8))) k 
+(sym_eq K k (Bind b1) H7))) c0 (sym_eq C c0 e1 H6))) H5)) H4))) c2 H1 H2 
+H0))) | (csuba_abst c1 c0 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C 
+(CHead c1 (Bind Abst) t) c2)).(\lambda (H4: (eq C (CHead c0 (Bind Abbr) u) 
+(CHead e1 (Bind b1) v1))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c: 
+C).((eq C (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1)) \to ((csuba g c1 
+c0) \to ((arity g c1 t (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind 
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 
+e1)))))))))) (\lambda (H5: (eq C (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) 
+v1))).(let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda 
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) 
+(CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def 
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with 
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return 
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
+Abbr])])) (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H8 
+\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) 
+with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 
+(Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in (eq_ind C e1 (\lambda (c: 
+C).((eq B Abbr b1) \to ((eq T u v1) \to ((csuba g c1 c) \to ((arity g c1 t 
+(asucc g a)) \to ((arity g c u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda 
+(e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) 
+v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 
+e1))))))))))) (\lambda (H9: (eq B Abbr b1)).(eq_ind B Abbr (\lambda (_: 
+B).((eq T u v1) \to ((csuba g c1 e1) \to ((arity g c1 t (asucc g a)) \to 
+((arity g e1 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda 
+(v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) v2))))) (\lambda 
+(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))))) (\lambda 
+(H10: (eq T u v1)).(eq_ind T v1 (\lambda (t0: T).((csuba g c1 e1) \to ((arity 
+g c1 t (asucc g a)) \to ((arity g e1 t0 a) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 
+(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g 
+e2 e1))))))))) (\lambda (H11: (csuba g c1 e1)).(\lambda (_: (arity g c1 t 
+(asucc g a))).(\lambda (_: (arity g e1 v1 a)).(let H14 \def (eq_ind_r C c2 
+(\lambda (c: C).(csuba g c (CHead e1 (Bind b1) v1))) H (CHead c1 (Bind Abst) 
+t) H3) in (let H15 \def (eq_ind_r B b1 (\lambda (b: B).(csuba g (CHead c1 
+(Bind Abst) t) (CHead e1 (Bind b) v1))) H14 Abbr H9) in (ex2_3_intro B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind 
+Abst) t) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csuba g e2 e1)))) Abst c1 t (refl_equal C (CHead c1 (Bind 
+Abst) t)) H11)))))) u (sym_eq T u v1 H10))) b1 H9)) c0 (sym_eq C c0 e1 H8))) 
+H7)) H6))) c2 H3 H4 H0 H1 H2)))]) in (H0 (refl_equal C c2) (refl_equal C 
+(CHead e1 (Bind b1) v1))))))))).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/getl.ma
index 137b41453..3a72c1aed 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/getl.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/getl.ma
@@ -473,3 +473,452 @@ c2 x7 x8 H20 (CHead x9 (Bind Abbr) x10) n H23) H24 H25 H26))))))))) H22))
 H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) x H1 
 H2)))) H0))))))).
 
+theorem csuba_getl_abst_rev:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall 
+(i: nat).((getl i c1 (CHead d1 (Bind Abst) u)) \to (\forall (c2: C).((csuba g 
+c2 c1) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda 
+(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u))).(let H0 \def 
+(getl_gen_all c1 (CHead d1 (Bind Abst) u) i H) in (ex2_ind C (\lambda (e: 
+C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u))) 
+(\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda (d2: C).(getl i c2 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda (x: 
+C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind 
+Abst) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1 
+(Bind Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda 
+(d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
+d1)))))))) (\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda 
+(H4: (clear (CSort n) (CHead d1 (Bind Abst) u))).(clear_gen_sort (CHead d1 
+(Bind Abst) u) n H4 (\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda 
+(d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
+d1))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 
+(CHead d1 (Bind Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (ex2 C 
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: 
+(drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 
+(Bind Abst) u))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to 
+((clear (CHead x0 k0 t) (CHead d1 (Bind Abst) u)) \to (\forall (c2: 
+C).((csuba g c2 c1) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) (\lambda (b: B).(\lambda 
+(H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0 
+(Bind b) t) (CHead d1 (Bind Abst) u))).(let H7 \def (f_equal C C (\lambda (e: 
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | 
+(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t) 
+(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in ((let H8 \def 
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with 
+[(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 in K 
+return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
+\Rightarrow Abst])])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t) 
+(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in ((let H9 \def 
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
+[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind 
+Abst) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u) 
+t H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda 
+(c2: C).(\lambda (H12: (csuba g c2 c1)).(let H13 \def (eq_ind_r T t (\lambda 
+(t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def 
+(eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u))) H13 Abst 
+H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c 
+(Bind Abst) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abst_rev i c1 d1 u 
+H15 g c2 H12) in (ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(getl i 
+c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda 
+(x1: C).(\lambda (H17: (drop i O c2 (CHead x1 (Bind Abst) u))).(\lambda (H18: 
+(csuba g x1 d1)).(ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x1 (getl_intro i c2 (CHead x1 
+(Bind Abst) u) (CHead x1 (Bind Abst) u) H17 (clear_bind Abst x1 u)) H18)))) 
+H16)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead 
+x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind 
+Abst) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c 
+(CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) \to (ex2 C 
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop n 
+O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (ex2 C 
+(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1)))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead 
+x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g c2 x1)).(let H10 
+\def (eq_ind C x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead x0 (Flat f) t) 
+(drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def (clear_flat x0 
+(CHead d1 (Bind Abst) u) (clear_gen_flat f x0 (CHead d1 (Bind Abst) u) t H6) 
+f t) in (let H11 \def (csuba_clear_trans g (CHead x0 (Flat f) t) c2 H10 
+(CHead d1 (Bind Abst) u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g e2 
+(CHead d1 (Bind Abst) u))) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda 
+(d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
+d1))) (\lambda (x2: C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abst) 
+u))).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abst_rev g d1 x2 
+u H12) in (let H14 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: 
+C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
+(\lambda (x3: C).(\lambda (H15: (eq C x2 (CHead x3 (Bind Abst) u))).(\lambda 
+(H16: (csuba g x3 d1)).(let H17 \def (eq_ind C x2 (\lambda (c: C).(clear c2 
+c)) H13 (CHead x3 (Bind Abst) u) H15) in (ex_intro2 C (\lambda (d2: C).(getl 
+O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x3 
+(getl_intro O c2 (CHead x3 (Bind Abst) u) c2 (drop_refl c2) H17) H16))))) 
+H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1: 
+C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) 
+\to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda 
+(d2: C).(csuba g d2 d1))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O 
+x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g c2 
+x1)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B 
+C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind 
+b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead 
+x0 (Flat f) t))))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x2: B).(\lambda (x3: 
+C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) 
+x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 \def 
+(csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C 
+(\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) (\lambda (e2: 
+C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x5: C).(\lambda (H15: 
+(csuba g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: (clear c2 x5)).(let H_x 
+\def (csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let H17 \def H_x in 
+(ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
+T).(csuba g e2 x3)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x6: B).(\lambda (x7: 
+C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) 
+x8))).(\lambda (H19: (csuba g x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: 
+C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 
+x7 H19) in (ex2_ind C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(getl (S n) c2 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x9: 
+C).(\lambda (H22: (getl n x7 (CHead x9 (Bind Abst) u))).(\lambda (H23: (csuba 
+g x9 d1)).(ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) 
+u))) (\lambda (d2: C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20 
+(CHead x9 (Bind Abst) u) n H22) H23)))) H21)))))))) H17)))))) H14))))))) 
+H11)))))))) i) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))).
+
+theorem csuba_getl_abbr_rev:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall 
+(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba 
+g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda 
+(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u1))).(let H0 \def 
+(getl_gen_all c1 (CHead d1 (Bind Abbr) u1) i H) in (ex2_ind C (\lambda (e: 
+C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u1))) 
+(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))))))) (\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: 
+(clear x (CHead d1 (Bind Abbr) u1))).(C_ind (\lambda (c: C).((drop i O c1 c) 
+\to ((clear c (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 
+c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))))) (\lambda 
+(n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) 
+(CHead d1 (Bind Abbr) u1))).(clear_gen_sort (CHead d1 (Bind Abbr) u1) n H4 
+(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear 
+x0 (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 c1) \to (or 
+(ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))))))))))).(\lambda (k: K).(\lambda 
+(t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear 
+(CHead x0 k t) (CHead d1 (Bind Abbr) u1))).(K_ind (\lambda (k0: K).((drop i O 
+c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind Abbr) u1)) 
+\to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i 
+c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))))))))) (\lambda (b: B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) 
+t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 (Bind Abbr) 
+u1))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda 
+(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) 
+(CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead 
+d1 (Bind Abbr) u1) t H6)) in ((let H8 \def (f_equal C B (\lambda (e: 
+C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | 
+(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with 
+[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d1 (Bind 
+Abbr) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) 
+u1) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e in C return 
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t0) 
+\Rightarrow t0])) (CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b) t) 
+(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) in (\lambda (H10: (eq B 
+Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: C).(\lambda (H12: (csuba 
+g c2 c1)).(let H13 \def (eq_ind_r T t (\lambda (t0: T).(drop i O c1 (CHead x0 
+(Bind b) t0))) H5 u1 H9) in (let H14 \def (eq_ind_r B b (\lambda (b0: 
+B).(drop i O c1 (CHead x0 (Bind b0) u1))) H13 Abbr H10) in (let H15 \def 
+(eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c (Bind Abbr) u1))) H14 d1 
+H11) in (let H16 \def (csuba_drop_abbr_rev i c1 d1 u1 H15 g c2 H12) in 
+(or_ind (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or (ex2 C 
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H17: (ex2 C (\lambda 
+(d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(getl i c2 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))))) (\lambda (x1: C).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abbr) 
+u1))).(\lambda (H19: (csuba g x1 d1)).(or_introl (ex2 C (\lambda (d2: 
+C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead 
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x1 (getl_intro i c2 
+(CHead x1 (Bind Abbr) u1) (CHead x1 (Bind Abbr) u1) H18 (clear_bind Abbr x1 
+u1)) H19))))) H17)) (\lambda (H17: (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x1: 
+C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1 
+(Bind Abst) x2))).(\lambda (H19: (csuba g x1 d1)).(\lambda (H20: (arity g x1 
+x2 (asucc g x3))).(\lambda (H21: (arity g d1 u1 x3)).(or_intror (ex2 C 
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x1 x2 x3 
+(getl_intro i c2 (CHead x1 (Bind Abst) x2) (CHead x1 (Bind Abst) x2) H18 
+(clear_bind Abst x1 x2)) H19 H20 H21))))))))) H17)) H16)))))))))) H8)) 
+H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f) 
+t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr) 
+u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0 
+(Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda 
+(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i 
+c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: 
+C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) 
+\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))))) (\lambda 
+(x1: C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: 
+C).(\lambda (H9: (csuba g c2 x1)).(let H10 \def (eq_ind C x1 (\lambda (c: 
+C).(csuba g c2 c)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat 
+f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abbr) u1) 
+(clear_gen_flat f x0 (CHead d1 (Bind Abbr) u1) t H6) f t) in (let H11 \def 
+(csuba_clear_trans g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u1) 
+H_y) in (ex2_ind C (\lambda (e2: C).(csuba g e2 (CHead d1 (Bind Abbr) u1))) 
+(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead 
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) 
+(\lambda (x2: C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abbr) 
+u1))).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abbr_rev g d1 x2 
+u1 H12) in (let H14 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C x2 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H15: 
+(ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: 
+C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead 
+x3 (Bind Abbr) u1))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C 
+x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) u1) H16) in 
+(or_introl (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C 
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abbr) u1) c2 
+(drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(eq C x2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or (ex2 C (\lambda (d2: C).(getl 
+O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H16: 
+(eq C x2 (CHead x3 (Bind Abst) x4))).(\lambda (H17: (csuba g x3 d1)).(\lambda 
+(H18: (arity g x3 x4 (asucc g x5))).(\lambda (H19: (arity g d1 u1 x5)).(let 
+H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst) 
+x4) H16) in (or_intror (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O 
+c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))) x3 x4 x5 (getl_intro O c2 (CHead x3 (Bind Abst) 
+x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) H14)))))) H11)))))))) 
+(\lambda (n: nat).(\lambda (H8: ((\forall (x1: C).((drop n O x1 (CHead x0 
+(Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or (ex2 C (\lambda 
+(d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n 
+c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S 
+n) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g c2 
+x1)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B 
+C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind 
+b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead 
+x0 (Flat f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) 
+(\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1 
+(CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) 
+t))).(let H14 \def (csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) 
+H12) in (ex2_ind C (\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) 
+(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))))) (\lambda (x5: C).(\lambda (H15: (csuba g x5 (CHead x3 (Bind x2) 
+x4))).(\lambda (H16: (clear c2 x5)).(let H_x \def (csuba_gen_bind_rev g x2 x3 
+x5 x4 H15) in (let H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda 
+(e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 x3)))) (or (ex2 C (\lambda 
+(d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g 
+d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl 
+(S n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x6: B).(\lambda (x7: 
+C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) 
+x8))).(\lambda (H19: (csuba g x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: 
+C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 
+x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or 
+(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H22: 
+(ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: 
+C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S 
+n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))))) (\lambda (x9: C).(\lambda (H23: (getl n x7 
+(CHead x9 (Bind Abbr) u1))).(\lambda (H24: (csuba g x9 d1)).(or_introl (ex2 C 
+(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(getl (S n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: 
+C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) u1) n H23) 
+H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) 
+(\lambda (x9: C).(\lambda (x10: T).(\lambda (x11: A).(\lambda (H23: (getl n 
+x7 (CHead x9 (Bind Abst) x10))).(\lambda (H24: (csuba g x9 d1)).(\lambda 
+(H25: (arity g x9 x10 (asucc g x11))).(\lambda (H26: (arity g d1 u1 
+x11)).(or_intror (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S 
+n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))) x9 x10 x11 (getl_clear_bind x6 c2 x7 x8 H20 
+(CHead x9 (Bind Abst) x10) n H23) H24 H25 H26))))))))) H22)) H21)))))))) 
+H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) x H1 H2)))) 
+H0))))))).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/arity.ma
new file mode 100644
index 000000000..2cad21c83
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/arity.ma
@@ -0,0 +1,38 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/arity".
+
+include "csubc/csuba.ma".
+
+include "arity/defs.ma".
+
+theorem csubc_arity_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to 
+(\forall (t: T).(\forall (a: A).((arity g c1 t a) \to (arity g c2 t a)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 
+c2)).(\lambda (t: T).(\lambda (a: A).(\lambda (H0: (arity g c1 t 
+a)).(csuba_arity g c1 t a H0 c2 (csubc_csuba g c1 c2 H)))))))).
+
+theorem csubc_arity_trans:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to 
+(\forall (t: T).(\forall (a: A).((arity g c2 t a) \to (arity g c1 t a)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 
+c2)).(\lambda (t: T).(\lambda (a: A).(\lambda (H0: (arity g c2 t 
+a)).(csuba_arity_rev g c2 t a H0 c1 (csubc_csuba g c1 c2 H)))))))).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/clear.ma
new file mode 100644
index 000000000..ab46f2baa
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/clear.ma
@@ -0,0 +1,149 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/clear".
+
+include "csubc/defs.ma".
+
+theorem csubc_clear_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (e1: C).((clear c1 e1) \to (\forall 
+(c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda 
+(e2: C).(csubc g e1 e2))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (e1: C).(\lambda (H: (clear c1 
+e1)).(clear_ind (\lambda (c: C).(\lambda (c0: C).(\forall (c2: C).((csubc g c 
+c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c0 
+e2))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (c2: 
+C).(\lambda (H0: (csubc g (CHead e (Bind b) u) c2)).(let H1 \def (match H0 in 
+csubc return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csubc ? c 
+c0)).((eq C c (CHead e (Bind b) u)) \to ((eq C c0 c2) \to (ex2 C (\lambda 
+(e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) 
+e2)))))))) with [(csubc_sort n) \Rightarrow (\lambda (H1: (eq C (CSort n) 
+(CHead e (Bind b) u))).(\lambda (H2: (eq C (CSort n) c2)).((let H3 \def 
+(eq_ind C (CSort n) (\lambda (e0: C).(match e0 in C return (\lambda (_: 
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
+False])) I (CHead e (Bind b) u) H1) in (False_ind ((eq C (CSort n) c2) \to 
+(ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead e 
+(Bind b) u) e2)))) H3)) H2))) | (csubc_head c0 c3 H1 k v) \Rightarrow 
+(\lambda (H2: (eq C (CHead c0 k v) (CHead e (Bind b) u))).(\lambda (H3: (eq C 
+(CHead c3 k v) c2)).((let H4 \def (f_equal C T (\lambda (e0: C).(match e0 in 
+C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) 
+\Rightarrow t])) (CHead c0 k v) (CHead e (Bind b) u) H2) in ((let H5 \def 
+(f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with 
+[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k v) 
+(CHead e (Bind b) u) H2) in ((let H6 \def (f_equal C C (\lambda (e0: 
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | 
+(CHead c _ _) \Rightarrow c])) (CHead c0 k v) (CHead e (Bind b) u) H2) in 
+(eq_ind C e (\lambda (c: C).((eq K k (Bind b)) \to ((eq T v u) \to ((eq C 
+(CHead c3 k v) c2) \to ((csubc g c c3) \to (ex2 C (\lambda (e2: C).(clear c2 
+e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2)))))))) (\lambda (H7: 
+(eq K k (Bind b))).(eq_ind K (Bind b) (\lambda (k0: K).((eq T v u) \to ((eq C 
+(CHead c3 k0 v) c2) \to ((csubc g e c3) \to (ex2 C (\lambda (e2: C).(clear c2 
+e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))))))) (\lambda (H8: 
+(eq T v u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c3 (Bind b) t) c2) \to 
+((csubc g e c3) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: 
+C).(csubc g (CHead e (Bind b) u) e2)))))) (\lambda (H9: (eq C (CHead c3 (Bind 
+b) u) c2)).(eq_ind C (CHead c3 (Bind b) u) (\lambda (c: C).((csubc g e c3) 
+\to (ex2 C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead e 
+(Bind b) u) e2))))) (\lambda (H10: (csubc g e c3)).(ex_intro2 C (\lambda (e2: 
+C).(clear (CHead c3 (Bind b) u) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind 
+b) u) e2)) (CHead c3 (Bind b) u) (clear_bind b c3 u) (csubc_head g e c3 H10 
+(Bind b) u))) c2 H9)) v (sym_eq T v u H8))) k (sym_eq K k (Bind b) H7))) c0 
+(sym_eq C c0 e H6))) H5)) H4)) H3 H1))) | (csubc_abst c0 c3 H1 v a H2 w H3) 
+\Rightarrow (\lambda (H4: (eq C (CHead c0 (Bind Abst) v) (CHead e (Bind b) 
+u))).(\lambda (H5: (eq C (CHead c3 (Bind Abbr) w) c2)).((let H6 \def (f_equal 
+C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) 
+\Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind Abst) v) 
+(CHead e (Bind b) u) H4) in ((let H7 \def (f_equal C B (\lambda (e0: 
+C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst 
+| (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with 
+[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (CHead c0 (Bind 
+Abst) v) (CHead e (Bind b) u) H4) in ((let H8 \def (f_equal C C (\lambda (e0: 
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | 
+(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind Abst) v) (CHead e (Bind b) u) 
+H4) in (eq_ind C e (\lambda (c: C).((eq B Abst b) \to ((eq T v u) \to ((eq C 
+(CHead c3 (Bind Abbr) w) c2) \to ((csubc g c c3) \to ((sc3 g (asucc g a) c v) 
+\to ((sc3 g a c3 w) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: 
+C).(csubc g (CHead e (Bind b) u) e2)))))))))) (\lambda (H9: (eq B Abst 
+b)).(eq_ind B Abst (\lambda (b0: B).((eq T v u) \to ((eq C (CHead c3 (Bind 
+Abbr) w) c2) \to ((csubc g e c3) \to ((sc3 g (asucc g a) e v) \to ((sc3 g a 
+c3 w) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g 
+(CHead e (Bind b0) u) e2))))))))) (\lambda (H10: (eq T v u)).(eq_ind T u 
+(\lambda (t: T).((eq C (CHead c3 (Bind Abbr) w) c2) \to ((csubc g e c3) \to 
+((sc3 g (asucc g a) e t) \to ((sc3 g a c3 w) \to (ex2 C (\lambda (e2: 
+C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead e (Bind Abst) u) 
+e2)))))))) (\lambda (H11: (eq C (CHead c3 (Bind Abbr) w) c2)).(eq_ind C 
+(CHead c3 (Bind Abbr) w) (\lambda (c: C).((csubc g e c3) \to ((sc3 g (asucc g 
+a) e u) \to ((sc3 g a c3 w) \to (ex2 C (\lambda (e2: C).(clear c e2)) 
+(\lambda (e2: C).(csubc g (CHead e (Bind Abst) u) e2))))))) (\lambda (H12: 
+(csubc g e c3)).(\lambda (H13: (sc3 g (asucc g a) e u)).(\lambda (H14: (sc3 g 
+a c3 w)).(ex_intro2 C (\lambda (e2: C).(clear (CHead c3 (Bind Abbr) w) e2)) 
+(\lambda (e2: C).(csubc g (CHead e (Bind Abst) u) e2)) (CHead c3 (Bind Abbr) 
+w) (clear_bind Abbr c3 w) (csubc_abst g e c3 H12 u a H13 w H14))))) c2 H11)) 
+v (sym_eq T v u H10))) b H9)) c0 (sym_eq C c0 e H8))) H7)) H6)) H5 H1 H2 
+H3)))]) in (H1 (refl_equal C (CHead e (Bind b) u)) (refl_equal C c2)))))))) 
+(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1: 
+((\forall (c2: C).((csubc g e c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) 
+(\lambda (e2: C).(csubc g c e2))))))).(\lambda (f: F).(\lambda (u: 
+T).(\lambda (c2: C).(\lambda (H2: (csubc g (CHead e (Flat f) u) c2)).(let H3 
+\def (match H2 in csubc return (\lambda (c0: C).(\lambda (c3: C).(\lambda (_: 
+(csubc ? c0 c3)).((eq C c0 (CHead e (Flat f) u)) \to ((eq C c3 c2) \to (ex2 C 
+(\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2)))))))) with 
+[(csubc_sort n) \Rightarrow (\lambda (H3: (eq C (CSort n) (CHead e (Flat f) 
+u))).(\lambda (H4: (eq C (CSort n) c2)).((let H5 \def (eq_ind C (CSort n) 
+(\lambda (e0: C).(match e0 in C return (\lambda (_: C).Prop) with [(CSort _) 
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e (Flat f) u) 
+H3) in (False_ind ((eq C (CSort n) c2) \to (ex2 C (\lambda (e2: C).(clear c2 
+e2)) (\lambda (e2: C).(csubc g c e2)))) H5)) H4))) | (csubc_head c0 c3 H3 k 
+v) \Rightarrow (\lambda (H4: (eq C (CHead c0 k v) (CHead e (Flat f) 
+u))).(\lambda (H5: (eq C (CHead c3 k v) c2)).((let H6 \def (f_equal C T 
+(\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) 
+\Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c0 k v) (CHead e (Flat 
+f) u) H4) in ((let H7 \def (f_equal C K (\lambda (e0: C).(match e0 in C 
+return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) 
+\Rightarrow k0])) (CHead c0 k v) (CHead e (Flat f) u) H4) in ((let H8 \def 
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with 
+[(CSort _) \Rightarrow c0 | (CHead c4 _ _) \Rightarrow c4])) (CHead c0 k v) 
+(CHead e (Flat f) u) H4) in (eq_ind C e (\lambda (c4: C).((eq K k (Flat f)) 
+\to ((eq T v u) \to ((eq C (CHead c3 k v) c2) \to ((csubc g c4 c3) \to (ex2 C 
+(\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2)))))))) 
+(\lambda (H9: (eq K k (Flat f))).(eq_ind K (Flat f) (\lambda (k0: K).((eq T v 
+u) \to ((eq C (CHead c3 k0 v) c2) \to ((csubc g e c3) \to (ex2 C (\lambda 
+(e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))))))) (\lambda (H10: 
+(eq T v u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c3 (Flat f) t) c2) \to 
+((csubc g e c3) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: 
+C).(csubc g c e2)))))) (\lambda (H11: (eq C (CHead c3 (Flat f) u) 
+c2)).(eq_ind C (CHead c3 (Flat f) u) (\lambda (c4: C).((csubc g e c3) \to 
+(ex2 C (\lambda (e2: C).(clear c4 e2)) (\lambda (e2: C).(csubc g c e2))))) 
+(\lambda (H12: (csubc g e c3)).(let H_x \def (H1 c3 H12) in (let H13 \def H_x 
+in (ex2_ind C (\lambda (e2: C).(clear c3 e2)) (\lambda (e2: C).(csubc g c 
+e2)) (ex2 C (\lambda (e2: C).(clear (CHead c3 (Flat f) u) e2)) (\lambda (e2: 
+C).(csubc g c e2))) (\lambda (x: C).(\lambda (H14: (clear c3 x)).(\lambda 
+(H15: (csubc g c x)).(ex_intro2 C (\lambda (e2: C).(clear (CHead c3 (Flat f) 
+u) e2)) (\lambda (e2: C).(csubc g c e2)) x (clear_flat c3 x H14 f u) H15)))) 
+H13)))) c2 H11)) v (sym_eq T v u H10))) k (sym_eq K k (Flat f) H9))) c0 
+(sym_eq C c0 e H8))) H7)) H6)) H5 H3))) | (csubc_abst c0 c3 H3 v a H4 w H5) 
+\Rightarrow (\lambda (H6: (eq C (CHead c0 (Bind Abst) v) (CHead e (Flat f) 
+u))).(\lambda (H7: (eq C (CHead c3 (Bind Abbr) w) c2)).((let H8 \def (eq_ind 
+C (CHead c0 (Bind Abst) v) (\lambda (e0: C).(match e0 in C return (\lambda 
+(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow 
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | 
+(Flat _) \Rightarrow False])])) I (CHead e (Flat f) u) H6) in (False_ind ((eq 
+C (CHead c3 (Bind Abbr) w) c2) \to ((csubc g c0 c3) \to ((sc3 g (asucc g a) 
+c0 v) \to ((sc3 g a c3 w) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda 
+(e2: C).(csubc g c e2))))))) H8)) H7 H3 H4 H5)))]) in (H3 (refl_equal C 
+(CHead e (Flat f) u)) (refl_equal C c2))))))))))) c1 e1 H)))).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/csuba.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/csuba.ma
new file mode 100644
index 000000000..783c668e6
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/csuba.ma
@@ -0,0 +1,38 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/csuba".
+
+include "csubc/defs.ma".
+
+include "sc3/props.ma".
+
+include "csuba/defs.ma".
+
+theorem csubc_csuba:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (csuba 
+g c1 c2))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 
+c2)).(csubc_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda 
+(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda 
+(_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda 
+(v: T).(csuba_head g c3 c4 H1 k v))))))) (\lambda (c3: C).(\lambda (c4: 
+C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (v: 
+T).(\lambda (a: A).(\lambda (H2: (sc3 g (asucc g a) c3 v)).(\lambda (w: 
+T).(\lambda (H3: (sc3 g a c4 w)).(csuba_abst g c3 c4 H1 v a (sc3_arity_gen g 
+c3 v (asucc g a) H2) w (sc3_arity_gen g c4 w a H3))))))))))) c1 c2 H)))).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/defs.ma
new file mode 100644
index 000000000..8f5830604
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/defs.ma
@@ -0,0 +1,29 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/defs".
+
+include "sc3/defs.ma".
+
+inductive csubc (g: G): C \to (C \to Prop) \def
+| csubc_sort: \forall (n: nat).(csubc g (CSort n) (CSort n))
+| csubc_head: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall 
+(k: K).(\forall (v: T).(csubc g (CHead c1 k v) (CHead c2 k v))))))
+| csubc_abst: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall 
+(v: T).(\forall (a: A).((sc3 g (asucc g a) c1 v) \to (\forall (w: T).((sc3 g 
+a c2 w) \to (csubc g (CHead c1 (Bind Abst) v) (CHead c2 (Bind Abbr) 
+w))))))))).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/drop.ma
new file mode 100644
index 000000000..f2d90423d
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/drop.ma
@@ -0,0 +1,450 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/drop".
+
+include "csubc/defs.ma".
+
+include "sc3/props.ma".
+
+theorem csubc_drop_conf_O:
+ \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (h: nat).((drop h 
+O c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: 
+C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (e1: 
+C).(\forall (h: nat).((drop h O c e1) \to (\forall (c2: C).((csubc g c c2) 
+\to (ex2 C (\lambda (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 
+e2))))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H: 
+(drop h O (CSort n) e1)).(\lambda (c2: C).(\lambda (H0: (csubc g (CSort n) 
+c2)).(and3_ind (eq C e1 (CSort n)) (eq nat h O) (eq nat O O) (ex2 C (\lambda 
+(e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (H1: 
+(eq C e1 (CSort n))).(\lambda (H2: (eq nat h O)).(\lambda (_: (eq nat O 
+O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (e2: C).(drop n0 O c2 
+e2)) (\lambda (e2: C).(csubc g e1 e2)))) (eq_ind_r C (CSort n) (\lambda (c: 
+C).(ex2 C (\lambda (e2: C).(drop O O c2 e2)) (\lambda (e2: C).(csubc g c 
+e2)))) (ex_intro2 C (\lambda (e2: C).(drop O O c2 e2)) (\lambda (e2: 
+C).(csubc g (CSort n) e2)) c2 (drop_refl c2) H0) e1 H1) h H2)))) 
+(drop_gen_sort n h O e1 H)))))))) (\lambda (c: C).(\lambda (H: ((\forall (e1: 
+C).(\forall (h: nat).((drop h O c e1) \to (\forall (c2: C).((csubc g c c2) 
+\to (ex2 C (\lambda (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 
+e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (h: 
+nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e1) \to (\forall 
+(c2: C).((csubc g (CHead c k t) c2) \to (ex2 C (\lambda (e2: C).(drop n O c2 
+e2)) (\lambda (e2: C).(csubc g e1 e2))))))) (\lambda (H0: (drop O O (CHead c 
+k t) e1)).(\lambda (c2: C).(\lambda (H1: (csubc g (CHead c k t) c2)).(eq_ind 
+C (CHead c k t) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop O O c2 e2)) 
+(\lambda (e2: C).(csubc g c0 e2)))) (ex_intro2 C (\lambda (e2: C).(drop O O 
+c2 e2)) (\lambda (e2: C).(csubc g (CHead c k t) e2)) c2 (drop_refl c2) H1) e1 
+(drop_gen_refl (CHead c k t) e1 H0))))) (\lambda (n: nat).(\lambda (H0: 
+(((drop n O (CHead c k t) e1) \to (\forall (c2: C).((csubc g (CHead c k t) 
+c2) \to (ex2 C (\lambda (e2: C).(drop n O c2 e2)) (\lambda (e2: C).(csubc g 
+e1 e2)))))))).(\lambda (H1: (drop (S n) O (CHead c k t) e1)).(\lambda (c2: 
+C).(\lambda (H2: (csubc g (CHead c k t) c2)).(let H3 \def (match H2 in csubc 
+return (\lambda (c0: C).(\lambda (c3: C).(\lambda (_: (csubc ? c0 c3)).((eq C 
+c0 (CHead c k t)) \to ((eq C c3 c2) \to (ex2 C (\lambda (e2: C).(drop (S n) O 
+c2 e2)) (\lambda (e2: C).(csubc g e1 e2)))))))) with [(csubc_sort n0) 
+\Rightarrow (\lambda (H3: (eq C (CSort n0) (CHead c k t))).(\lambda (H4: (eq 
+C (CSort n0) c2)).((let H5 \def (eq_ind C (CSort n0) (\lambda (e: C).(match e 
+in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ 
+_ _) \Rightarrow False])) I (CHead c k t) H3) in (False_ind ((eq C (CSort n0) 
+c2) \to (ex2 C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc 
+g e1 e2)))) H5)) H4))) | (csubc_head c0 c3 H3 k0 v) \Rightarrow (\lambda (H4: 
+(eq C (CHead c0 k0 v) (CHead c k t))).(\lambda (H5: (eq C (CHead c3 k0 v) 
+c2)).((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda 
+(_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t0) \Rightarrow t0])) 
+(CHead c0 k0 v) (CHead c k t) H4) in ((let H7 \def (f_equal C K (\lambda (e: 
+C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | 
+(CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 v) (CHead c k t) H4) in ((let 
+H8 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) 
+with [(CSort _) \Rightarrow c0 | (CHead c4 _ _) \Rightarrow c4])) (CHead c0 
+k0 v) (CHead c k t) H4) in (eq_ind C c (\lambda (c4: C).((eq K k0 k) \to ((eq 
+T v t) \to ((eq C (CHead c3 k0 v) c2) \to ((csubc g c4 c3) \to (ex2 C 
+(\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 
+e2)))))))) (\lambda (H9: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T v 
+t) \to ((eq C (CHead c3 k1 v) c2) \to ((csubc g c c3) \to (ex2 C (\lambda 
+(e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))))))) (\lambda 
+(H10: (eq T v t)).(eq_ind T t (\lambda (t0: T).((eq C (CHead c3 k t0) c2) \to 
+((csubc g c c3) \to (ex2 C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda 
+(e2: C).(csubc g e1 e2)))))) (\lambda (H11: (eq C (CHead c3 k t) c2)).(eq_ind 
+C (CHead c3 k t) (\lambda (c4: C).((csubc g c c3) \to (ex2 C (\lambda (e2: 
+C).(drop (S n) O c4 e2)) (\lambda (e2: C).(csubc g e1 e2))))) (\lambda (H12: 
+(csubc g c c3)).(let H_x \def (H e1 (r k n) (drop_gen_drop k c e1 t n H1) c3 
+H12) in (let H13 \def H_x in (ex2_ind C (\lambda (e2: C).(drop (r k n) O c3 
+e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: C).(drop (S n) O 
+(CHead c3 k t) e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: 
+C).(\lambda (H14: (drop (r k n) O c3 x)).(\lambda (H15: (csubc g e1 
+x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead c3 k t) e2)) (\lambda 
+(e2: C).(csubc g e1 e2)) x (drop_drop k n c3 x H14 t) H15)))) H13)))) c2 
+H11)) v (sym_eq T v t H10))) k0 (sym_eq K k0 k H9))) c0 (sym_eq C c0 c H8))) 
+H7)) H6)) H5 H3))) | (csubc_abst c0 c3 H3 v a H4 w H5) \Rightarrow (\lambda 
+(H6: (eq C (CHead c0 (Bind Abst) v) (CHead c k t))).(\lambda (H7: (eq C 
+(CHead c3 (Bind Abbr) w) c2)).((let H8 \def (f_equal C T (\lambda (e: 
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | 
+(CHead _ _ t0) \Rightarrow t0])) (CHead c0 (Bind Abst) v) (CHead c k t) H6) 
+in ((let H9 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda 
+(_: C).K) with [(CSort _) \Rightarrow (Bind Abst) | (CHead _ k0 _) 
+\Rightarrow k0])) (CHead c0 (Bind Abst) v) (CHead c k t) H6) in ((let H10 
+\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) 
+with [(CSort _) \Rightarrow c0 | (CHead c4 _ _) \Rightarrow c4])) (CHead c0 
+(Bind Abst) v) (CHead c k t) H6) in (eq_ind C c (\lambda (c4: C).((eq K (Bind 
+Abst) k) \to ((eq T v t) \to ((eq C (CHead c3 (Bind Abbr) w) c2) \to ((csubc 
+g c4 c3) \to ((sc3 g (asucc g a) c4 v) \to ((sc3 g a c3 w) \to (ex2 C 
+(\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 
+e2)))))))))) (\lambda (H11: (eq K (Bind Abst) k)).(eq_ind K (Bind Abst) 
+(\lambda (_: K).((eq T v t) \to ((eq C (CHead c3 (Bind Abbr) w) c2) \to 
+((csubc g c c3) \to ((sc3 g (asucc g a) c v) \to ((sc3 g a c3 w) \to (ex2 C 
+(\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 
+e2))))))))) (\lambda (H12: (eq T v t)).(eq_ind T t (\lambda (t0: T).((eq C 
+(CHead c3 (Bind Abbr) w) c2) \to ((csubc g c c3) \to ((sc3 g (asucc g a) c 
+t0) \to ((sc3 g a c3 w) \to (ex2 C (\lambda (e2: C).(drop (S n) O c2 e2)) 
+(\lambda (e2: C).(csubc g e1 e2)))))))) (\lambda (H13: (eq C (CHead c3 (Bind 
+Abbr) w) c2)).(eq_ind C (CHead c3 (Bind Abbr) w) (\lambda (c4: C).((csubc g c 
+c3) \to ((sc3 g (asucc g a) c t) \to ((sc3 g a c3 w) \to (ex2 C (\lambda (e2: 
+C).(drop (S n) O c4 e2)) (\lambda (e2: C).(csubc g e1 e2))))))) (\lambda 
+(H14: (csubc g c c3)).(\lambda (_: (sc3 g (asucc g a) c t)).(\lambda (_: (sc3 
+g a c3 w)).(let H17 \def (eq_ind_r K k (\lambda (k0: K).(drop (r k0 n) O c 
+e1)) (drop_gen_drop k c e1 t n H1) (Bind Abst) H11) in (let H18 \def 
+(eq_ind_r K k (\lambda (k0: K).((drop n O (CHead c k0 t) e1) \to (\forall 
+(c4: C).((csubc g (CHead c k0 t) c4) \to (ex2 C (\lambda (e2: C).(drop n O c4 
+e2)) (\lambda (e2: C).(csubc g e1 e2))))))) H0 (Bind Abst) H11) in (let H_x 
+\def (H e1 (r (Bind Abst) n) H17 c3 H14) in (let H19 \def H_x in (ex2_ind C 
+(\lambda (e2: C).(drop (r (Bind Abst) n) O c3 e2)) (\lambda (e2: C).(csubc g 
+e1 e2)) (ex2 C (\lambda (e2: C).(drop (S n) O (CHead c3 (Bind Abbr) w) e2)) 
+(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: C).(\lambda (H20: (drop (r 
+(Bind Abst) n) O c3 x)).(\lambda (H21: (csubc g e1 x)).(ex_intro2 C (\lambda 
+(e2: C).(drop (S n) O (CHead c3 (Bind Abbr) w) e2)) (\lambda (e2: C).(csubc g 
+e1 e2)) x (drop_drop (Bind Abbr) n c3 x H20 w) H21)))) H19)))))))) c2 H13)) v 
+(sym_eq T v t H12))) k H11)) c0 (sym_eq C c0 c H10))) H9)) H8)) H7 H3 H4 
+H5)))]) in (H3 (refl_equal C (CHead c k t)) (refl_equal C c2)))))))) h))))))) 
+c1)).
+
+theorem drop_csubc_trans:
+ \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall 
+(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C 
+(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))))))))
+\def
+ \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: 
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: 
+C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda 
+(c1: C).(csubc g c c1)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda 
+(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda 
+(e1: C).(\lambda (H0: (csubc g e2 e1)).(and3_ind (eq C e2 (CSort n)) (eq nat 
+h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: 
+C).(csubc g (CSort n) c1))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: 
+(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: 
+nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g 
+(CSort n) c1)))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: 
+C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)))) (let H4 \def 
+(eq_ind C e2 (\lambda (c: C).(csubc g c e1)) H0 (CSort n) H1) in (ex_intro2 C 
+(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)) 
+e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) 
+(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: 
+nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C 
+(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c 
+c1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: 
+nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) 
+e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h 
+n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) (\lambda (h: 
+nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall 
+(e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) 
+(\lambda (c1: C).(csubc g (CHead c k t) c1))))))) (\lambda (H0: (drop O O 
+(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2 
+\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g c0 e1)) H1 (CHead c k t) 
+(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O 
+O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)) e1 (drop_refl e1) 
+H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to 
+(\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 
+e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))).(\lambda (H1: (drop 
+(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 
+e1)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in 
+(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) 
+(\lambda (c1: C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 
+e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))) (\lambda (x: C).(\lambda 
+(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g c x)).(ex_intro2 C 
+(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k 
+t) c1)) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g c x H5 k t))))) 
+H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n 
+(CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda 
+(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) 
+c1))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) 
+e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 e1)).(ex3_2_ind C T (\lambda 
+(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: 
+T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k 
+n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: 
+C).(csubc g (CHead c k t) c1))) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
+(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) 
+x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda 
+(c0: C).(csubc g c0 e1)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 
+(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to 
+(\forall (e3: C).((csubc g c0 e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 
+e3)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) H0 (CHead x0 k x1) 
+H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 
+n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g (CHead x0 k 
+x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: 
+C).(csubc g (CHead c k t0) c1)))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r 
+T (lift h (r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) 
+c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t0) c1)))) (let H9 \def (match 
+H6 in csubc return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csubc ? c0 
+c1)).((eq C c0 (CHead x0 k x1)) \to ((eq C c1 e1) \to (ex2 C (\lambda (c3: 
+C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) 
+x1)) c3)))))))) with [(csubc_sort n0) \Rightarrow (\lambda (H9: (eq C (CSort 
+n0) (CHead x0 k x1))).(\lambda (H10: (eq C (CSort n0) e1)).((let H11 \def 
+(eq_ind C (CSort n0) (\lambda (e: C).(match e in C return (\lambda (_: 
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
+False])) I (CHead x0 k x1) H9) in (False_ind ((eq C (CSort n0) e1) \to (ex2 C 
+(\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g (CHead c k 
+(lift h (r k n) x1)) c1)))) H11)) H10))) | (csubc_head c1 c0 H9 k0 v) 
+\Rightarrow (\lambda (H10: (eq C (CHead c1 k0 v) (CHead x0 k x1))).(\lambda 
+(H11: (eq C (CHead c0 k0 v) e1)).((let H12 \def (f_equal C T (\lambda (e: 
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | 
+(CHead _ _ t0) \Rightarrow t0])) (CHead c1 k0 v) (CHead x0 k x1) H10) in 
+((let H13 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: 
+C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) 
+(CHead c1 k0 v) (CHead x0 k x1) H10) in ((let H14 \def (f_equal C C (\lambda 
+(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 
+| (CHead c3 _ _) \Rightarrow c3])) (CHead c1 k0 v) (CHead x0 k x1) H10) in 
+(eq_ind C x0 (\lambda (c3: C).((eq K k0 k) \to ((eq T v x1) \to ((eq C (CHead 
+c0 k0 v) e1) \to ((csubc g c3 c0) \to (ex2 C (\lambda (c4: C).(drop h (S n) 
+c4 e1)) (\lambda (c4: C).(csubc g (CHead c k (lift h (r k n) x1)) c4)))))))) 
+(\lambda (H15: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T v x1) \to 
+((eq C (CHead c0 k1 v) e1) \to ((csubc g x0 c0) \to (ex2 C (\lambda (c3: 
+C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) 
+x1)) c3))))))) (\lambda (H16: (eq T v x1)).(eq_ind T x1 (\lambda (t0: T).((eq 
+C (CHead c0 k t0) e1) \to ((csubc g x0 c0) \to (ex2 C (\lambda (c3: C).(drop 
+h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) 
+c3)))))) (\lambda (H17: (eq C (CHead c0 k x1) e1)).(eq_ind C (CHead c0 k x1) 
+(\lambda (c3: C).((csubc g x0 c0) \to (ex2 C (\lambda (c4: C).(drop h (S n) 
+c4 c3)) (\lambda (c4: C).(csubc g (CHead c k (lift h (r k n) x1)) c4))))) 
+(\lambda (H18: (csubc g x0 c0)).(let H_x \def (H x0 (r k n) h H5 c0 H18) in 
+(let H19 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r k n) c3 c0)) 
+(\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 
+(CHead c0 k x1))) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) 
+c3))) (\lambda (x: C).(\lambda (H20: (drop h (r k n) x c0)).(\lambda (H21: 
+(csubc g c x)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c0 k 
+x1))) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3)) (CHead x 
+k (lift h (r k n) x1)) (drop_skip k h n x c0 H20 x1) (csubc_head g c x H21 k 
+(lift h (r k n) x1)))))) H19)))) e1 H17)) v (sym_eq T v x1 H16))) k0 (sym_eq 
+K k0 k H15))) c1 (sym_eq C c1 x0 H14))) H13)) H12)) H11 H9))) | (csubc_abst 
+c1 c0 H9 v a H10 w H11) \Rightarrow (\lambda (H12: (eq C (CHead c1 (Bind 
+Abst) v) (CHead x0 k x1))).(\lambda (H13: (eq C (CHead c0 (Bind Abbr) w) 
+e1)).((let H14 \def (f_equal C T (\lambda (e: C).(match e in C return 
+(\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t0) \Rightarrow 
+t0])) (CHead c1 (Bind Abst) v) (CHead x0 k x1) H12) in ((let H15 \def 
+(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with 
+[(CSort _) \Rightarrow (Bind Abst) | (CHead _ k0 _) \Rightarrow k0])) (CHead 
+c1 (Bind Abst) v) (CHead x0 k x1) H12) in ((let H16 \def (f_equal C C 
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
+\Rightarrow c1 | (CHead c3 _ _) \Rightarrow c3])) (CHead c1 (Bind Abst) v) 
+(CHead x0 k x1) H12) in (eq_ind C x0 (\lambda (c3: C).((eq K (Bind Abst) k) 
+\to ((eq T v x1) \to ((eq C (CHead c0 (Bind Abbr) w) e1) \to ((csubc g c3 c0) 
+\to ((sc3 g (asucc g a) c3 v) \to ((sc3 g a c0 w) \to (ex2 C (\lambda (c4: 
+C).(drop h (S n) c4 e1)) (\lambda (c4: C).(csubc g (CHead c k (lift h (r k n) 
+x1)) c4)))))))))) (\lambda (H17: (eq K (Bind Abst) k)).(eq_ind K (Bind Abst) 
+(\lambda (k0: K).((eq T v x1) \to ((eq C (CHead c0 (Bind Abbr) w) e1) \to 
+((csubc g x0 c0) \to ((sc3 g (asucc g a) x0 v) \to ((sc3 g a c0 w) \to (ex2 C 
+(\lambda (c3: C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k0 
+(lift h (r k0 n) x1)) c3))))))))) (\lambda (H18: (eq T v x1)).(eq_ind T x1 
+(\lambda (t0: T).((eq C (CHead c0 (Bind Abbr) w) e1) \to ((csubc g x0 c0) \to 
+((sc3 g (asucc g a) x0 t0) \to ((sc3 g a c0 w) \to (ex2 C (\lambda (c3: 
+C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c (Bind Abst) (lift 
+h (r (Bind Abst) n) x1)) c3)))))))) (\lambda (H19: (eq C (CHead c0 (Bind 
+Abbr) w) e1)).(eq_ind C (CHead c0 (Bind Abbr) w) (\lambda (c3: C).((csubc g 
+x0 c0) \to ((sc3 g (asucc g a) x0 x1) \to ((sc3 g a c0 w) \to (ex2 C (\lambda 
+(c4: C).(drop h (S n) c4 c3)) (\lambda (c4: C).(csubc g (CHead c (Bind Abst) 
+(lift h (r (Bind Abst) n) x1)) c4))))))) (\lambda (H20: (csubc g x0 
+c0)).(\lambda (H21: (sc3 g (asucc g a) x0 x1)).(\lambda (H22: (sc3 g a c0 
+w)).(let H23 \def (eq_ind_r K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 
+n (CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: 
+C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c3: C).(drop h0 n c3 
+e3)) (\lambda (c3: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c3)))))))) 
+H8 (Bind Abst) H17) in (let H24 \def (eq_ind_r K k (\lambda (k0: K).(drop h 
+(r k0 n) c x0)) H5 (Bind Abst) H17) in (let H_x \def (H x0 (r (Bind Abst) n) 
+h H24 c0 H20) in (let H25 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r 
+(Bind Abst) n) c3 c0)) (\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (c3: 
+C).(drop h (S n) c3 (CHead c0 (Bind Abbr) w))) (\lambda (c3: C).(csubc g 
+(CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c3))) (\lambda (x: 
+C).(\lambda (H26: (drop h (r (Bind Abst) n) x c0)).(\lambda (H27: (csubc g c 
+x)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c0 (Bind Abbr) w))) 
+(\lambda (c3: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) 
+c3)) (CHead x (Bind Abbr) (lift h n w)) (drop_skip_bind h n x c0 H26 Abbr w) 
+(csubc_abst g c x H27 (lift h (r (Bind Abst) n) x1) a (sc3_lift g (asucc g a) 
+x0 x1 H21 c h (r (Bind Abst) n) H24) (lift h n w) (sc3_lift g a c0 w H22 x h 
+n H26)))))) H25)))))))) e1 H19)) v (sym_eq T v x1 H18))) k H17)) c1 (sym_eq C 
+c1 x0 H16))) H15)) H14)) H13 H9 H10 H11)))]) in (H9 (refl_equal C (CHead x0 k 
+x1)) (refl_equal C e1))) t H4))))))))) (drop_gen_skip_l c e2 t h n k 
+H1)))))))) d))))))) c2)).
+
+theorem csubc_drop_conf_rev:
+ \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall 
+(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C 
+(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))))))))
+\def
+ \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: 
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: 
+C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda 
+(c1: C).(csubc g c1 c)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda 
+(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda 
+(e1: C).(\lambda (H0: (csubc g e1 e2)).(and3_ind (eq C e2 (CSort n)) (eq nat 
+h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: 
+C).(csubc g c1 (CSort n)))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: 
+(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: 
+nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g c1 
+(CSort n))))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: 
+C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))))) (let H4 \def 
+(eq_ind C e2 (\lambda (c: C).(csubc g e1 c)) H0 (CSort n) H1) in (ex_intro2 C 
+(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))) 
+e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) 
+(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: 
+nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C 
+(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 
+c))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: 
+nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) 
+e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h 
+n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) (\lambda (h: 
+nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall 
+(e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) 
+(\lambda (c1: C).(csubc g c1 (CHead c k t)))))))) (\lambda (H0: (drop O O 
+(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2 
+\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g e1 c0)) H1 (CHead c k t) 
+(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O 
+O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))) e1 (drop_refl e1) 
+H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to 
+(\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 
+e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))).(\lambda (H1: (drop 
+(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 
+e2)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in 
+(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) 
+(\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 
+e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))) (\lambda (x: C).(\lambda 
+(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g x c)).(ex_intro2 C 
+(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c 
+k t))) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g x c H5 k t))))) 
+H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n 
+(CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda 
+(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k 
+t)))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) 
+e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 e2)).(ex3_2_ind C T (\lambda 
+(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: 
+T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k 
+n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: 
+C).(csubc g c1 (CHead c k t)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
+(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) 
+x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda 
+(c0: C).(csubc g e1 c0)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 
+(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to 
+(\forall (e3: C).((csubc g e3 c0) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 
+e3)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) H0 (CHead x0 k x1) 
+H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 
+n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g e3 (CHead x0 
+k x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc 
+g c1 (CHead c k t0))))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h 
+(r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) 
+(\lambda (c1: C).(csubc g c1 (CHead c k t0))))) (let H9 \def (match H6 in 
+csubc return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csubc ? c0 
+c1)).((eq C c0 e1) \to ((eq C c1 (CHead x0 k x1)) \to (ex2 C (\lambda (c3: 
+C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k 
+n) x1)))))))))) with [(csubc_sort n0) \Rightarrow (\lambda (H9: (eq C (CSort 
+n0) e1)).(\lambda (H10: (eq C (CSort n0) (CHead x0 k x1))).(eq_ind C (CSort 
+n0) (\lambda (c0: C).((eq C (CSort n0) (CHead x0 k x1)) \to (ex2 C (\lambda 
+(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h 
+(r k n) x1))))))) (\lambda (H11: (eq C (CSort n0) (CHead x0 k x1))).(let H12 
+\def (eq_ind C (CSort n0) (\lambda (e: C).(match e in C return (\lambda (_: 
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
+False])) I (CHead x0 k x1) H11) in (False_ind (ex2 C (\lambda (c1: C).(drop h 
+(S n) c1 (CSort n0))) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) 
+x1))))) H12))) e1 H9 H10))) | (csubc_head c1 c0 H9 k0 v) \Rightarrow (\lambda 
+(H10: (eq C (CHead c1 k0 v) e1)).(\lambda (H11: (eq C (CHead c0 k0 v) (CHead 
+x0 k x1))).(eq_ind C (CHead c1 k0 v) (\lambda (c3: C).((eq C (CHead c0 k0 v) 
+(CHead x0 k x1)) \to ((csubc g c1 c0) \to (ex2 C (\lambda (c4: C).(drop h (S 
+n) c4 c3)) (\lambda (c4: C).(csubc g c4 (CHead c k (lift h (r k n) x1)))))))) 
+(\lambda (H12: (eq C (CHead c0 k0 v) (CHead x0 k x1))).(let H13 \def (f_equal 
+C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
+\Rightarrow v | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k0 v) (CHead x0 k 
+x1) H12) in ((let H14 \def (f_equal C K (\lambda (e: C).(match e in C return 
+(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) 
+\Rightarrow k1])) (CHead c0 k0 v) (CHead x0 k x1) H12) in ((let H15 \def 
+(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with 
+[(CSort _) \Rightarrow c0 | (CHead c3 _ _) \Rightarrow c3])) (CHead c0 k0 v) 
+(CHead x0 k x1) H12) in (eq_ind C x0 (\lambda (c3: C).((eq K k0 k) \to ((eq T 
+v x1) \to ((csubc g c1 c3) \to (ex2 C (\lambda (c4: C).(drop h (S n) c4 
+(CHead c1 k0 v))) (\lambda (c4: C).(csubc g c4 (CHead c k (lift h (r k n) 
+x1))))))))) (\lambda (H16: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T v 
+x1) \to ((csubc g c1 x0) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead 
+c1 k1 v))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))))))) 
+(\lambda (H17: (eq T v x1)).(eq_ind T x1 (\lambda (t0: T).((csubc g c1 x0) 
+\to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k t0))) (\lambda (c3: 
+C).(csubc g c3 (CHead c k (lift h (r k n) x1))))))) (\lambda (H18: (csubc g 
+c1 x0)).(let H19 \def (eq_ind T v (\lambda (t0: T).(eq C (CHead c1 k0 t0) 
+e1)) H10 x1 H17) in (let H20 \def (eq_ind K k0 (\lambda (k1: K).(eq C (CHead 
+c1 k1 x1) e1)) H19 k H16) in (let H_x \def (H x0 (r k n) h H5 c1 H18) in (let 
+H21 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r k n) c3 c1)) (\lambda 
+(c3: C).(csubc g c3 c)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k 
+x1))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1))))) 
+(\lambda (x: C).(\lambda (H22: (drop h (r k n) x c1)).(\lambda (H23: (csubc g 
+x c)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k x1))) 
+(\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))) (CHead x k 
+(lift h (r k n) x1)) (drop_skip k h n x c1 H22 x1) (csubc_head g x c H23 k 
+(lift h (r k n) x1)))))) H21)))))) v (sym_eq T v x1 H17))) k0 (sym_eq K k0 k 
+H16))) c0 (sym_eq C c0 x0 H15))) H14)) H13))) e1 H10 H11 H9))) | (csubc_abst 
+c1 c0 H9 v a H10 w H11) \Rightarrow (\lambda (H12: (eq C (CHead c1 (Bind 
+Abst) v) e1)).(\lambda (H13: (eq C (CHead c0 (Bind Abbr) w) (CHead x0 k 
+x1))).(eq_ind C (CHead c1 (Bind Abst) v) (\lambda (c3: C).((eq C (CHead c0 
+(Bind Abbr) w) (CHead x0 k x1)) \to ((csubc g c1 c0) \to ((sc3 g (asucc g a) 
+c1 v) \to ((sc3 g a c0 w) \to (ex2 C (\lambda (c4: C).(drop h (S n) c4 c3)) 
+(\lambda (c4: C).(csubc g c4 (CHead c k (lift h (r k n) x1)))))))))) (\lambda 
+(H14: (eq C (CHead c0 (Bind Abbr) w) (CHead x0 k x1))).(let H15 \def (f_equal 
+C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
+\Rightarrow w | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 (Bind Abbr) w) 
+(CHead x0 k x1) H14) in ((let H16 \def (f_equal C K (\lambda (e: C).(match e 
+in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow (Bind Abbr) | 
+(CHead _ k0 _) \Rightarrow k0])) (CHead c0 (Bind Abbr) w) (CHead x0 k x1) 
+H14) in ((let H17 \def (f_equal C C (\lambda (e: C).(match e in C return 
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c3 _ _) 
+\Rightarrow c3])) (CHead c0 (Bind Abbr) w) (CHead x0 k x1) H14) in (eq_ind C 
+x0 (\lambda (c3: C).((eq K (Bind Abbr) k) \to ((eq T w x1) \to ((csubc g c1 
+c3) \to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a c3 w) \to (ex2 C (\lambda 
+(c4: C).(drop h (S n) c4 (CHead c1 (Bind Abst) v))) (\lambda (c4: C).(csubc g 
+c4 (CHead c k (lift h (r k n) x1))))))))))) (\lambda (H18: (eq K (Bind Abbr) 
+k)).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T w x1) \to ((csubc g c1 x0) 
+\to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a x0 w) \to (ex2 C (\lambda (c3: 
+C).(drop h (S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: C).(csubc g c3 
+(CHead c k0 (lift h (r k0 n) x1)))))))))) (\lambda (H19: (eq T w x1)).(eq_ind 
+T x1 (\lambda (t0: T).((csubc g c1 x0) \to ((sc3 g (asucc g a) c1 v) \to 
+((sc3 g a x0 t0) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 (Bind 
+Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c (Bind Abbr) (lift h (r (Bind 
+Abbr) n) x1))))))))) (\lambda (H20: (csubc g c1 x0)).(\lambda (H21: (sc3 g 
+(asucc g a) c1 v)).(\lambda (H22: (sc3 g a x0 x1)).(let H23 \def (eq_ind_r K 
+k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c k0 (lift h (r k0 
+n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3 (CHead x0 k0 x1)) 
+\to (ex2 C (\lambda (c3: C).(drop h0 n c3 e3)) (\lambda (c3: C).(csubc g c3 
+(CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind Abbr) H18) in (let H24 
+\def (eq_ind_r K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5 (Bind Abbr) 
+H18) in (let H_x \def (H x0 (r (Bind Abbr) n) h H24 c1 H20) in (let H25 \def 
+H_x in (ex2_ind C (\lambda (c3: C).(drop h (r (Bind Abbr) n) c3 c1)) (\lambda 
+(c3: C).(csubc g c3 c)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 
+(Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c (Bind Abbr) (lift h (r 
+(Bind Abbr) n) x1))))) (\lambda (x: C).(\lambda (H26: (drop h (r (Bind Abbr) 
+n) x c1)).(\lambda (H27: (csubc g x c)).(ex_intro2 C (\lambda (c3: C).(drop h 
+(S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c 
+(Bind Abbr) (lift h (r (Bind Abbr) n) x1)))) (CHead x (Bind Abst) (lift h n 
+v)) (drop_skip_bind h n x c1 H26 Abst v) (csubc_abst g x c H27 (lift h n v) a 
+(sc3_lift g (asucc g a) c1 v H21 x h n H26) (lift h (r (Bind Abbr) n) x1) 
+(sc3_lift g a x0 x1 H22 c h (r (Bind Abbr) n) H24)))))) H25)))))))) w (sym_eq 
+T w x1 H19))) k H18)) c0 (sym_eq C c0 x0 H17))) H16)) H15))) e1 H12 H13 H9 
+H10 H11)))]) in (H9 (refl_equal C e1) (refl_equal C (CHead x0 k x1)))) t 
+H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/drop1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/drop1.ma
new file mode 100644
index 000000000..9c6355244
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/drop1.ma
@@ -0,0 +1,198 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/drop1".
+
+include "csubc/drop.ma".
+
+theorem drop1_csubc_trans:
+ \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: 
+C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C 
+(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))
+\def
+ \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
+(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 
+e1) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 
+c1))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 
+e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e2 e1)).(let H1 \def (match H in 
+drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda 
+(_: (drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 e2) \to 
+(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c2 
+c1)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil 
+PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c e2)).(eq_ind C c2 
+(\lambda (c0: C).((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1 
+e1)) (\lambda (c1: C).(csubc g c2 c1))))) (\lambda (H4: (eq C c2 e2)).(eq_ind 
+C e2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda 
+(c1: C).(csubc g c0 c1)))) (let H5 \def (eq_ind_r C e2 (\lambda (c0: 
+C).(csubc g c0 e1)) H0 c2 H4) in (eq_ind C c2 (\lambda (c0: C).(ex2 C 
+(\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c0 c1)))) 
+(ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g 
+c2 c1)) e1 (drop1_nil e1) H5) e2 H4)) c2 (sym_eq C c2 e2 H4))) c (sym_eq C c 
+c2 H2) H3)))) | (drop1_cons c1 c0 h d H1 c3 hds0 H2) \Rightarrow (\lambda 
+(H3: (eq PList (PCons h d hds0) PNil)).(\lambda (H4: (eq C c1 c2)).(\lambda 
+(H5: (eq C c3 e2)).((let H6 \def (eq_ind PList (PCons h d hds0) (\lambda (e: 
+PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil 
+\Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in 
+(False_ind ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop h d c1 c0) \to ((drop1 
+hds0 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 PNil c4 e1)) (\lambda (c4: 
+C).(csubc g c2 c4))))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList 
+PNil) (refl_equal C c2) (refl_equal C e2)))))))) (\lambda (n: nat).(\lambda 
+(n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: C).(\forall (e2: 
+C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda 
+(c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))).(\lambda 
+(c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n n0 p) c2 e2)).(\lambda 
+(e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2 \def (match H0 in drop1 return 
+(\lambda (p0: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p0 
+c c0)).((eq PList p0 (PCons n n0 p)) \to ((eq C c c2) \to ((eq C c0 e2) \to 
+(ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc 
+g c2 c1)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList 
+PNil (PCons n n0 p))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c 
+e2)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e in PList 
+return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) 
+\Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c2) \to ((eq 
+C c e2) \to (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda 
+(c1: C).(csubc g c2 c1))))) H5)) H3 H4)))) | (drop1_cons c1 c0 h d H2 c3 hds0 
+H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds0) (PCons n n0 
+p))).(\lambda (H5: (eq C c1 c2)).(\lambda (H6: (eq C c3 e2)).((let H7 \def 
+(f_equal PList PList (\lambda (e: PList).(match e in PList return (\lambda 
+(_: PList).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p0) \Rightarrow 
+p0])) (PCons h d hds0) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat 
+(\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with 
+[PNil \Rightarrow d | (PCons _ n1 _) \Rightarrow n1])) (PCons h d hds0) 
+(PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e: 
+PList).(match e in PList return (\lambda (_: PList).nat) with [PNil 
+\Rightarrow h | (PCons n1 _ _) \Rightarrow n1])) (PCons h d hds0) (PCons n n0 
+p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds0 
+p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n1 d c1 c0) \to ((drop1 
+hds0 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) 
+(\lambda (c4: C).(csubc g c2 c4)))))))))) (\lambda (H10: (eq nat d 
+n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds0 p) \to ((eq C c1 c2) 
+\to ((eq C c3 e2) \to ((drop n n1 c1 c0) \to ((drop1 hds0 c0 c3) \to (ex2 C 
+(\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c2 
+c4))))))))) (\lambda (H11: (eq PList hds0 p)).(eq_ind PList p (\lambda (p0: 
+PList).((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n n0 c1 c0) \to ((drop1 p0 
+c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda 
+(c4: C).(csubc g c2 c4)))))))) (\lambda (H12: (eq C c1 c2)).(eq_ind C c2 
+(\lambda (c: C).((eq C c3 e2) \to ((drop n n0 c c0) \to ((drop1 p c0 c3) \to 
+(ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc 
+g c2 c4))))))) (\lambda (H13: (eq C c3 e2)).(eq_ind C e2 (\lambda (c: 
+C).((drop n n0 c2 c0) \to ((drop1 p c0 c) \to (ex2 C (\lambda (c4: C).(drop1 
+(PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c2 c4)))))) (\lambda (H14: 
+(drop n n0 c2 c0)).(\lambda (H15: (drop1 p c0 e2)).(let H_x \def (H c0 e2 H15 
+e1 H1) in (let H16 \def H_x in (ex2_ind C (\lambda (c4: C).(drop1 p c4 e1)) 
+(\lambda (c4: C).(csubc g c0 c4)) (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 
+p) c4 e1)) (\lambda (c4: C).(csubc g c2 c4))) (\lambda (x: C).(\lambda (H17: 
+(drop1 p x e1)).(\lambda (H18: (csubc g c0 x)).(let H_x0 \def 
+(drop_csubc_trans g c2 c0 n0 n H14 x H18) in (let H19 \def H_x0 in (ex2_ind C 
+(\lambda (c4: C).(drop n n0 c4 x)) (\lambda (c4: C).(csubc g c2 c4)) (ex2 C 
+(\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c2 
+c4))) (\lambda (x0: C).(\lambda (H20: (drop n n0 x0 x)).(\lambda (H21: (csubc 
+g c2 x0)).(ex_intro2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) 
+(\lambda (c4: C).(csubc g c2 c4)) x0 (drop1_cons x0 x n n0 H20 e1 p H17) 
+H21)))) H19)))))) H16))))) c3 (sym_eq C c3 e2 H13))) c1 (sym_eq C c1 c2 
+H12))) hds0 (sym_eq PList hds0 p H11))) d (sym_eq nat d n0 H10))) h (sym_eq 
+nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n 
+n0 p)) (refl_equal C c2) (refl_equal C e2)))))))))))) hds)).
+
+theorem csubc_drop1_conf_rev:
+ \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: 
+C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C 
+(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))
+\def
+ \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
+(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 
+e2) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 
+c2))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 
+e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e1 e2)).(let H1 \def (match H in 
+drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda 
+(_: (drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 e2) \to 
+(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 
+c2)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil 
+PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c e2)).(eq_ind C c2 
+(\lambda (c0: C).((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1 
+e1)) (\lambda (c1: C).(csubc g c1 c2))))) (\lambda (H4: (eq C c2 e2)).(eq_ind 
+C e2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda 
+(c1: C).(csubc g c1 c0)))) (let H5 \def (eq_ind_r C e2 (\lambda (c0: 
+C).(csubc g e1 c0)) H0 c2 H4) in (eq_ind C c2 (\lambda (c0: C).(ex2 C 
+(\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 c0)))) 
+(ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g 
+c1 c2)) e1 (drop1_nil e1) H5) e2 H4)) c2 (sym_eq C c2 e2 H4))) c (sym_eq C c 
+c2 H2) H3)))) | (drop1_cons c1 c0 h d H1 c3 hds0 H2) \Rightarrow (\lambda 
+(H3: (eq PList (PCons h d hds0) PNil)).(\lambda (H4: (eq C c1 c2)).(\lambda 
+(H5: (eq C c3 e2)).((let H6 \def (eq_ind PList (PCons h d hds0) (\lambda (e: 
+PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil 
+\Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in 
+(False_ind ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop h d c1 c0) \to ((drop1 
+hds0 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 PNil c4 e1)) (\lambda (c4: 
+C).(csubc g c4 c2))))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList 
+PNil) (refl_equal C c2) (refl_equal C e2)))))))) (\lambda (n: nat).(\lambda 
+(n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: C).(\forall (e2: 
+C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda 
+(c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))).(\lambda 
+(c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n n0 p) c2 e2)).(\lambda 
+(e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2 \def (match H0 in drop1 return 
+(\lambda (p0: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p0 
+c c0)).((eq PList p0 (PCons n n0 p)) \to ((eq C c c2) \to ((eq C c0 e2) \to 
+(ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc 
+g c1 c2)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList 
+PNil (PCons n n0 p))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c 
+e2)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e in PList 
+return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) 
+\Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c2) \to ((eq 
+C c e2) \to (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda 
+(c1: C).(csubc g c1 c2))))) H5)) H3 H4)))) | (drop1_cons c1 c0 h d H2 c3 hds0 
+H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds0) (PCons n n0 
+p))).(\lambda (H5: (eq C c1 c2)).(\lambda (H6: (eq C c3 e2)).((let H7 \def 
+(f_equal PList PList (\lambda (e: PList).(match e in PList return (\lambda 
+(_: PList).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p0) \Rightarrow 
+p0])) (PCons h d hds0) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat 
+(\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with 
+[PNil \Rightarrow d | (PCons _ n1 _) \Rightarrow n1])) (PCons h d hds0) 
+(PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e: 
+PList).(match e in PList return (\lambda (_: PList).nat) with [PNil 
+\Rightarrow h | (PCons n1 _ _) \Rightarrow n1])) (PCons h d hds0) (PCons n n0 
+p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds0 
+p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n1 d c1 c0) \to ((drop1 
+hds0 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) 
+(\lambda (c4: C).(csubc g c4 c2)))))))))) (\lambda (H10: (eq nat d 
+n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds0 p) \to ((eq C c1 c2) 
+\to ((eq C c3 e2) \to ((drop n n1 c1 c0) \to ((drop1 hds0 c0 c3) \to (ex2 C 
+(\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c4 
+c2))))))))) (\lambda (H11: (eq PList hds0 p)).(eq_ind PList p (\lambda (p0: 
+PList).((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n n0 c1 c0) \to ((drop1 p0 
+c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda 
+(c4: C).(csubc g c4 c2)))))))) (\lambda (H12: (eq C c1 c2)).(eq_ind C c2 
+(\lambda (c: C).((eq C c3 e2) \to ((drop n n0 c c0) \to ((drop1 p c0 c3) \to 
+(ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc 
+g c4 c2))))))) (\lambda (H13: (eq C c3 e2)).(eq_ind C e2 (\lambda (c: 
+C).((drop n n0 c2 c0) \to ((drop1 p c0 c) \to (ex2 C (\lambda (c4: C).(drop1 
+(PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c4 c2)))))) (\lambda (H14: 
+(drop n n0 c2 c0)).(\lambda (H15: (drop1 p c0 e2)).(let H_x \def (H c0 e2 H15 
+e1 H1) in (let H16 \def H_x in (ex2_ind C (\lambda (c4: C).(drop1 p c4 e1)) 
+(\lambda (c4: C).(csubc g c4 c0)) (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 
+p) c4 e1)) (\lambda (c4: C).(csubc g c4 c2))) (\lambda (x: C).(\lambda (H17: 
+(drop1 p x e1)).(\lambda (H18: (csubc g x c0)).(let H_x0 \def 
+(csubc_drop_conf_rev g c2 c0 n0 n H14 x H18) in (let H19 \def H_x0 in 
+(ex2_ind C (\lambda (c4: C).(drop n n0 c4 x)) (\lambda (c4: C).(csubc g c4 
+c2)) (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: 
+C).(csubc g c4 c2))) (\lambda (x0: C).(\lambda (H20: (drop n n0 x0 
+x)).(\lambda (H21: (csubc g x0 c2)).(ex_intro2 C (\lambda (c4: C).(drop1 
+(PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c4 c2)) x0 (drop1_cons x0 x 
+n n0 H20 e1 p H17) H21)))) H19)))))) H16))))) c3 (sym_eq C c3 e2 H13))) c1 
+(sym_eq C c1 c2 H12))) hds0 (sym_eq PList hds0 p H11))) d (sym_eq nat d n0 
+H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal 
+PList (PCons n n0 p)) (refl_equal C c2) (refl_equal C e2)))))))))))) hds)).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/getl.ma
new file mode 100644
index 000000000..6e07e138e
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/getl.ma
@@ -0,0 +1,44 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/getl".
+
+include "csubc/drop.ma".
+
+include "csubc/clear.ma".
+
+theorem csubc_getl_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (i: nat).((getl i 
+c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: 
+C).(getl i c2 e2)) (\lambda (e2: C).(csubc g e1 e2)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (e1: C).(\lambda (i: nat).(\lambda 
+(H: (getl i c1 e1)).(\lambda (c2: C).(\lambda (H0: (csubc g c1 c2)).(let H1 
+\def (getl_gen_all c1 e1 i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) 
+(\lambda (e: C).(clear e e1)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) 
+(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: C).(\lambda (H2: (drop i O c1 
+x)).(\lambda (H3: (clear x e1)).(let H_x \def (csubc_drop_conf_O g c1 x i H2 
+c2 H0) in (let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(drop i O c2 e2)) 
+(\lambda (e2: C).(csubc g x e2)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) 
+(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O 
+c2 x0)).(\lambda (H6: (csubc g x x0)).(let H_x0 \def (csubc_clear_conf g x e1 
+H3 x0 H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (e2: C).(clear x0 e2)) 
+(\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) 
+(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x1: C).(\lambda (H8: (clear x0 
+x1)).(\lambda (H9: (csubc g e1 x1)).(ex_intro2 C (\lambda (e2: C).(getl i c2 
+e2)) (\lambda (e2: C).(csubc g e1 e2)) x1 (getl_intro i c2 x1 x0 H5 H8) 
+H9)))) H7)))))) H4)))))) H1)))))))).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/props.ma
new file mode 100644
index 000000000..84fa2ec98
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/props.ma
@@ -0,0 +1,29 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csubc/props".
+
+include "csubc/defs.ma".
+
+include "sc3/props.ma".
+
+theorem csubc_refl:
+ \forall (g: G).(\forall (c: C).(csubc g c c))
+\def
+ \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubc g c0 c0)) 
+(\lambda (n: nat).(csubc_sort g n)) (\lambda (c0: C).(\lambda (H: (csubc g c0 
+c0)).(\lambda (k: K).(\lambda (t: T).(csubc_head g c0 c0 H k t))))) c)).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubst1/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubst1/defs.ma
index 00a6abca1..c4950fb62 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubst1/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubst1/defs.ma
@@ -18,7 +18,7 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csubst1/defs".
 
 include "csubst0/defs.ma".
 
-inductive csubst1 (i:nat) (v:T) (c1:C): C \to Prop \def
+inductive csubst1 (i: nat) (v: T) (c1: C): C \to Prop \def
 | csubst1_refl: csubst1 i v c1 c1
 | csubst1_sing: \forall (c2: C).((csubst0 i v c1 c2) \to (csubst1 i v c1 c2)).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/clear.ma
new file mode 100644
index 000000000..845371e0f
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/clear.ma
@@ -0,0 +1,72 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/clear".
+
+include "csubt/defs.ma".
+
+include "clear/fwd.ma".
+
+theorem csubt_clear_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to 
+(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) 
+(\lambda (e2: C).(clear c2 e2))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 
+c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c 
+e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear c0 
+e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) 
+e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csubt g e1 e2)) 
+(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: 
+C).(\lambda (H0: (csubt g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 
+e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear c4 
+e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: 
+(clear (CHead c3 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c3 k0 u) 
+e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear 
+(CHead c4 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c3 (Bind 
+b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda 
+(e2: C).(csubt g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)))) 
+(ex_intro2 C (\lambda (e2: C).(csubt g (CHead c3 (Bind b) u) e2)) (\lambda 
+(e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csubt_head g 
+c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3)))) 
+(\lambda (f: F).(\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def 
+(H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csubt g 
+e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csubt g e1 
+e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: 
+C).(\lambda (H5: (csubt g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C 
+(\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) 
+u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: 
+C).(\lambda (c4: C).(\lambda (H0: (csubt g c3 c4)).(\lambda (_: ((\forall 
+(e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda 
+(e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b 
+Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: 
+(clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1) 
+(\lambda (c: C).(ex2 C (\lambda (e2: C).(csubt g c e2)) (\lambda (e2: 
+C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csubt 
+g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) 
+u2) e2)) (CHead c4 (Bind b) u2) (csubt_void g c3 c4 H0 b H2 u1 u2) 
+(clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3)))))))))))) 
+(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubt g c3 c4)).(\lambda (_: 
+((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) 
+(\lambda (e2: C).(clear c4 e2))))))).(\lambda (u: T).(\lambda (t: T).(\lambda 
+(H2: (ty3 g c4 u t)).(\lambda (e1: C).(\lambda (H3: (clear (CHead c3 (Bind 
+Abst) t) e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda (c: C).(ex2 C 
+(\lambda (e2: C).(csubt g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind 
+Abbr) u) e2)))) (ex_intro2 C (\lambda (e2: C).(csubt g (CHead c3 (Bind Abst) 
+t) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)) (CHead c4 (Bind 
+Abbr) u) (csubt_abst g c3 c4 H0 u t H2) (clear_bind Abbr c4 u)) e1 
+(clear_gen_bind Abst c3 e1 t H3))))))))))) c1 c2 H)))).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/defs.ma
new file mode 100644
index 000000000..bbdaaae50
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/defs.ma
@@ -0,0 +1,31 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/defs".
+
+include "ty3/defs.ma".
+
+inductive csubt (g: G): C \to (C \to Prop) \def
+| csubt_sort: \forall (n: nat).(csubt g (CSort n) (CSort n))
+| csubt_head: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall 
+(k: K).(\forall (u: T).(csubt g (CHead c1 k u) (CHead c2 k u))))))
+| csubt_void: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall 
+(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csubt g 
+(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2))))))))
+| csubt_abst: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall 
+(u: T).(\forall (t: T).((ty3 g c2 u t) \to (csubt g (CHead c1 (Bind Abst) t) 
+(CHead c2 (Bind Abbr) u))))))).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/drop.ma
new file mode 100644
index 000000000..c6216627e
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/drop.ma
@@ -0,0 +1,805 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/drop".
+
+include "csubt/defs.ma".
+
+include "drop/fwd.ma".
+
+theorem csubt_drop_flat:
+ \forall (g: G).(\forall (f: F).(\forall (n: nat).(\forall (c1: C).(\forall 
+(c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 
+(CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(drop n O c2 (CHead d2 (Flat f) u))))))))))))
+\def
+ \lambda (g: G).(\lambda (f: F).(\lambda (n: nat).(nat_ind (\lambda (n0: 
+nat).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: 
+C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda 
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f) 
+u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 
+c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 
+(Flat f) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H 
+(CHead d1 (Flat f) u) (drop_gen_refl c1 (CHead d1 (Flat f) u) H0)) in (let H2 
+\def (match H1 in csubt return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: 
+(csubt ? c c0)).((eq C c (CHead d1 (Flat f) u)) \to ((eq C c0 c2) \to (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 
+(Flat f) u))))))))) with [(csubt_sort n0) \Rightarrow (\lambda (H2: (eq C 
+(CSort n0) (CHead d1 (Flat f) u))).(\lambda (H3: (eq C (CSort n0) c2)).((let 
+H4 \def (eq_ind C (CSort n0) (\lambda (e: C).(match e in C return (\lambda 
+(_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
+False])) I (CHead d1 (Flat f) u) H2) in (False_ind ((eq C (CSort n0) c2) \to 
+(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead 
+d2 (Flat f) u))))) H4)) H3))) | (csubt_head c0 c3 H2 k u0) \Rightarrow 
+(\lambda (H3: (eq C (CHead c0 k u0) (CHead d1 (Flat f) u))).(\lambda (H4: (eq 
+C (CHead c3 k u0) c2)).((let H5 \def (f_equal C T (\lambda (e: C).(match e in 
+C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) 
+\Rightarrow t])) (CHead c0 k u0) (CHead d1 (Flat f) u) H3) in ((let H6 \def 
+(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with 
+[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u0) 
+(CHead d1 (Flat f) u) H3) in ((let H7 \def (f_equal C C (\lambda (e: 
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | 
+(CHead c _ _) \Rightarrow c])) (CHead c0 k u0) (CHead d1 (Flat f) u) H3) in 
+(eq_ind C d1 (\lambda (c: C).((eq K k (Flat f)) \to ((eq T u0 u) \to ((eq C 
+(CHead c3 k u0) c2) \to ((csubt g c c3) \to (ex2 C (\lambda (d2: C).(csubt g 
+d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))))) (\lambda 
+(H8: (eq K k (Flat f))).(eq_ind K (Flat f) (\lambda (k0: K).((eq T u0 u) \to 
+((eq C (CHead c3 k0 u0) c2) \to ((csubt g d1 c3) \to (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) 
+u)))))))) (\lambda (H9: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C 
+(CHead c3 (Flat f) t) c2) \to ((csubt g d1 c3) \to (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))) 
+(\lambda (H10: (eq C (CHead c3 (Flat f) u) c2)).(eq_ind C (CHead c3 (Flat f) 
+u) (\lambda (c: C).((csubt g d1 c3) \to (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(drop O O c (CHead d2 (Flat f) u)))))) (\lambda (H11: 
+(csubt g d1 c3)).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop O O (CHead c3 (Flat f) u) (CHead d2 (Flat f) u))) c3 H11 (drop_refl 
+(CHead c3 (Flat f) u)))) c2 H10)) u0 (sym_eq T u0 u H9))) k (sym_eq K k (Flat 
+f) H8))) c0 (sym_eq C c0 d1 H7))) H6)) H5)) H4 H2))) | (csubt_void c0 c3 H2 b 
+H3 u1 u2) \Rightarrow (\lambda (H4: (eq C (CHead c0 (Bind Void) u1) (CHead d1 
+(Flat f) u))).(\lambda (H5: (eq C (CHead c3 (Bind b) u2) c2)).((let H6 \def 
+(eq_ind C (CHead c0 (Bind Void) u1) (\lambda (e: C).(match e in C return 
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 (Flat f) u) 
+H4) in (False_ind ((eq C (CHead c3 (Bind b) u2) c2) \to ((csubt g c0 c3) \to 
+((not (eq B b Void)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))) H6)) H5 H2 H3))) | 
+(csubt_abst c0 c3 H2 u0 t H3) \Rightarrow (\lambda (H4: (eq C (CHead c0 (Bind 
+Abst) t) (CHead d1 (Flat f) u))).(\lambda (H5: (eq C (CHead c3 (Bind Abbr) 
+u0) c2)).((let H6 \def (eq_ind C (CHead c0 (Bind Abst) t) (\lambda (e: 
+C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow 
+False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) 
+with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 
+(Flat f) u) H4) in (False_ind ((eq C (CHead c3 (Bind Abbr) u0) c2) \to 
+((csubt g c0 c3) \to ((ty3 g c3 u0 t) \to (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))) H6)) H5 H2 
+H3)))]) in (H2 (refl_equal C (CHead d1 (Flat f) u)) (refl_equal C 
+c2)))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: 
+C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 
+(CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(drop n0 O c2 (CHead d2 (Flat f) u)))))))))))).(\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (H0: (csubt g c1 c2)).(csubt_ind g (\lambda (c: 
+C).(\lambda (c0: C).(\forall (d1: C).(\forall (u: T).((drop (S n0) O c (CHead 
+d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O c0 (CHead d2 (Flat f) u))))))))) (\lambda (n1: 
+nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n0) O (CSort n1) 
+(CHead d1 (Flat f) u))).(let H2 \def (match H1 in drop return (\lambda (n2: 
+nat).(\lambda (n3: nat).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop 
+n2 n3 c c0)).((eq nat n2 (S n0)) \to ((eq nat n3 O) \to ((eq C c (CSort n1)) 
+\to ((eq C c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) 
+u))))))))))))) with [(drop_refl c) \Rightarrow (\lambda (H2: (eq nat O (S 
+n0))).(\lambda (H3: (eq nat O O)).(\lambda (H4: (eq C c (CSort n1))).(\lambda 
+(H5: (eq C c (CHead d1 (Flat f) u))).((let H6 \def (eq_ind nat O (\lambda (e: 
+nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True 
+| (S _) \Rightarrow False])) I (S n0) H2) in (False_ind ((eq nat O O) \to 
+((eq C c (CSort n1)) \to ((eq C c (CHead d1 (Flat f) u)) \to (ex2 C (\lambda 
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 
+(Flat f) u))))))) H6)) H3 H4 H5))))) | (drop_drop k h c e H2 u0) \Rightarrow 
+(\lambda (H3: (eq nat (S h) (S n0))).(\lambda (H4: (eq nat O O)).(\lambda 
+(H5: (eq C (CHead c k u0) (CSort n1))).(\lambda (H6: (eq C e (CHead d1 (Flat 
+f) u))).((let H7 \def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat 
+return (\lambda (_: nat).nat) with [O \Rightarrow h | (S n2) \Rightarrow 
+n2])) (S h) (S n0) H3) in (eq_ind nat n0 (\lambda (n2: nat).((eq nat O O) \to 
+((eq C (CHead c k u0) (CSort n1)) \to ((eq C e (CHead d1 (Flat f) u)) \to 
+((drop (r k n2) O c e) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u))))))))) (\lambda (_: 
+(eq nat O O)).(\lambda (H9: (eq C (CHead c k u0) (CSort n1))).(let H10 \def 
+(eq_ind C (CHead c k u0) (\lambda (e0: C).(match e0 in C return (\lambda (_: 
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow 
+True])) I (CSort n1) H9) in (False_ind ((eq C e (CHead d1 (Flat f) u)) \to 
+((drop (r k n0) O c e) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u)))))) H10)))) h 
+(sym_eq nat h n0 H7))) H4 H5 H6 H2))))) | (drop_skip k h d c e H2 u0) 
+\Rightarrow (\lambda (H3: (eq nat h (S n0))).(\lambda (H4: (eq nat (S d) 
+O)).(\lambda (H5: (eq C (CHead c k (lift h (r k d) u0)) (CSort n1))).(\lambda 
+(H6: (eq C (CHead e k u0) (CHead d1 (Flat f) u))).(eq_ind nat (S n0) (\lambda 
+(n2: nat).((eq nat (S d) O) \to ((eq C (CHead c k (lift n2 (r k d) u0)) 
+(CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 (Flat f) u)) \to ((drop n2 (r 
+k d) c e) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop 
+(S n0) O (CSort n1) (CHead d2 (Flat f) u))))))))) (\lambda (H7: (eq nat (S d) 
+O)).(let H8 \def (eq_ind nat (S d) (\lambda (e0: nat).(match e0 in nat return 
+(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) 
+I O H7) in (False_ind ((eq C (CHead c k (lift (S n0) (r k d) u0)) (CSort n1)) 
+\to ((eq C (CHead e k u0) (CHead d1 (Flat f) u)) \to ((drop (S n0) (r k d) c 
+e) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) 
+O (CSort n1) (CHead d2 (Flat f) u))))))) H8))) h (sym_eq nat h (S n0) H3) H4 
+H5 H6 H2)))))]) in (H2 (refl_equal nat (S n0)) (refl_equal nat O) (refl_equal 
+C (CSort n1)) (refl_equal C (CHead d1 (Flat f) u)))))))) (\lambda (c0: 
+C).(\lambda (c3: C).(\lambda (H1: (csubt g c0 c3)).(\lambda (H2: ((\forall 
+(d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead 
+d2 (Flat f) u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: 
+T).(\forall (d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead 
+d1 (Flat f) u0)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Flat f) u0))))))))) (\lambda (b: 
+B).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S 
+n0) O (CHead c0 (Bind b) u) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) 
+(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
+(CHead c3 (Bind b) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: 
+(csubt g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x (Flat f) 
+u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop 
+(S n0) O (CHead c3 (Bind b) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Bind 
+b) n0 c3 (CHead x (Flat f) u0) H5 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop 
+(Bind b) c0 (CHead d1 (Flat f) u0) u n0 H3)))))))) (\lambda (f0: F).(\lambda 
+(u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead 
+c0 (Flat f0) u) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g 
+d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u0))) (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Flat f0) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g 
+d1 x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Flat f) u0))).(ex_intro2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Flat f0) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Flat f0) n0 c3 (CHead 
+x (Flat f) u0) H5 u))))) (H2 d1 u0 (drop_gen_drop (Flat f0) c0 (CHead d1 
+(Flat f) u0) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: 
+C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: 
+T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) 
+u))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S 
+n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Flat f) u))).(ex2_ind C (\lambda 
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) 
+u))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
+(CHead c3 (Bind b) u2) (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H5: 
+(csubt g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u))).(ex_intro2 
+C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind b) u2) (CHead d2 (Flat f) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead x 
+(Flat f) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 (CHead 
+d1 (Flat f) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: 
+C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: 
+T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) 
+u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c3 u 
+t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H4: (drop (S n0) O (CHead c0 
+(Bind Abst) t) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g 
+d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind Abbr) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H5: (csubt 
+g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind Abbr) u) (CHead d2 (Flat f) u0))) x H5 (drop_drop (Bind Abbr) n0 c3 
+(CHead x (Flat f) u0) H6 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) 
+c0 (CHead d1 (Flat f) u0) t n0 H4))))))))))))) c1 c2 H0)))))) n))).
+
+theorem csubt_drop_abbr:
+ \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g 
+c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind 
+Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop 
+n O c2 (CHead d2 (Bind Abbr) u)))))))))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: 
+C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: 
+T).((drop n0 O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) 
+u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 
+c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 
+(Bind Abbr) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H 
+(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H0)) in 
+(let H2 \def (match H1 in csubt return (\lambda (c: C).(\lambda (c0: 
+C).(\lambda (_: (csubt ? c c0)).((eq C c (CHead d1 (Bind Abbr) u)) \to ((eq C 
+c0 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O 
+O c2 (CHead d2 (Bind Abbr) u))))))))) with [(csubt_sort n0) \Rightarrow 
+(\lambda (H2: (eq C (CSort n0) (CHead d1 (Bind Abbr) u))).(\lambda (H3: (eq C 
+(CSort n0) c2)).((let H4 \def (eq_ind C (CSort n0) (\lambda (e: C).(match e 
+in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ 
+_ _) \Rightarrow False])) I (CHead d1 (Bind Abbr) u) H2) in (False_ind ((eq C 
+(CSort n0) c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop O O c2 (CHead d2 (Bind Abbr) u))))) H4)) H3))) | (csubt_head c0 c3 
+H2 k u0) \Rightarrow (\lambda (H3: (eq C (CHead c0 k u0) (CHead d1 (Bind 
+Abbr) u))).(\lambda (H4: (eq C (CHead c3 k u0) c2)).((let H5 \def (f_equal C 
+T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
+\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u0) (CHead d1 
+(Bind Abbr) u) H3) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in 
+C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) 
+\Rightarrow k0])) (CHead c0 k u0) (CHead d1 (Bind Abbr) u) H3) in ((let H7 
+\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) 
+with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k 
+u0) (CHead d1 (Bind Abbr) u) H3) in (eq_ind C d1 (\lambda (c: C).((eq K k 
+(Bind Abbr)) \to ((eq T u0 u) \to ((eq C (CHead c3 k u0) c2) \to ((csubt g c 
+c3) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O 
+c2 (CHead d2 (Bind Abbr) u))))))))) (\lambda (H8: (eq K k (Bind 
+Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead 
+c3 k0 u0) c2) \to ((csubt g d1 c3) \to (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u)))))))) (\lambda 
+(H9: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c3 (Bind Abbr) t) 
+c2) \to ((csubt g d1 c3) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))))))) (\lambda (H10: 
+(eq C (CHead c3 (Bind Abbr) u) c2)).(eq_ind C (CHead c3 (Bind Abbr) u) 
+(\lambda (c: C).((csubt g d1 c3) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u)))))) (\lambda (H11: 
+(csubt g d1 c3)).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) c3 H11 
+(drop_refl (CHead c3 (Bind Abbr) u)))) c2 H10)) u0 (sym_eq T u0 u H9))) k 
+(sym_eq K k (Bind Abbr) H8))) c0 (sym_eq C c0 d1 H7))) H6)) H5)) H4 H2))) | 
+(csubt_void c0 c3 H2 b H3 u1 u2) \Rightarrow (\lambda (H4: (eq C (CHead c0 
+(Bind Void) u1) (CHead d1 (Bind Abbr) u))).(\lambda (H5: (eq C (CHead c3 
+(Bind b) u2) c2)).((let H6 \def (eq_ind C (CHead c0 (Bind Void) u1) (\lambda 
+(e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow 
+False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) 
+with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: B).Prop) with 
+[Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | 
+(Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H4) in (False_ind 
+((eq C (CHead c3 (Bind b) u2) c2) \to ((csubt g c0 c3) \to ((not (eq B b 
+Void)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O 
+O c2 (CHead d2 (Bind Abbr) u))))))) H6)) H5 H2 H3))) | (csubt_abst c0 c3 H2 
+u0 t H3) \Rightarrow (\lambda (H4: (eq C (CHead c0 (Bind Abst) t) (CHead d1 
+(Bind Abbr) u))).(\lambda (H5: (eq C (CHead c3 (Bind Abbr) u0) c2)).((let H6 
+\def (eq_ind C (CHead c0 (Bind Abst) t) (\lambda (e: C).(match e in C return 
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) 
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow 
+False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) 
+\Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H4) in (False_ind ((eq C 
+(CHead c3 (Bind Abbr) u0) c2) \to ((csubt g c0 c3) \to ((ty3 g c3 u0 t) \to 
+(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead 
+d2 (Bind Abbr) u))))))) H6)) H5 H2 H3)))]) in (H2 (refl_equal C (CHead d1 
+(Bind Abbr) u)) (refl_equal C c2)))))))))) (\lambda (n0: nat).(\lambda (H: 
+((\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: 
+C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 
+(Bind Abbr) u)))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H0: 
+(csubt g c1 c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (d1: 
+C).(\forall (u: T).((drop (S n0) O c (CHead d1 (Bind Abbr) u)) \to (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c0 (CHead 
+d2 (Bind Abbr) u))))))))) (\lambda (n1: nat).(\lambda (d1: C).(\lambda (u: 
+T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Bind Abbr) u))).(let H2 
+\def (match H1 in drop return (\lambda (n2: nat).(\lambda (n3: nat).(\lambda 
+(c: C).(\lambda (c0: C).(\lambda (_: (drop n2 n3 c c0)).((eq nat n2 (S n0)) 
+\to ((eq nat n3 O) \to ((eq C c (CSort n1)) \to ((eq C c0 (CHead d1 (Bind 
+Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop 
+(S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))))))))) with [(drop_refl c) 
+\Rightarrow (\lambda (H2: (eq nat O (S n0))).(\lambda (H3: (eq nat O 
+O)).(\lambda (H4: (eq C c (CSort n1))).(\lambda (H5: (eq C c (CHead d1 (Bind 
+Abbr) u))).((let H6 \def (eq_ind nat O (\lambda (e: nat).(match e in nat 
+return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow 
+False])) I (S n0) H2) in (False_ind ((eq nat O O) \to ((eq C c (CSort n1)) 
+\to ((eq C c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g 
+d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) 
+u))))))) H6)) H3 H4 H5))))) | (drop_drop k h c e H2 u0) \Rightarrow (\lambda 
+(H3: (eq nat (S h) (S n0))).(\lambda (H4: (eq nat O O)).(\lambda (H5: (eq C 
+(CHead c k u0) (CSort n1))).(\lambda (H6: (eq C e (CHead d1 (Bind Abbr) 
+u))).((let H7 \def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat 
+return (\lambda (_: nat).nat) with [O \Rightarrow h | (S n2) \Rightarrow 
+n2])) (S h) (S n0) H3) in (eq_ind nat n0 (\lambda (n2: nat).((eq nat O O) \to 
+((eq C (CHead c k u0) (CSort n1)) \to ((eq C e (CHead d1 (Bind Abbr) u)) \to 
+((drop (r k n2) O c e) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))))) (\lambda 
+(_: (eq nat O O)).(\lambda (H9: (eq C (CHead c k u0) (CSort n1))).(let H10 
+\def (eq_ind C (CHead c k u0) (\lambda (e0: C).(match e0 in C return (\lambda 
+(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow 
+True])) I (CSort n1) H9) in (False_ind ((eq C e (CHead d1 (Bind Abbr) u)) \to 
+((drop (r k n0) O c e) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))))) H10)))) h 
+(sym_eq nat h n0 H7))) H4 H5 H6 H2))))) | (drop_skip k h d c e H2 u0) 
+\Rightarrow (\lambda (H3: (eq nat h (S n0))).(\lambda (H4: (eq nat (S d) 
+O)).(\lambda (H5: (eq C (CHead c k (lift h (r k d) u0)) (CSort n1))).(\lambda 
+(H6: (eq C (CHead e k u0) (CHead d1 (Bind Abbr) u))).(eq_ind nat (S n0) 
+(\lambda (n2: nat).((eq nat (S d) O) \to ((eq C (CHead c k (lift n2 (r k d) 
+u0)) (CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 (Bind Abbr) u)) \to 
+((drop n2 (r k d) c e) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))))) (\lambda 
+(H7: (eq nat (S d) O)).(let H8 \def (eq_ind nat (S d) (\lambda (e0: 
+nat).(match e0 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow 
+False | (S _) \Rightarrow True])) I O H7) in (False_ind ((eq C (CHead c k 
+(lift (S n0) (r k d) u0)) (CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 
+(Bind Abbr) u)) \to ((drop (S n0) (r k d) c e) \to (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 
+(Bind Abbr) u))))))) H8))) h (sym_eq nat h (S n0) H3) H4 H5 H6 H2)))))]) in 
+(H2 (refl_equal nat (S n0)) (refl_equal nat O) (refl_equal C (CSort n1)) 
+(refl_equal C (CHead d1 (Bind Abbr) u)))))))) (\lambda (c0: C).(\lambda (c3: 
+C).(\lambda (H1: (csubt g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall 
+(u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) 
+u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall 
+(d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Bind 
+Abbr) u0)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abbr) u0))))))))) (\lambda 
+(b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop 
+(S n0) O (CHead c0 (Bind b) u) (CHead d1 (Bind Abbr) u0))).(ex2_ind C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 
+(Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda 
+(x: C).(\lambda (H4: (csubt g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x 
+(Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0))) x H4 
+(drop_drop (Bind b) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) (H c0 c3 H1 d1 
+u0 (drop_gen_drop (Bind b) c0 (CHead d1 (Bind Abbr) u0) u n0 H3)))))))) 
+(\lambda (f: F).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda 
+(H3: (drop (S n0) O (CHead c0 (Flat f) u) (CHead d1 (Bind Abbr) 
+u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S 
+n0) O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) 
+u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 x)).(\lambda (H5: (drop (S 
+n0) O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind 
+Abbr) u0))) x H4 (drop_drop (Flat f) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) 
+(H2 d1 u0 (drop_gen_drop (Flat f) c0 (CHead d1 (Bind Abbr) u0) u n0 
+H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csubt g 
+c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 
+(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda 
+(b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S n0) O (CHead c0 
+(Bind Void) u1) (CHead d1 (Bind Abbr) u))).(ex2_ind C (\lambda (d2: C).(csubt 
+g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u))) (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (x: C).(\lambda (H5: (csubt 
+g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind Abbr) u))).(ex_intro2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind b) u2) (CHead d2 (Bind Abbr) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead 
+x (Bind Abbr) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 
+(CHead d1 (Bind Abbr) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda 
+(c3: C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: 
+C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead 
+d2 (Bind Abbr) u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g 
+c3 u t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H4: (drop (S n0) O 
+(CHead c0 (Bind Abst) t) (CHead d1 (Bind Abbr) u0))).(ex2_ind C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) 
+u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) 
+O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (x: 
+C).(\lambda (H5: (csubt g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind 
+Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0))) x H5 
+(drop_drop (Bind Abbr) n0 c3 (CHead x (Bind Abbr) u0) H6 u))))) (H c0 c3 H1 
+d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Bind Abbr) u0) t n0 
+H4))))))))))))) c1 c2 H0)))))) n)).
+
+theorem csubt_drop_abst:
+ \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g 
+c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n O c1 (CHead d1 (Bind 
+Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n 
+O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))))))))))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: 
+C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (t: 
+T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst) 
+t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
+(d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda 
+(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) (\lambda (c1: C).(\lambda 
+(c2: C).(\lambda (H: (csubt g c1 c2)).(\lambda (d1: C).(\lambda (t: 
+T).(\lambda (H0: (drop O O c1 (CHead d1 (Bind Abst) t))).(let H1 \def (eq_ind 
+C c1 (\lambda (c: C).(csubt g c c2)) H (CHead d1 (Bind Abst) t) 
+(drop_gen_refl c1 (CHead d1 (Bind Abst) t) H0)) in (let H2 \def (match H1 in 
+csubt return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csubt ? c 
+c0)).((eq C c (CHead d1 (Bind Abst) t)) \to ((eq C c0 c2) \to (or (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 
+(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
+d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) 
+u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) with 
+[(csubt_sort n0) \Rightarrow (\lambda (H2: (eq C (CSort n0) (CHead d1 (Bind 
+Abst) t))).(\lambda (H3: (eq C (CSort n0) c2)).((let H4 \def (eq_ind C (CSort 
+n0) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort 
+_) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind 
+Abst) t) H2) in (False_ind ((eq C (CSort n0) c2) \to (or (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) 
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
+C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
+C).(\lambda (u: T).(ty3 g d2 u t)))))) H4)) H3))) | (csubt_head c0 c3 H2 k u) 
+\Rightarrow (\lambda (H3: (eq C (CHead c0 k u) (CHead d1 (Bind Abst) 
+t))).(\lambda (H4: (eq C (CHead c3 k u) c2)).((let H5 \def (f_equal C T 
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
+\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k u) (CHead d1 
+(Bind Abst) t) H3) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in 
+C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) 
+\Rightarrow k0])) (CHead c0 k u) (CHead d1 (Bind Abst) t) H3) in ((let H7 
+\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) 
+with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k 
+u) (CHead d1 (Bind Abst) t) H3) in (eq_ind C d1 (\lambda (c: C).((eq K k 
+(Bind Abst)) \to ((eq T u t) \to ((eq C (CHead c3 k u) c2) \to ((csubt g c 
+c3) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O 
+O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 
+(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) 
+(\lambda (H8: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda (k0: 
+K).((eq T u t) \to ((eq C (CHead c3 k0 u) c2) \to ((csubt g d1 c3) \to (or 
+(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead 
+d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
+d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) 
+u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))))) (\lambda 
+(H9: (eq T u t)).(eq_ind T t (\lambda (t0: T).((eq C (CHead c3 (Bind Abst) 
+t0) c2) \to ((csubt g d1 c3) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda 
+(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: 
+T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
+T).(ty3 g d2 u0 t)))))))) (\lambda (H10: (eq C (CHead c3 (Bind Abst) t) 
+c2)).(eq_ind C (CHead c3 (Bind Abst) t) (\lambda (c: C).((csubt g d1 c3) \to 
+(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c 
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c (CHead d2 
+(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))) 
+(\lambda (H11: (csubt g d1 c3)).(or_introl (ex2 C (\lambda (d2: C).(csubt g 
+d1 d2)) (\lambda (d2: C).(drop O O (CHead c3 (Bind Abst) t) (CHead d2 (Bind 
+Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) 
+(\lambda (d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abst) t) (CHead 
+d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) 
+(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O 
+(CHead c3 (Bind Abst) t) (CHead d2 (Bind Abst) t))) c3 H11 (drop_refl (CHead 
+c3 (Bind Abst) t))))) c2 H10)) u (sym_eq T u t H9))) k (sym_eq K k (Bind 
+Abst) H8))) c0 (sym_eq C c0 d1 H7))) H6)) H5)) H4 H2))) | (csubt_void c0 c3 
+H2 b H3 u1 u2) \Rightarrow (\lambda (H4: (eq C (CHead c0 (Bind Void) u1) 
+(CHead d1 (Bind Abst) t))).(\lambda (H5: (eq C (CHead c3 (Bind b) u2) 
+c2)).((let H6 \def (eq_ind C (CHead c0 (Bind Void) u1) (\lambda (e: C).(match 
+e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | 
+(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
+[(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr 
+\Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat 
+_) \Rightarrow False])])) I (CHead d1 (Bind Abst) t) H4) in (False_ind ((eq C 
+(CHead c3 (Bind b) u2) c2) \to ((csubt g c0 c3) \to ((not (eq B b Void)) \to 
+(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c2 (CHead d2 
+(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))) H6)) 
+H5 H2 H3))) | (csubt_abst c0 c3 H2 u t0 H3) \Rightarrow (\lambda (H4: (eq C 
+(CHead c0 (Bind Abst) t0) (CHead d1 (Bind Abst) t))).(\lambda (H5: (eq C 
+(CHead c3 (Bind Abbr) u) c2)).((let H6 \def (f_equal C T (\lambda (e: 
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t0 | 
+(CHead _ _ t1) \Rightarrow t1])) (CHead c0 (Bind Abst) t0) (CHead d1 (Bind 
+Abst) t) H4) in ((let H7 \def (f_equal C C (\lambda (e: C).(match e in C 
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) 
+\Rightarrow c])) (CHead c0 (Bind Abst) t0) (CHead d1 (Bind Abst) t) H4) in 
+(eq_ind C d1 (\lambda (c: C).((eq T t0 t) \to ((eq C (CHead c3 (Bind Abbr) u) 
+c2) \to ((csubt g c c3) \to ((ty3 g c3 u t0) \to (or (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) 
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
+C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
+C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) (\lambda (H8: (eq T t0 
+t)).(eq_ind T t (\lambda (t1: T).((eq C (CHead c3 (Bind Abbr) u) c2) \to 
+((csubt g d1 c3) \to ((ty3 g c3 u t1) \to (or (ex2 C (\lambda (d2: C).(csubt 
+g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C 
+T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
+C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
+C).(\lambda (u0: T).(ty3 g d2 u0 t))))))))) (\lambda (H9: (eq C (CHead c3 
+(Bind Abbr) u) c2)).(eq_ind C (CHead c3 (Bind Abbr) u) (\lambda (c: 
+C).((csubt g d1 c3) \to ((ty3 g c3 u t) \to (or (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) t)))) 
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
+C).(\lambda (u0: T).(drop O O c (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
+C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))) (\lambda (H10: (csubt g d1 
+c3)).(\lambda (H11: (ty3 g c3 u t)).(or_intror (ex2 C (\lambda (d2: C).(csubt 
+g d1 d2)) (\lambda (d2: C).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind 
+Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) 
+(\lambda (d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abbr) u) (CHead 
+d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) 
+(ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
+(d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind 
+Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) c3 u H10 
+(drop_refl (CHead c3 (Bind Abbr) u)) H11)))) c2 H9)) t0 (sym_eq T t0 t H8))) 
+c0 (sym_eq C c0 d1 H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal C (CHead d1 
+(Bind Abst) t)) (refl_equal C c2)))))))))) (\lambda (n0: nat).(\lambda (H: 
+((\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: 
+C).(\forall (t: T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 
+(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
+d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) 
+u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))))).(\lambda 
+(c1: C).(\lambda (c2: C).(\lambda (H0: (csubt g c1 c2)).(csubt_ind g (\lambda 
+(c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (t: T).((drop (S n0) O c 
+(CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(drop (S n0) O c0 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
+C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (n1: nat).(\lambda (d1: 
+C).(\lambda (t: T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Bind 
+Abst) t))).(let H2 \def (match H1 in drop return (\lambda (n2: nat).(\lambda 
+(n3: nat).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop n2 n3 c 
+c0)).((eq nat n2 (S n0)) \to ((eq nat n3 O) \to ((eq C c (CSort n1)) \to ((eq 
+C c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) 
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
+C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) 
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))))) with [(drop_refl 
+c) \Rightarrow (\lambda (H2: (eq nat O (S n0))).(\lambda (H3: (eq nat O 
+O)).(\lambda (H4: (eq C c (CSort n1))).(\lambda (H5: (eq C c (CHead d1 (Bind 
+Abst) t))).((let H6 \def (eq_ind nat O (\lambda (e: nat).(match e in nat 
+return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow 
+False])) I (S n0) H2) in (False_ind ((eq nat O O) \to ((eq C c (CSort n1)) 
+\to ((eq C c (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt 
+g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) 
+t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
+(d2: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) 
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))) H6)) H3 H4 H5))))) | 
+(drop_drop k h c e H2 u) \Rightarrow (\lambda (H3: (eq nat (S h) (S 
+n0))).(\lambda (H4: (eq nat O O)).(\lambda (H5: (eq C (CHead c k u) (CSort 
+n1))).(\lambda (H6: (eq C e (CHead d1 (Bind Abst) t))).((let H7 \def (f_equal 
+nat nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) 
+with [O \Rightarrow h | (S n2) \Rightarrow n2])) (S h) (S n0) H3) in (eq_ind 
+nat n0 (\lambda (n2: nat).((eq nat O O) \to ((eq C (CHead c k u) (CSort n1)) 
+\to ((eq C e (CHead d1 (Bind Abst) t)) \to ((drop (r k n2) O c e) \to (or 
+(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
+(CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda 
+(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O 
+(CSort n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
+T).(ty3 g d2 u0 t)))))))))) (\lambda (_: (eq nat O O)).(\lambda (H9: (eq C 
+(CHead c k u) (CSort n1))).(let H10 \def (eq_ind C (CHead c k u) (\lambda 
+(e0: C).(match e0 in C return (\lambda (_: C).Prop) with [(CSort _) 
+\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n1) H9) in 
+(False_ind ((eq C e (CHead d1 (Bind Abst) t)) \to ((drop (r k n0) O c e) \to 
+(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
+(CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda 
+(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O 
+(CSort n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
+T).(ty3 g d2 u0 t))))))) H10)))) h (sym_eq nat h n0 H7))) H4 H5 H6 H2))))) | 
+(drop_skip k h d c e H2 u) \Rightarrow (\lambda (H3: (eq nat h (S 
+n0))).(\lambda (H4: (eq nat (S d) O)).(\lambda (H5: (eq C (CHead c k (lift h 
+(r k d) u)) (CSort n1))).(\lambda (H6: (eq C (CHead e k u) (CHead d1 (Bind 
+Abst) t))).(eq_ind nat (S n0) (\lambda (n2: nat).((eq nat (S d) O) \to ((eq C 
+(CHead c k (lift n2 (r k d) u)) (CSort n1)) \to ((eq C (CHead e k u) (CHead 
+d1 (Bind Abst) t)) \to ((drop n2 (r k d) c e) \to (or (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 
+(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
+d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CSort n1) (CHead d2 
+(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) 
+(\lambda (H7: (eq nat (S d) O)).(let H8 \def (eq_ind nat (S d) (\lambda (e0: 
+nat).(match e0 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow 
+False | (S _) \Rightarrow True])) I O H7) in (False_ind ((eq C (CHead c k 
+(lift (S n0) (r k d) u)) (CSort n1)) \to ((eq C (CHead e k u) (CHead d1 (Bind 
+Abst) t)) \to ((drop (S n0) (r k d) c e) \to (or (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 
+(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
+d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CSort n1) (CHead d2 
+(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))) 
+H8))) h (sym_eq nat h (S n0) H3) H4 H5 H6 H2)))))]) in (H2 (refl_equal nat (S 
+n0)) (refl_equal nat O) (refl_equal C (CSort n1)) (refl_equal C (CHead d1 
+(Bind Abst) t)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csubt 
+g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (t: T).((drop (S n0) O c0 
+(CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
+C).(\lambda (u: T).(ty3 g d2 u t)))))))))).(\lambda (k: K).(K_ind (\lambda 
+(k0: K).(\forall (u: T).(\forall (d1: C).(\forall (t: T).((drop (S n0) O 
+(CHead c0 k0 u) (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 
+(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
+d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 k0 u) (CHead 
+d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 
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+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
+c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
+T).(ty3 g d2 u0 t0))) x0 x1 H6 (drop_drop (Bind Abbr) n0 c3 (CHead x0 (Bind 
+Abbr) x1) H7 u) H8))))))) H5)) (H c0 c3 H1 d1 t0 (drop_gen_drop (Bind Abst) 
+c0 (CHead d1 (Bind Abst) t0) t n0 H4))))))))))))) c1 c2 H0)))))) n)).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/fwd.ma
new file mode 100644
index 000000000..14a5c05ec
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/fwd.ma
@@ -0,0 +1,42 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/fwd".
+
+include "csubt/defs.ma".
+
+axiom csubt_gen_abbr:
+ \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csubt g 
+(CHead e1 (Bind Abbr) v) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 
+(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))))
+.
+
+axiom csubt_gen_abst:
+ \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csubt g 
+(CHead e1 (Bind Abst) v1) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead 
+e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda 
+(e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
+C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g 
+e2 v2 v1)))))))))
+.
+
+axiom csubt_gen_bind:
+ \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall 
+(v1: T).((csubt g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))))))
+.
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/getl.ma
new file mode 100644
index 000000000..4210416c0
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/getl.ma
@@ -0,0 +1,398 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/getl".
+
+include "csubt/fwd.ma".
+
+include "csubt/clear.ma".
+
+include "csubt/drop.ma".
+
+include "getl/clear.ma".
+
+theorem csubt_getl_abbr:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall 
+(n: nat).((getl n c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g 
+c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n 
+c2 (CHead d2 (Bind Abbr) u)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda 
+(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abbr) u))).(let H0 \def 
+(getl_gen_all c1 (CHead d1 (Bind Abbr) u) n H) in (ex2_ind C (\lambda (e: 
+C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u))) 
+(\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))) (\lambda (x: 
+C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind 
+Abbr) u))).(C_ind (\lambda (c: C).((drop n O c1 c) \to ((clear c (CHead d1 
+(Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda 
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) 
+u))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 (CSort n0))).(\lambda 
+(H4: (clear (CSort n0) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1 
+(Bind Abbr) u) n0 H4 (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda 
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) 
+u)))))))))) (\lambda (x0: C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0 
+(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 c2) \to (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind 
+Abbr) u)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: (drop n O c1 
+(CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 (Bind Abbr) 
+u))).(K_ind (\lambda (k0: K).((drop n O c1 (CHead x0 k0 t)) \to ((clear 
+(CHead x0 k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 
+c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 
+(CHead d2 (Bind Abbr) u))))))))) (\lambda (b: B).(\lambda (H5: (drop n O c1 
+(CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 
+(Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C 
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) 
+\Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) 
+(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def 
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with 
+[(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K 
+return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
+\Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) 
+(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def 
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
+[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind 
+Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) 
+t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda 
+(c2: C).(\lambda (H12: (csubt g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda 
+(t0: T).(drop n O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def 
+(eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) u))) H13 Abbr 
+H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1 (CHead c 
+(Bind Abbr) u))) H14 d1 H11) in (ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))) 
+(\lambda (x1: C).(\lambda (H16: (csubt g d1 x1)).(\lambda (H17: (drop n O c2 
+(CHead x1 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) x1 H16 (getl_intro n 
+c2 (CHead x1 (Bind Abbr) u) (CHead x1 (Bind Abbr) u) H17 (clear_bind Abbr x1 
+u)))))) (csubt_drop_abbr g n c1 c2 H12 d1 u H15)))))))))) H8)) H7))))) 
+(\lambda (f: F).(\lambda (H5: (drop n O c1 (CHead x0 (Flat f) t))).(\lambda 
+(H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr) u))).(let H7 \def H5 
+in (unintro C c1 (\lambda (c: C).((drop n O c (CHead x0 (Flat f) t)) \to 
+(\forall (c2: C).((csubt g c c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))))) (nat_ind (\lambda 
+(n0: nat).(\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall 
+(c2: C).((csubt g x1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u))))))))) (\lambda (x1: 
+C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: 
+C).(\lambda (H9: (csubt g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: 
+C).(csubt g c c2)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat 
+f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abbr) u) 
+(clear_gen_flat f x0 (CHead d1 (Bind Abbr) u) t H6) f t) in (let H11 \def 
+(csubt_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u) 
+H_y) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead d1 (Bind Abbr) u) e2)) 
+(\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x2: 
+C).(\lambda (H12: (csubt g (CHead d1 (Bind Abbr) u) x2)).(\lambda (H13: 
+(clear c2 x2)).(let H14 \def (csubt_gen_abbr g d1 x2 u H12) in (ex2_ind C 
+(\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csubt 
+g d1 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x3: C).(\lambda (H15: (eq C x2 
+(CHead x3 (Bind Abbr) u))).(\lambda (H16: (csubt g d1 x3)).(let H17 \def 
+(eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) u) H15) 
+in (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 
+(CHead d2 (Bind Abbr) u))) x3 H16 (getl_intro O c2 (CHead x3 (Bind Abbr) u) 
+c2 (drop_refl c2) H17)))))) H14))))) H11)))))))) (\lambda (n0: nat).(\lambda 
+(H8: ((\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall 
+(c2: C).((csubt g x1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u)))))))))).(\lambda (x1: 
+C).(\lambda (H9: (drop (S n0) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: 
+C).(\lambda (H10: (csubt g x1 c2)).(let H11 \def (drop_clear x1 (CHead x0 
+(Flat f) t) n0 H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: 
+C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) (\lambda (_: 
+B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead x0 (Flat f) t))))) 
+(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 
+(CHead d2 (Bind Abbr) u)))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: 
+T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0 
+O x3 (CHead x0 (Flat f) t))).(let H14 \def (csubt_clear_conf g x1 c2 H10 
+(CHead x3 (Bind x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead 
+x3 (Bind x2) x4) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) 
+u)))) (\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 (Bind x2) x4) 
+x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def (csubt_gen_bind g x2 x3 x5 
+x4 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
+T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csubt g x3 e2)))) (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda 
+(x6: B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 
+(Bind x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def (eq_ind C x5 
+(\lambda (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 
+\def (H8 x3 H13 x7 H19) in (ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) 
+u)))) (\lambda (x9: C).(\lambda (H22: (csubt g d1 x9)).(\lambda (H23: (getl 
+n0 x7 (CHead x9 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u))) x9 H22 
+(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) u) n0 H23))))) 
+H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))) k H3 H4))))))) x H1 
+H2)))) H0))))))).
+
+theorem csubt_getl_abst:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (t: T).(\forall 
+(n: nat).((getl n c1 (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g 
+c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (t: T).(\lambda 
+(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abst) t))).(let H0 \def 
+(getl_gen_all c1 (CHead d1 (Bind Abst) t) n H) in (ex2_ind C (\lambda (e: 
+C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) t))) 
+(\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t))))))) (\lambda (x: C).(\lambda (H1: (drop n O c1 
+x)).(\lambda (H2: (clear x (CHead d1 (Bind Abst) t))).(C_ind (\lambda (c: 
+C).((drop n O c1 c) \to ((clear c (CHead d1 (Bind Abst) t)) \to (\forall (c2: 
+C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t)))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 (CSort 
+n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abst) 
+t))).(clear_gen_sort (CHead d1 (Bind Abst) t) n0 H4 (\forall (c2: C).((csubt 
+g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))))))))))) (\lambda (x0: C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0 
+(CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 
+C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 
+(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
+d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) 
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))).(\lambda (k: 
+K).(\lambda (t0: T).(\lambda (H3: (drop n O c1 (CHead x0 k t0))).(\lambda 
+(H4: (clear (CHead x0 k t0) (CHead d1 (Bind Abst) t))).(K_ind (\lambda (k0: 
+K).((drop n O c1 (CHead x0 k0 t0)) \to ((clear (CHead x0 k0 t0) (CHead d1 
+(Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda 
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) 
+t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
+(d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
+C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (b: B).(\lambda (H5: 
+(drop n O c1 (CHead x0 (Bind b) t0))).(\lambda (H6: (clear (CHead x0 (Bind b) 
+t0) (CHead d1 (Bind Abst) t))).(let H7 \def (f_equal C C (\lambda (e: 
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | 
+(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) t) (CHead x0 (Bind b) 
+t0) (clear_gen_bind b x0 (CHead d1 (Bind Abst) t) t0 H6)) in ((let H8 \def 
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with 
+[(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 in K 
+return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
+\Rightarrow Abst])])) (CHead d1 (Bind Abst) t) (CHead x0 (Bind b) t0) 
+(clear_gen_bind b x0 (CHead d1 (Bind Abst) t) t0 H6)) in ((let H9 \def 
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
+[(CSort _) \Rightarrow t | (CHead _ _ t1) \Rightarrow t1])) (CHead d1 (Bind 
+Abst) t) (CHead x0 (Bind b) t0) (clear_gen_bind b x0 (CHead d1 (Bind Abst) t) 
+t0 H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 
+x0)).(\lambda (c2: C).(\lambda (H12: (csubt g c1 c2)).(let H13 \def (eq_ind_r 
+T t0 (\lambda (t1: T).(drop n O c1 (CHead x0 (Bind b) t1))) H5 t H9) in (let 
+H14 \def (eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) t))) 
+H13 Abst H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1 
+(CHead c (Bind Abst) t))) H14 d1 H11) in (or_ind (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t)))) 
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
+C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
+C).(\lambda (u: T).(ty3 g d2 u t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t))))) (\lambda (H16: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t))))).(ex2_ind C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 
+(Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))))) (\lambda (x1: C).(\lambda (H17: (csubt g d1 x1)).(\lambda (H18: (drop 
+n O c2 (CHead x1 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g 
+d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(getl n c2 (CHead d2 (Bind Abst) t))) x1 H17 (getl_intro n c2 (CHead 
+x1 (Bind Abst) t) (CHead x1 (Bind Abst) t) H18 (clear_bind Abst x1 t))))))) 
+H16)) (\lambda (H16: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
+d2))) (\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) 
+u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (H17: (csubt g d1 
+x1)).(\lambda (H18: (drop n O c2 (CHead x1 (Bind Abbr) x2))).(\lambda (H19: 
+(ty3 g x1 x2 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) 
+(\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) 
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x1 x2 H17 (getl_intro n c2 
+(CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2) H18 (clear_bind Abbr x1 
+x2)) H19))))))) H16)) (csubt_drop_abst g n c1 c2 H12 d1 t H15)))))))))) H8)) 
+H7))))) (\lambda (f: F).(\lambda (H5: (drop n O c1 (CHead x0 (Flat f) 
+t0))).(\lambda (H6: (clear (CHead x0 (Flat f) t0) (CHead d1 (Bind Abst) 
+t))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop n O c (CHead x0 
+(Flat f) t0)) \to (\forall (c2: C).((csubt g c c2) \to (or (ex2 C (\lambda 
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) 
+t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
+(d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
+C).(\lambda (u: T).(ty3 g d2 u t))))))))) (nat_ind (\lambda (n0: 
+nat).(\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t0)) \to (\forall 
+(c2: C).((csubt g x1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda 
+(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: 
+T).(getl n0 c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t)))))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead 
+x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H9: (csubt g x1 c2)).(let H10 
+\def (eq_ind C x1 (\lambda (c: C).(csubt g c c2)) H9 (CHead x0 (Flat f) t0) 
+(drop_gen_refl x1 (CHead x0 (Flat f) t0) H8)) in (let H_y \def (clear_flat x0 
+(CHead d1 (Bind Abst) t) (clear_gen_flat f x0 (CHead d1 (Bind Abst) t) t0 H6) 
+f t0) in (let H11 \def (csubt_clear_conf g (CHead x0 (Flat f) t0) c2 H10 
+(CHead d1 (Bind Abst) t) H_y) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead 
+d1 (Bind Abst) t) e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda 
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) 
+t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
+(d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
+C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x2: C).(\lambda (H12: (csubt 
+g (CHead d1 (Bind Abst) t) x2)).(\lambda (H13: (clear c2 x2)).(let H14 \def 
+(csubt_gen_abst g d1 x2 t H12) in (or_ind (ex2 C (\lambda (e2: C).(eq C x2 
+(CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2))) (ex3_2 C T 
+(\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 (Bind Abbr) v2)))) 
+(\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (e2: C).(\lambda 
+(v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda 
+(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: 
+T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t))))) (\lambda (H15: (ex2 C (\lambda (e2: C).(eq C x2 (CHead 
+e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2)))).(ex2_ind C (\lambda 
+(e2: C).(eq C x2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2)) 
+(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 
+(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
+(\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst) t))).(\lambda 
+(H17: (csubt g d1 x3)).(let H18 \def (eq_ind C x2 (\lambda (c: C).(clear c2 
+c)) H13 (CHead x3 (Bind Abst) t) H16) in (or_introl (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) 
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
+C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
+C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g 
+d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))) x3 H17 
+(getl_intro O c2 (CHead x3 (Bind Abst) t) c2 (drop_refl c2) H18))))))) H15)) 
+(\lambda (H15: (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead 
+e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) 
+(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))))).(ex3_2_ind C T (\lambda 
+(e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
+C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g 
+e2 v2 t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 
+(Bind Abbr) x4))).(\lambda (H17: (csubt g d1 x3)).(\lambda (H18: (ty3 g x3 x4 
+t)).(let H19 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 
+(Bind Abbr) x4) H16) in (or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda 
+(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: 
+T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt 
+g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) 
+u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x3 x4 H17 (getl_intro 
+O c2 (CHead x3 (Bind Abbr) x4) c2 (drop_refl c2) H19) H18)))))))) H15)) 
+H14))))) H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x1: 
+C).((drop n0 O x1 (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g x1 
+c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl 
+n0 c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 
+(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n0) O x1 (CHead x0 (Flat 
+f) t0))).(\lambda (c2: C).(\lambda (H10: (csubt g x1 c2)).(let H11 \def 
+(drop_clear x1 (CHead x0 (Flat f) t0) n0 H9) in (ex2_3_ind B C T (\lambda (b: 
+B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) 
+(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead x0 (Flat 
+f) t0))))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl 
+(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
+d2 u t))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: 
+(clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0 O x3 (CHead x0 
+(Flat f) t0))).(let H14 \def (csubt_clear_conf g x1 c2 H10 (CHead x3 (Bind 
+x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead x3 (Bind x2) x4) 
+e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda 
+(u: T).(ty3 g d2 u t))))) (\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 
+(Bind x2) x4) x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def 
+(csubt_gen_bind g x2 x3 x5 x4 H15) in (ex2_3_ind B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g x3 e2)))) (or (ex2 
+C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead 
+d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
+d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) 
+u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x6: 
+B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind 
+x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def (eq_ind C x5 (\lambda 
+(c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 
+H13 x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(getl n0 x7 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl 
+n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 
+u t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl 
+(S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda 
+(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 
+(CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(getl n0 x7 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abst) t))) 
+(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead 
+d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
+(\lambda (x9: C).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24: (getl n0 x7 
+(CHead x9 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda 
+(u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t))) x9 H23 
+(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abst) t) n0 H24)))))) H22)) 
+(\lambda (H22: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) 
+(\lambda (d2: C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) 
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda 
+(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: 
+T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl 
+(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
+d2 u t))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (H23: (csubt g d1 
+x9)).(\lambda (H24: (getl n0 x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H25: 
+(ty3 g x9 x10 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda 
+(u: T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: 
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead 
+d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x9 x10 
+H23 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) x10) n0 H24) 
+H25))))))) H22)) H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))) k H3 
+H4))))))) x H1 H2)))) H0))))))).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/pc3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/pc3.ma
new file mode 100644
index 000000000..94e98b52a
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/pc3.ma
@@ -0,0 +1,58 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/pc3".
+
+include "csubt/getl.ma".
+
+include "pc3/left.ma".
+
+theorem csubt_pr2:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 
+t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (pr2 c2 t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: 
+T).(\forall (c2: C).((csubt g c c2) \to (pr2 c2 t t0)))))) (\lambda (c: 
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c2: 
+C).(\lambda (_: (csubt g c c2)).(pr2_free c2 t3 t4 H0))))))) (\lambda (c: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c 
+(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
+(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2: 
+C).(\lambda (H3: (csubt g c c2)).(let H4 \def (csubt_getl_abbr g c d u i H0 
+c2 H3) in (ex2_ind C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2: C).(getl 
+i c2 (CHead d2 (Bind Abbr) u))) (pr2 c2 t3 t) (\lambda (x: C).(\lambda (_: 
+(csubt g d x)).(\lambda (H6: (getl i c2 (CHead x (Bind Abbr) u))).(pr2_delta 
+c2 x u i H6 t3 t4 H1 t H2)))) H4)))))))))))))) c1 t1 t2 H))))).
+
+theorem csubt_pc3:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1 
+t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (pc3 c1 t1 t2)).(pc3_ind_left c1 (\lambda (t: T).(\lambda (t0: 
+T).(\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t t0))))) (\lambda (t: 
+T).(\lambda (c2: C).(\lambda (_: (csubt g c1 c2)).(pc3_refl c2 t)))) (\lambda 
+(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 t0 t3)).(\lambda (t4: 
+T).(\lambda (_: (pc3 c1 t3 t4)).(\lambda (H2: ((\forall (c2: C).((csubt g c1 
+c2) \to (pc3 c2 t3 t4))))).(\lambda (c2: C).(\lambda (H3: (csubt g c1 
+c2)).(pc3_pr2_u c2 t3 t0 (csubt_pr2 g c1 t0 t3 H0 c2 H3) t4 (H2 c2 
+H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 t0 
+t3)).(\lambda (t4: T).(\lambda (_: (pc3 c1 t0 t4)).(\lambda (H2: ((\forall 
+(c2: C).((csubt g c1 c2) \to (pc3 c2 t0 t4))))).(\lambda (c2: C).(\lambda 
+(H3: (csubt g c1 c2)).(pc3_t t0 c2 t3 (pc3_pr2_x c2 t3 t0 (csubt_pr2 g c1 t0 
+t3 H0 c2 H3)) t4 (H2 c2 H3)))))))))) t1 t2 H))))).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/props.ma
new file mode 100644
index 000000000..c4321b61b
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/props.ma
@@ -0,0 +1,27 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/props".
+
+include "csubt/defs.ma".
+
+theorem csubt_refl:
+ \forall (g: G).(\forall (c: C).(csubt g c c))
+\def
+ \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubt g c0 c0)) 
+(\lambda (n: nat).(csubt_sort g n)) (\lambda (c0: C).(\lambda (H: (csubt g c0 
+c0)).(\lambda (k: K).(\lambda (t: T).(csubt_head g c0 c0 H k t))))) c)).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/ty3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/ty3.ma
new file mode 100644
index 000000000..9e0eb6534
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/ty3.ma
@@ -0,0 +1,99 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csubt/ty3".
+
+include "csubt/pc3.ma".
+
+include "csubt/props.ma".
+
+theorem csubt_ty3:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 
+t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (ty3 g c2 t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda 
+(t0: T).(\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t t0)))))) (\lambda 
+(c: C).(\lambda (t0: T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda 
+(H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t0 t))))).(\lambda (u: 
+T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2: 
+C).((csubt g c c2) \to (ty3 g c2 u t3))))).(\lambda (H4: (pc3 c t3 
+t0)).(\lambda (c2: C).(\lambda (H5: (csubt g c c2)).(ty3_conv g c2 t0 t (H1 
+c2 H5) u t3 (H3 c2 H5) (csubt_pc3 g c t3 t0 H4 c2 H5)))))))))))))) (\lambda 
+(c: C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (csubt g c 
+c2)).(ty3_sort g c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda 
+(t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((csubt g 
+d c2) \to (ty3 g c2 u t))))).(\lambda (c2: C).(\lambda (H3: (csubt g c 
+c2)).(let H4 \def (csubt_getl_abbr g c d u n H0 c2 H3) in (ex2_ind C (\lambda 
+(d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) 
+u))) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x: C).(\lambda (H5: 
+(csubt g d x)).(\lambda (H6: (getl n c2 (CHead x (Bind Abbr) u))).(ty3_abbr g 
+n c2 x u H6 t (H2 x H5))))) H4)))))))))))) (\lambda (n: nat).(\lambda (c: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind 
+Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: 
+((\forall (c2: C).((csubt g d c2) \to (ty3 g c2 u t))))).(\lambda (c2: 
+C).(\lambda (H3: (csubt g c c2)).(let H4 \def (csubt_getl_abst g c d u n H0 
+c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2: 
+C).(getl n c2 (CHead d2 (Bind Abst) u)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 
+u0 u)))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (H5: (ex2 C (\lambda 
+(d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) 
+u))))).(ex2_ind C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n 
+c2 (CHead d2 (Bind Abst) u))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda 
+(x: C).(\lambda (H6: (csubt g d x)).(\lambda (H7: (getl n c2 (CHead x (Bind 
+Abst) u))).(ty3_abst g n c2 x u H7 t (H2 x H6))))) H5)) (\lambda (H5: (ex3_2 
+C T (\lambda (d2: C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: 
+C).(\lambda (u0: T).(getl n c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
+C).(\lambda (u0: T).(ty3 g d2 u0 u))))).(ex3_2_ind C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 
+u0 u))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (_: (csubt g d x0)).(\lambda (H7: (getl n c2 (CHead x0 (Bind 
+Abbr) x1))).(\lambda (H8: (ty3 g x0 x1 u)).(ty3_abbr g n c2 x0 x1 H7 u 
+H8)))))) H5)) H4)))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t: 
+T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c2: C).((csubt g c 
+c2) \to (ty3 g c2 u t))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t0 t3)).(\lambda (H3: ((\forall 
+(c2: C).((csubt g (CHead c (Bind b) u) c2) \to (ty3 g c2 t0 t3))))).(\lambda 
+(t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t3 t4)).(\lambda (H5: 
+((\forall (c2: C).((csubt g (CHead c (Bind b) u) c2) \to (ty3 g c2 t3 
+t4))))).(\lambda (c2: C).(\lambda (H6: (csubt g c c2)).(ty3_bind g c2 u t (H1 
+c2 H6) b t0 t3 (H3 (CHead c2 (Bind b) u) (csubt_head g c c2 H6 (Bind b) u)) 
+t4 (H5 (CHead c2 (Bind b) u) (csubt_head g c c2 H6 (Bind b) 
+u)))))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda 
+(_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g 
+c2 w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead 
+(Bind Abst) u t))).(\lambda (H3: ((\forall (c2: C).((csubt g c c2) \to (ty3 g 
+c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (csubt g c 
+c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c: 
+C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t0 t3)).(\lambda 
+(H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t0 t3))))).(\lambda (t4: 
+T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3: ((\forall (c2: C).((csubt g c 
+c2) \to (ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (csubt g c 
+c2)).(ty3_cast g c2 t0 t3 (H1 c2 H4) t4 (H3 c2 H4)))))))))))) c1 t1 t2 H))))).
+
+theorem csubt_ty3_ld:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (v: T).((ty3 g c u 
+v) \to (\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind Abst) v) t1 
+t2) \to (ty3 g (CHead c (Bind Abbr) u) t1 t2))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (H: 
+(ty3 g c u v)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead 
+c (Bind Abst) v) t1 t2)).(csubt_ty3 g (CHead c (Bind Abst) v) t1 t2 H0 (CHead 
+c (Bind Abbr) u) (csubt_abst g c c (csubt_refl g c) u v H))))))))).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/drop1/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/drop1/defs.ma
index 164dab30a..dafc9d015 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/drop1/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/drop1/defs.ma
@@ -26,12 +26,12 @@ inductive drop1: PList \to (C \to (C \to Prop)) \def
 nat).((drop h d c1 c2) \to (\forall (c3: C).(\forall (hds: PList).((drop1 hds 
 c2 c3) \to (drop1 (PCons h d hds) c1 c3)))))))).
 
-definition ctrans:
- PList \to (nat \to (T \to T))
+definition ptrans:
+ PList \to (nat \to PList)
 \def
- let rec ctrans (hds: PList) on hds: (nat \to (T \to T)) \def (\lambda (i: 
-nat).(\lambda (t: T).(match hds with [PNil \Rightarrow t | (PCons h d hds0) 
-\Rightarrow (let j \def (trans hds0 i) in (let u \def (ctrans hds0 i t) in 
-(match (blt j d) with [true \Rightarrow (lift h (minus d (S j)) u) | false 
-\Rightarrow u])))]))) in ctrans.
+ let rec ptrans (hds: PList) on hds: (nat \to PList) \def (\lambda (i: 
+nat).(match hds with [PNil \Rightarrow PNil | (PCons h d hds0) \Rightarrow 
+(let j \def (trans hds0 i) in (let q \def (ptrans hds0 i) in (match (blt j d) 
+with [true \Rightarrow (PCons h (minus d (S j)) q) | false \Rightarrow 
+q])))])) in ptrans.
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/drop1/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/drop1/getl.ma
index 4adda6f18..fd5a2b8be 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/drop1/getl.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/drop1/getl.ma
@@ -23,59 +23,69 @@ include "getl/drop.ma".
 theorem drop1_getl_trans:
  \forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1) 
 \to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl 
-i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl (trans hds i) c2 
-(CHead e2 (Bind b) (ctrans hds i v)))))))))))))
+i c1 (CHead e1 (Bind b) v)) \to (ex2 C (\lambda (e2: C).(drop1 (ptrans hds i) 
+e2 e1)) (\lambda (e2: C).(getl (trans hds i) c2 (CHead e2 (Bind b) (lift1 
+(ptrans hds i) v)))))))))))))
 \def
  \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1: 
 C).(\forall (c2: C).((drop1 p c2 c1) \to (\forall (b: B).(\forall (e1: 
 C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to 
-(ex C (\lambda (e2: C).(getl (trans p i) c2 (CHead e2 (Bind b) (ctrans p i 
-v)))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c2 
-c1)).(\lambda (b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: 
-nat).(\lambda (H0: (getl i c1 (CHead e1 (Bind b) v))).(let H1 \def (match H 
-in drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda 
-(_: (drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 c1) \to 
-(ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v))))))))))) with 
-[(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: 
-(eq C c c2)).(\lambda (H3: (eq C c c1)).(eq_ind C c2 (\lambda (c0: C).((eq C 
-c0 c1) \to (ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v)))))) 
-(\lambda (H4: (eq C c2 c1)).(eq_ind C c1 (\lambda (c0: C).(ex C (\lambda (e2: 
-C).(getl i c0 (CHead e2 (Bind b) v))))) (ex_intro C (\lambda (e2: C).(getl i 
-c1 (CHead e2 (Bind b) v))) e1 H0) c2 (sym_eq C c2 c1 H4))) c (sym_eq C c c2 
+(ex2 C (\lambda (e2: C).(drop1 (ptrans p i) e2 e1)) (\lambda (e2: C).(getl 
+(trans p i) c2 (CHead e2 (Bind b) (lift1 (ptrans p i) v)))))))))))))) 
+(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c2 c1)).(\lambda 
+(b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl 
+i c1 (CHead e1 (Bind b) v))).(let H1 \def (match H in drop1 return (\lambda 
+(p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p c c0)).((eq 
+PList p PNil) \to ((eq C c c2) \to ((eq C c0 c1) \to (ex2 C (\lambda (e2: 
+C).(drop1 PNil e2 e1)) (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) 
+v))))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil 
+PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c c1)).(eq_ind C c2 
+(\lambda (c0: C).((eq C c0 c1) \to (ex2 C (\lambda (e2: C).(drop1 PNil e2 
+e1)) (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v)))))) (\lambda (H4: (eq 
+C c2 c1)).(eq_ind C c1 (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop1 PNil 
+e2 e1)) (\lambda (e2: C).(getl i c0 (CHead e2 (Bind b) v))))) (ex_intro2 C 
+(\lambda (e2: C).(drop1 PNil e2 e1)) (\lambda (e2: C).(getl i c1 (CHead e2 
+(Bind b) v))) e1 (drop1_nil e1) H0) c2 (sym_eq C c2 c1 H4))) c (sym_eq C c c2 
 H2) H3)))) | (drop1_cons c0 c3 h d H1 c4 hds0 H2) \Rightarrow (\lambda (H3: 
 (eq PList (PCons h d hds0) PNil)).(\lambda (H4: (eq C c0 c2)).(\lambda (H5: 
 (eq C c4 c1)).((let H6 \def (eq_ind PList (PCons h d hds0) (\lambda (e: 
 PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil 
 \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in 
 (False_ind ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop h d c0 c3) \to ((drop1 
-hds0 c3 c4) \to (ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) 
-v)))))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C 
-c2) (refl_equal C c1))))))))))) (\lambda (h: nat).(\lambda (d: nat).(\lambda 
-(hds0: PList).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((drop1 hds0 c2 
-c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: 
-nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl 
-(trans hds0 i) c2 (CHead e2 (Bind b) (ctrans hds0 i v))))))))))))))).(\lambda 
-(c1: C).(\lambda (c2: C).(\lambda (H0: (drop1 (PCons h d hds0) c2 
-c1)).(\lambda (b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: 
-nat).(\lambda (H1: (getl i c1 (CHead e1 (Bind b) v))).(let H2 \def (match H0 
-in drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda 
-(_: (drop1 p c c0)).((eq PList p (PCons h d hds0)) \to ((eq C c c2) \to ((eq 
-C c0 c1) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with 
-[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) 
-h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true 
-\Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false 
-\Rightarrow (ctrans hds0 i v)])))))))))))) with [(drop1_nil c) \Rightarrow 
-(\lambda (H2: (eq PList PNil (PCons h d hds0))).(\lambda (H3: (eq C c 
-c2)).(\lambda (H4: (eq C c c1)).((let H5 \def (eq_ind PList PNil (\lambda (e: 
-PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil 
-\Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons h d hds0) H2) 
-in (False_ind ((eq C c c2) \to ((eq C c c1) \to (ex C (\lambda (e2: C).(getl 
-(match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false 
-\Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt 
-(trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) 
-(ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))) H5)) H3 
-H4)))) | (drop1_cons c0 c3 h0 d0 H2 c4 hds1 H3) \Rightarrow (\lambda (H4: (eq 
-PList (PCons h0 d0 hds1) (PCons h d hds0))).(\lambda (H5: (eq C c0 
+hds0 c3 c4) \to (ex2 C (\lambda (e2: C).(drop1 PNil e2 e1)) (\lambda (e2: 
+C).(getl i c2 (CHead e2 (Bind b) v)))))))) H6)) H4 H5 H1 H2))))]) in (H1 
+(refl_equal PList PNil) (refl_equal C c2) (refl_equal C c1))))))))))) 
+(\lambda (h: nat).(\lambda (d: nat).(\lambda (hds0: PList).(\lambda (H: 
+((\forall (c1: C).(\forall (c2: C).((drop1 hds0 c2 c1) \to (\forall (b: 
+B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 
+(Bind b) v)) \to (ex2 C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1)) 
+(\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift1 (ptrans 
+hds0 i) v))))))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H0: 
+(drop1 (PCons h d hds0) c2 c1)).(\lambda (b: B).(\lambda (e1: C).(\lambda (v: 
+T).(\lambda (i: nat).(\lambda (H1: (getl i c1 (CHead e1 (Bind b) v))).(let H2 
+\def (match H0 in drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda 
+(c0: C).(\lambda (_: (drop1 p c c0)).((eq PList p (PCons h d hds0)) \to ((eq 
+C c c2) \to ((eq C c0 c1) \to (ex2 C (\lambda (e2: C).(drop1 (match (blt 
+(trans hds0 i) d) with [true \Rightarrow (PCons h (minus d (S (trans hds0 
+i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) e2 e1)) (\lambda 
+(e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans 
+hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) 
+(lift1 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d 
+(S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) 
+v)))))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil 
+(PCons h d hds0))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c 
+c1)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e in PList 
+return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) 
+\Rightarrow False])) I (PCons h d hds0) H2) in (False_ind ((eq C c c2) \to 
+((eq C c c1) \to (ex2 C (\lambda (e2: C).(drop1 (match (blt (trans hds0 i) d) 
+with [true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) 
+| false \Rightarrow (ptrans hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match 
+(blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false 
+\Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (lift1 (match 
+(blt (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d (S (trans 
+hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) v))))))) 
+H5)) H3 H4)))) | (drop1_cons c0 c3 h0 d0 H2 c4 hds1 H3) \Rightarrow (\lambda 
+(H4: (eq PList (PCons h0 d0 hds1) (PCons h d hds0))).(\lambda (H5: (eq C c0 
 c2)).(\lambda (H6: (eq C c4 c1)).((let H7 \def (f_equal PList PList (\lambda 
 (e: PList).(match e in PList return (\lambda (_: PList).PList) with [PNil 
 \Rightarrow hds1 | (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds1) (PCons h 
@@ -86,74 +96,101 @@ n _) \Rightarrow n])) (PCons h0 d0 hds1) (PCons h d hds0) H4) in ((let H9
 (_: PList).nat) with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n])) 
 (PCons h0 d0 hds1) (PCons h d hds0) H4) in (eq_ind nat h (\lambda (n: 
 nat).((eq nat d0 d) \to ((eq PList hds1 hds0) \to ((eq C c0 c2) \to ((eq C c4 
-c1) \to ((drop n d0 c0 c3) \to ((drop1 hds1 c3 c4) \to (ex C (\lambda (e2: 
+c1) \to ((drop n d0 c0 c3) \to ((drop1 hds1 c3 c4) \to (ex2 C (\lambda (e2: 
+C).(drop1 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h 
+(minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans 
+hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with 
+[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) 
+h)]) c2 (CHead e2 (Bind b) (lift1 (match (blt (trans hds0 i) d) with [true 
+\Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false 
+\Rightarrow (ptrans hds0 i)]) v)))))))))))) (\lambda (H10: (eq nat d0 
+d)).(eq_ind nat d (\lambda (n: nat).((eq PList hds1 hds0) \to ((eq C c0 c2) 
+\to ((eq C c4 c1) \to ((drop h n c0 c3) \to ((drop1 hds1 c3 c4) \to (ex2 C 
+(\lambda (e2: C).(drop1 (match (blt (trans hds0 i) d) with [true \Rightarrow 
+(PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow 
+(ptrans hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match (blt (trans hds0 i) 
+d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans 
+hds0 i) h)]) c2 (CHead e2 (Bind b) (lift1 (match (blt (trans hds0 i) d) with 
+[true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | 
+false \Rightarrow (ptrans hds0 i)]) v))))))))))) (\lambda (H11: (eq PList 
+hds1 hds0)).(eq_ind PList hds0 (\lambda (p: PList).((eq C c0 c2) \to ((eq C 
+c4 c1) \to ((drop h d c0 c3) \to ((drop1 p c3 c4) \to (ex2 C (\lambda (e2: 
+C).(drop1 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h 
+(minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans 
+hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with 
+[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) 
+h)]) c2 (CHead e2 (Bind b) (lift1 (match (blt (trans hds0 i) d) with [true 
+\Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false 
+\Rightarrow (ptrans hds0 i)]) v)))))))))) (\lambda (H12: (eq C c0 
+c2)).(eq_ind C c2 (\lambda (c: C).((eq C c4 c1) \to ((drop h d c c3) \to 
+((drop1 hds0 c3 c4) \to (ex2 C (\lambda (e2: C).(drop1 (match (blt (trans 
+hds0 i) d) with [true \Rightarrow (PCons h (minus d (S (trans hds0 i))) 
+(ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) e2 e1)) (\lambda (e2: 
 C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) 
-| false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match 
-(blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 
-i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)])))))))))))) 
-(\lambda (H10: (eq nat d0 d)).(eq_ind nat d (\lambda (n: nat).((eq PList hds1 
-hds0) \to ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop h n c0 c3) \to ((drop1 
-hds1 c3 c4) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) 
-with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 
-i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true 
-\Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false 
-\Rightarrow (ctrans hds0 i v)]))))))))))) (\lambda (H11: (eq PList hds1 
-hds0)).(eq_ind PList hds0 (\lambda (p: PList).((eq C c0 c2) \to ((eq C c4 c1) 
-\to ((drop h d c0 c3) \to ((drop1 p c3 c4) \to (ex C (\lambda (e2: C).(getl 
-(match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false 
-\Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt 
-(trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) 
-(ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)])))))))))) (\lambda 
-(H12: (eq C c0 c2)).(eq_ind C c2 (\lambda (c: C).((eq C c4 c1) \to ((drop h d 
-c c3) \to ((drop1 hds0 c3 c4) \to (ex C (\lambda (e2: C).(getl (match (blt 
-(trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow 
-(plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) 
-with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i 
-v)) | false \Rightarrow (ctrans hds0 i v)]))))))))) (\lambda (H13: (eq C c4 
-c1)).(eq_ind C c1 (\lambda (c: C).((drop h d c2 c3) \to ((drop1 hds0 c3 c) 
-\to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true 
-\Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 
-(CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift 
-h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans 
-hds0 i v)])))))))) (\lambda (H14: (drop h d c2 c3)).(\lambda (H15: (drop1 
-hds0 c3 c1)).(xinduction bool (blt (trans hds0 i) d) (\lambda (b0: bool).(ex 
-C (\lambda (e2: C).(getl (match b0 with [true \Rightarrow (trans hds0 i) | 
-false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match b0 
-with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i 
-v)) | false \Rightarrow (ctrans hds0 i v)])))))) (\lambda (x_x: 
+| false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (lift1 
+(match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d (S 
+(trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) 
+v))))))))) (\lambda (H13: (eq C c4 c1)).(eq_ind C c1 (\lambda (c: C).((drop h 
+d c2 c3) \to ((drop1 hds0 c3 c) \to (ex2 C (\lambda (e2: C).(drop1 (match 
+(blt (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d (S (trans 
+hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) e2 e1)) 
+(\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow 
+(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 
+(Bind b) (lift1 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h 
+(minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans 
+hds0 i)]) v)))))))) (\lambda (H14: (drop h d c2 c3)).(\lambda (H15: (drop1 
+hds0 c3 c1)).(xinduction bool (blt (trans hds0 i) d) (\lambda (b0: bool).(ex2 
+C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow (PCons h (minus d 
+(S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) e2 
+e1)) (\lambda (e2: C).(getl (match b0 with [true \Rightarrow (trans hds0 i) | 
+false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (lift1 
+(match b0 with [true \Rightarrow (PCons h (minus d (S (trans hds0 i))) 
+(ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) v)))))) (\lambda (x_x: 
 bool).(bool_ind (\lambda (b0: bool).((eq bool (blt (trans hds0 i) d) b0) \to 
-(ex C (\lambda (e2: C).(getl (match b0 with [true \Rightarrow (trans hds0 i) 
-| false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match b0 
-with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i 
-v)) | false \Rightarrow (ctrans hds0 i v)]))))))) (\lambda (H16: (eq bool 
-(blt (trans hds0 i) d) true)).(let H_x \def (H c1 c3 H15 b e1 v i H1) in (let 
-H17 \def H_x in (ex_ind C (\lambda (e2: C).(getl (trans hds0 i) c3 (CHead e2 
-(Bind b) (ctrans hds0 i v)))) (ex C (\lambda (e2: C).(getl (trans hds0 i) c2 
-(CHead e2 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i 
-v)))))) (\lambda (x: C).(\lambda (H18: (getl (trans hds0 i) c3 (CHead x (Bind 
-b) (ctrans hds0 i v)))).(let H_x0 \def (drop_getl_trans_lt (trans hds0 i) d 
-(le_S_n (S (trans hds0 i)) d (lt_le_S (S (trans hds0 i)) (S d) (blt_lt (S d) 
-(S (trans hds0 i)) H16))) c2 c3 h H14 b x (ctrans hds0 i v) H18) in (let H19 
-\def H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 
-(Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v))))) (\lambda 
-(e2: C).(drop h (minus d (S (trans hds0 i))) e2 x)) (ex C (\lambda (e2: 
-C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans hds0 
-i))) (ctrans hds0 i v)))))) (\lambda (x0: C).(\lambda (H20: (getl (trans hds0 
-i) c2 (CHead x0 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i 
-v))))).(\lambda (_: (drop h (minus d (S (trans hds0 i))) x0 x)).(ex_intro C 
-(\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d 
-(S (trans hds0 i))) (ctrans hds0 i v))))) x0 H20)))) H19))))) H17)))) 
-(\lambda (H16: (eq bool (blt (trans hds0 i) d) false)).(let H_x \def (H c1 c3 
-H15 b e1 v i H1) in (let H17 \def H_x in (ex_ind C (\lambda (e2: C).(getl 
-(trans hds0 i) c3 (CHead e2 (Bind b) (ctrans hds0 i v)))) (ex C (\lambda (e2: 
-C).(getl (plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (ctrans hds0 i v))))) 
-(\lambda (x: C).(\lambda (H18: (getl (trans hds0 i) c3 (CHead x (Bind b) 
-(ctrans hds0 i v)))).(let H19 \def (drop_getl_trans_ge (trans hds0 i) c2 c3 d 
-h H14 (CHead x (Bind b) (ctrans hds0 i v)) H18) in (ex_intro C (\lambda (e2: 
-C).(getl (plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (ctrans hds0 i v)))) x 
-(H19 (bge_le d (trans hds0 i) H16)))))) H17)))) x_x))))) c4 (sym_eq C c4 c1 
-H13))) c0 (sym_eq C c0 c2 H12))) hds1 (sym_eq PList hds1 hds0 H11))) d0 
-(sym_eq nat d0 d H10))) h0 (sym_eq nat h0 h H9))) H8)) H7)) H5 H6 H2 H3))))]) 
-in (H2 (refl_equal PList (PCons h d hds0)) (refl_equal C c2) (refl_equal C 
-c1))))))))))))))) hds).
+(ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow (PCons h 
+(minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans 
+hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match b0 with [true \Rightarrow 
+(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 
+(Bind b) (lift1 (match b0 with [true \Rightarrow (PCons h (minus d (S (trans 
+hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) v))))))) 
+(\lambda (H16: (eq bool (blt (trans hds0 i) d) true)).(let H_x \def (H c1 c3 
+H15 b e1 v i H1) in (let H17 \def H_x in (ex2_ind C (\lambda (e2: C).(drop1 
+(ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) c3 (CHead e2 
+(Bind b) (lift1 (ptrans hds0 i) v)))) (ex2 C (\lambda (e2: C).(drop1 (PCons h 
+(minus d (S (trans hds0 i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl 
+(trans hds0 i) c2 (CHead e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0 
+i))) (ptrans hds0 i)) v))))) (\lambda (x: C).(\lambda (H18: (drop1 (ptrans 
+hds0 i) x e1)).(\lambda (H19: (getl (trans hds0 i) c3 (CHead x (Bind b) 
+(lift1 (ptrans hds0 i) v)))).(let H_x0 \def (drop_getl_trans_lt (trans hds0 
+i) d (le_S_n (S (trans hds0 i)) d (lt_le_S (S (trans hds0 i)) (S d) (blt_lt 
+(S d) (S (trans hds0 i)) H16))) c2 c3 h H14 b x (lift1 (ptrans hds0 i) v) 
+H19) in (let H20 \def H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans hds0 
+i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans hds0 i))) (lift1 (ptrans 
+hds0 i) v))))) (\lambda (e2: C).(drop h (minus d (S (trans hds0 i))) e2 x)) 
+(ex2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S (trans hds0 i))) (ptrans 
+hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) 
+(lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) v))))) (\lambda 
+(x0: C).(\lambda (H21: (getl (trans hds0 i) c2 (CHead x0 (Bind b) (lift h 
+(minus d (S (trans hds0 i))) (lift1 (ptrans hds0 i) v))))).(\lambda (H22: 
+(drop h (minus d (S (trans hds0 i))) x0 x)).(ex_intro2 C (\lambda (e2: 
+C).(drop1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) e2 e1)) 
+(\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift1 (PCons h 
+(minus d (S (trans hds0 i))) (ptrans hds0 i)) v)))) x0 (drop1_cons x0 x h 
+(minus d (S (trans hds0 i))) H22 e1 (ptrans hds0 i) H18) H21)))) H20)))))) 
+H17)))) (\lambda (H16: (eq bool (blt (trans hds0 i) d) false)).(let H_x \def 
+(H c1 c3 H15 b e1 v i H1) in (let H17 \def H_x in (ex2_ind C (\lambda (e2: 
+C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) c3 
+(CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) (ex2 C (\lambda (e2: 
+C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (plus (trans hds0 i) 
+h) c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))) (\lambda (x: 
+C).(\lambda (H18: (drop1 (ptrans hds0 i) x e1)).(\lambda (H19: (getl (trans 
+hds0 i) c3 (CHead x (Bind b) (lift1 (ptrans hds0 i) v)))).(let H20 \def 
+(drop_getl_trans_ge (trans hds0 i) c2 c3 d h H14 (CHead x (Bind b) (lift1 
+(ptrans hds0 i) v)) H19) in (ex_intro2 C (\lambda (e2: C).(drop1 (ptrans hds0 
+i) e2 e1)) (\lambda (e2: C).(getl (plus (trans hds0 i) h) c2 (CHead e2 (Bind 
+b) (lift1 (ptrans hds0 i) v)))) x H18 (H20 (bge_le d (trans hds0 i) 
+H16))))))) H17)))) x_x))))) c4 (sym_eq C c4 c1 H13))) c0 (sym_eq C c0 c2 
+H12))) hds1 (sym_eq PList hds1 hds0 H11))) d0 (sym_eq nat d0 d H10))) h0 
+(sym_eq nat h0 h H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList 
+(PCons h d hds0)) (refl_equal C c2) (refl_equal C c1))))))))))))))) hds).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/drop1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/drop1/props.ma
index dc61e4c29..ee7723013 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/drop1/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/drop1/props.ma
@@ -163,3 +163,72 @@ PList hds0 p H11))) d0 (sym_eq nat d0 n0 H10))) h0 (sym_eq nat h0 n H9)))
 H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p)) 
 (refl_equal C c1) (refl_equal C c2))))))))) hds)))))).
 
+theorem drop1_trans:
+ \forall (is1: PList).(\forall (c1: C).(\forall (c0: C).((drop1 is1 c1 c0) 
+\to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0 c2) \to (drop1 
+(papp is1 is2) c1 c2)))))))
+\def
+ \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (c1: 
+C).(\forall (c0: C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2: 
+C).((drop1 is2 c0 c2) \to (drop1 (papp p is2) c1 c2)))))))) (\lambda (c1: 
+C).(\lambda (c0: C).(\lambda (H: (drop1 PNil c1 c0)).(\lambda (is2: 
+PList).(\lambda (c2: C).(\lambda (H0: (drop1 is2 c0 c2)).(let H1 \def (match 
+H in drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c3: 
+C).(\lambda (_: (drop1 p c c3)).((eq PList p PNil) \to ((eq C c c1) \to ((eq 
+C c3 c0) \to (drop1 is2 c1 c2)))))))) with [(drop1_nil c) \Rightarrow 
+(\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c c1)).(\lambda (H3: 
+(eq C c c0)).(eq_ind C c1 (\lambda (c3: C).((eq C c3 c0) \to (drop1 is2 c1 
+c2))) (\lambda (H4: (eq C c1 c0)).(eq_ind C c0 (\lambda (c3: C).(drop1 is2 c3 
+c2)) (let H5 \def (eq_ind_r C c0 (\lambda (c3: C).(drop1 is2 c3 c2)) H0 c1 
+H4) in (eq_ind C c1 (\lambda (c3: C).(drop1 is2 c3 c2)) H5 c0 H4)) c1 (sym_eq 
+C c1 c0 H4))) c (sym_eq C c c1 H2) H3)))) | (drop1_cons c3 c4 h d H1 c5 hds 
+H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds) PNil)).(\lambda (H4: 
+(eq C c3 c1)).(\lambda (H5: (eq C c5 c0)).((let H6 \def (eq_ind PList (PCons 
+h d hds) (\lambda (e: PList).(match e in PList return (\lambda (_: 
+PList).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) 
+I PNil H3) in (False_ind ((eq C c3 c1) \to ((eq C c5 c0) \to ((drop h d c3 
+c4) \to ((drop1 hds c4 c5) \to (drop1 is2 c1 c2))))) H6)) H4 H5 H1 H2))))]) 
+in (H1 (refl_equal PList PNil) (refl_equal C c1) (refl_equal C c0))))))))) 
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: 
+((\forall (c1: C).(\forall (c0: C).((drop1 p c1 c0) \to (\forall (is2: 
+PList).(\forall (c2: C).((drop1 is2 c0 c2) \to (drop1 (papp p is2) c1 
+c2))))))))).(\lambda (c1: C).(\lambda (c0: C).(\lambda (H0: (drop1 (PCons n 
+n0 p) c1 c0)).(\lambda (is2: PList).(\lambda (c2: C).(\lambda (H1: (drop1 is2 
+c0 c2)).(let H2 \def (match H0 in drop1 return (\lambda (p0: PList).(\lambda 
+(c: C).(\lambda (c3: C).(\lambda (_: (drop1 p0 c c3)).((eq PList p0 (PCons n 
+n0 p)) \to ((eq C c c1) \to ((eq C c3 c0) \to (drop1 (PCons n n0 (papp p 
+is2)) c1 c2)))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList 
+PNil (PCons n n0 p))).(\lambda (H3: (eq C c c1)).(\lambda (H4: (eq C c 
+c0)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e in PList 
+return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) 
+\Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c1) \to ((eq 
+C c c0) \to (drop1 (PCons n n0 (papp p is2)) c1 c2))) H5)) H3 H4)))) | 
+(drop1_cons c3 c4 h d H2 c5 hds H3) \Rightarrow (\lambda (H4: (eq PList 
+(PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C c3 c1)).(\lambda (H6: 
+(eq C c5 c0)).((let H7 \def (f_equal PList PList (\lambda (e: PList).(match e 
+in PList return (\lambda (_: PList).PList) with [PNil \Rightarrow hds | 
+(PCons _ _ p0) \Rightarrow p0])) (PCons h d hds) (PCons n n0 p) H4) in ((let 
+H8 \def (f_equal PList nat (\lambda (e: PList).(match e in PList return 
+(\lambda (_: PList).nat) with [PNil \Rightarrow d | (PCons _ n1 _) 
+\Rightarrow n1])) (PCons h d hds) (PCons n n0 p) H4) in ((let H9 \def 
+(f_equal PList nat (\lambda (e: PList).(match e in PList return (\lambda (_: 
+PList).nat) with [PNil \Rightarrow h | (PCons n1 _ _) \Rightarrow n1])) 
+(PCons h d hds) (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq 
+nat d n0) \to ((eq PList hds p) \to ((eq C c3 c1) \to ((eq C c5 c0) \to 
+((drop n1 d c3 c4) \to ((drop1 hds c4 c5) \to (drop1 (PCons n n0 (papp p 
+is2)) c1 c2)))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 (\lambda 
+(n1: nat).((eq PList hds p) \to ((eq C c3 c1) \to ((eq C c5 c0) \to ((drop n 
+n1 c3 c4) \to ((drop1 hds c4 c5) \to (drop1 (PCons n n0 (papp p is2)) c1 
+c2))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: 
+PList).((eq C c3 c1) \to ((eq C c5 c0) \to ((drop n n0 c3 c4) \to ((drop1 p0 
+c4 c5) \to (drop1 (PCons n n0 (papp p is2)) c1 c2)))))) (\lambda (H12: (eq C 
+c3 c1)).(eq_ind C c1 (\lambda (c: C).((eq C c5 c0) \to ((drop n n0 c c4) \to 
+((drop1 p c4 c5) \to (drop1 (PCons n n0 (papp p is2)) c1 c2))))) (\lambda 
+(H13: (eq C c5 c0)).(eq_ind C c0 (\lambda (c: C).((drop n n0 c1 c4) \to 
+((drop1 p c4 c) \to (drop1 (PCons n n0 (papp p is2)) c1 c2)))) (\lambda (H14: 
+(drop n n0 c1 c4)).(\lambda (H15: (drop1 p c4 c0)).(drop1_cons c1 c4 n n0 H14 
+c2 (papp p is2) (H c4 c0 H15 is2 c2 H1)))) c5 (sym_eq C c5 c0 H13))) c3 
+(sym_eq C c3 c1 H12))) hds (sym_eq PList hds p H11))) d (sym_eq nat d n0 
+H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal 
+PList (PCons n n0 p)) (refl_equal C c1) (refl_equal C c0))))))))))))) is1).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/fsubst0/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/fsubst0/defs.ma
index 71ccebcd6..f5f76deec 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/fsubst0/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/fsubst0/defs.ma
@@ -20,7 +20,7 @@ include "csubst0/defs.ma".
 
 include "subst0/defs.ma".
 
-inductive fsubst0 (i:nat) (v:T) (c1:C) (t1:T): C \to (T \to Prop) \def
+inductive fsubst0 (i: nat) (v: T) (c1: C) (t1: T): C \to (T \to Prop) \def
 | fsubst0_snd: \forall (t2: T).((subst0 i v t1 t2) \to (fsubst0 i v c1 t1 c1 
 t2))
 | fsubst0_fst: \forall (c2: C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/defs.ma
index 40845209b..8c40f8832 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/defs.ma
@@ -20,7 +20,7 @@ include "drop/defs.ma".
 
 include "clear/defs.ma".
 
-inductive getl (h:nat) (c1:C) (c2:C): Prop \def
+inductive getl (h: nat) (c1: C) (c2: C): Prop \def
 | getl_intro: \forall (e: C).((drop h O c1 e) \to ((clear e c2) \to (getl h 
 c1 c2))).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/iso/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/iso/fwd.ma
index 9402e8cbf..61a965d15 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/iso/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/iso/fwd.ma
@@ -18,6 +18,8 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/iso/fwd".
 
 include "iso/defs.ma".
 
+include "tlist/defs.ma".
+
 theorem iso_flats_lref_bind_false:
  \forall (f: F).(\forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall 
 (t: T).(\forall (vs: TList).((iso (THeads (Flat f) vs (TLRef i)) (THead (Bind 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/leq/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/leq/defs.ma
index 8015e7132..51db1202c 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/leq/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/leq/defs.ma
@@ -18,7 +18,7 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/leq/defs".
 
 include "aplus/defs.ma".
 
-inductive leq (g:G): A \to (A \to Prop) \def
+inductive leq (g: G): A \to (A \to Prop) \def
 | leq_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall 
 (n2: nat).(\forall (k: nat).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort 
 h2 n2) k)) \to (leq g (ASort h1 n1) (ASort h2 n2)))))))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/defs.ma
index 41c5d1e46..66b000e77 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/defs.ma
@@ -18,6 +18,8 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/defs".
 
 include "T/defs.ma".
 
+include "tlist/defs.ma".
+
 include "s/defs.ma".
 
 definition lref_map:
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props.ma
index 3e2301883..8987e40f0 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props.ma
@@ -16,6 +16,8 @@
 
 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props".
 
+include "tlist/defs.ma".
+
 include "lift/fwd.ma".
 
 include "s/props.ma".
@@ -116,6 +118,20 @@ theorem lift_lref_gt:
 (pred n) (eq_ind nat n (\lambda (n0: nat).(le (S d) n0)) H (S (pred n)) 
 (S_pred n d H))))))).
 
+theorem lifts_tapp:
+ \forall (h: nat).(\forall (d: nat).(\forall (v: T).(\forall (vs: TList).(eq 
+TList (lifts h d (TApp vs v)) (TApp (lifts h d vs) (lift h d v))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (v: T).(\lambda (vs: 
+TList).(TList_ind (\lambda (t: TList).(eq TList (lifts h d (TApp t v)) (TApp 
+(lifts h d t) (lift h d v)))) (refl_equal TList (TCons (lift h d v) TNil)) 
+(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq TList (lifts h d (TApp 
+t0 v)) (TApp (lifts h d t0) (lift h d v)))).(eq_ind_r TList (TApp (lifts h d 
+t0) (lift h d v)) (\lambda (t1: TList).(eq TList (TCons (lift h d t) t1) 
+(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v))))) (refl_equal TList 
+(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v)))) (lifts h d (TApp t0 
+v)) H)))) vs)))).
+
 theorem lift_inj:
  \forall (x: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((eq T 
 (lift h d x) (lift h d t)) \to (eq T x t)))))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift1/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift1/fwd.ma
index 4b548cbff..8400f1f34 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift1/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift1/fwd.ma
@@ -20,20 +20,28 @@ include "lift1/defs.ma".
 
 include "lift/fwd.ma".
 
+theorem lift1_sort:
+ \forall (n: nat).(\forall (is: PList).(eq T (lift1 is (TSort n)) (TSort n)))
+\def
+ \lambda (n: nat).(\lambda (is: PList).(PList_ind (\lambda (p: PList).(eq T 
+(lift1 p (TSort n)) (TSort n))) (refl_equal T (TSort n)) (\lambda (n0: 
+nat).(\lambda (n1: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1 p 
+(TSort n)) (TSort n))).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (lift n0 
+n1 t) (TSort n))) (refl_equal T (TSort n)) (lift1 p (TSort n)) H))))) is)).
+
 theorem lift1_lref:
  \forall (hds: PList).(\forall (i: nat).(eq T (lift1 hds (TLRef i)) (TLRef 
 (trans hds i))))
 \def
  \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: nat).(eq T 
 (lift1 p (TLRef i)) (TLRef (trans p i))))) (\lambda (i: nat).(refl_equal T 
-(TLRef (trans PNil i)))) (\lambda (h: nat).(\lambda (d: nat).(\lambda (p: 
-PList).(\lambda (H: ((\forall (i: nat).(eq T (lift1 p (TLRef i)) (TLRef 
-(trans p i)))))).(\lambda (i: nat).(eq_ind_r T (TLRef (trans p i)) (\lambda 
-(t: T).(eq T (lift h d t) (TLRef (match (blt (trans p i) d) with [true 
-\Rightarrow (trans p i) | false \Rightarrow (plus (trans p i) h)])))) 
-(refl_equal T (TLRef (match (blt (trans p i) d) with [true \Rightarrow (trans 
-p i) | false \Rightarrow (plus (trans p i) h)]))) (lift1 p (TLRef i)) (H 
-i))))))) hds).
+(TLRef i))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda 
+(H: ((\forall (i: nat).(eq T (lift1 p (TLRef i)) (TLRef (trans p 
+i)))))).(\lambda (i: nat).(eq_ind_r T (TLRef (trans p i)) (\lambda (t: T).(eq 
+T (lift n n0 t) (TLRef (match (blt (trans p i) n0) with [true \Rightarrow 
+(trans p i) | false \Rightarrow (plus (trans p i) n)])))) (refl_equal T 
+(TLRef (match (blt (trans p i) n0) with [true \Rightarrow (trans p i) | false 
+\Rightarrow (plus (trans p i) n)]))) (lift1 p (TLRef i)) (H i))))))) hds).
 
 theorem lift1_bind:
  \forall (b: B).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T 
@@ -43,18 +51,18 @@ hds) t))))))
  \lambda (b: B).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
 (u: T).(\forall (t: T).(eq T (lift1 p (THead (Bind b) u t)) (THead (Bind b) 
 (lift1 p u) (lift1 (Ss p) t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal 
-T (THead (Bind b) (lift1 PNil u) (lift1 (Ss PNil) t))))) (\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (p: PList).(\lambda (H: ((\forall (u: 
-T).(\forall (t: T).(eq T (lift1 p (THead (Bind b) u t)) (THead (Bind b) 
-(lift1 p u) (lift1 (Ss p) t))))))).(\lambda (u: T).(\lambda (t: T).(eq_ind_r 
-T (THead (Bind b) (lift1 p u) (lift1 (Ss p) t)) (\lambda (t0: T).(eq T (lift 
-h d t0) (THead (Bind b) (lift h d (lift1 p u)) (lift h (S d) (lift1 (Ss p) 
-t))))) (eq_ind_r T (THead (Bind b) (lift h d (lift1 p u)) (lift h (S d) 
-(lift1 (Ss p) t))) (\lambda (t0: T).(eq T t0 (THead (Bind b) (lift h d (lift1 
-p u)) (lift h (S d) (lift1 (Ss p) t))))) (refl_equal T (THead (Bind b) (lift 
-h d (lift1 p u)) (lift h (S d) (lift1 (Ss p) t)))) (lift h d (THead (Bind b) 
-(lift1 p u) (lift1 (Ss p) t))) (lift_bind b (lift1 p u) (lift1 (Ss p) t) h 
-d)) (lift1 p (THead (Bind b) u t)) (H u t)))))))) hds)).
+T (THead (Bind b) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: 
+PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead 
+(Bind b) u t)) (THead (Bind b) (lift1 p u) (lift1 (Ss p) t))))))).(\lambda 
+(u: T).(\lambda (t: T).(eq_ind_r T (THead (Bind b) (lift1 p u) (lift1 (Ss p) 
+t)) (\lambda (t0: T).(eq T (lift n n0 t0) (THead (Bind b) (lift n n0 (lift1 p 
+u)) (lift n (S n0) (lift1 (Ss p) t))))) (eq_ind_r T (THead (Bind b) (lift n 
+n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))) (\lambda (t0: T).(eq T t0 
+(THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))))) 
+(refl_equal T (THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1 
+(Ss p) t)))) (lift n n0 (THead (Bind b) (lift1 p u) (lift1 (Ss p) t))) 
+(lift_bind b (lift1 p u) (lift1 (Ss p) t) n n0)) (lift1 p (THead (Bind b) u 
+t)) (H u t)))))))) hds)).
 
 theorem lift1_flat:
  \forall (f: F).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T 
@@ -64,17 +72,17 @@ t))))))
  \lambda (f: F).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
 (u: T).(\forall (t: T).(eq T (lift1 p (THead (Flat f) u t)) (THead (Flat f) 
 (lift1 p u) (lift1 p t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal T 
-(THead (Flat f) (lift1 PNil u) (lift1 PNil t))))) (\lambda (h: nat).(\lambda 
-(d: nat).(\lambda (p: PList).(\lambda (H: ((\forall (u: T).(\forall (t: 
-T).(eq T (lift1 p (THead (Flat f) u t)) (THead (Flat f) (lift1 p u) (lift1 p 
-t))))))).(\lambda (u: T).(\lambda (t: T).(eq_ind_r T (THead (Flat f) (lift1 p 
-u) (lift1 p t)) (\lambda (t0: T).(eq T (lift h d t0) (THead (Flat f) (lift h 
-d (lift1 p u)) (lift h d (lift1 p t))))) (eq_ind_r T (THead (Flat f) (lift h 
-d (lift1 p u)) (lift h d (lift1 p t))) (\lambda (t0: T).(eq T t0 (THead (Flat 
-f) (lift h d (lift1 p u)) (lift h d (lift1 p t))))) (refl_equal T (THead 
-(Flat f) (lift h d (lift1 p u)) (lift h d (lift1 p t)))) (lift h d (THead 
-(Flat f) (lift1 p u) (lift1 p t))) (lift_flat f (lift1 p u) (lift1 p t) h d)) 
-(lift1 p (THead (Flat f) u t)) (H u t)))))))) hds)).
+(THead (Flat f) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: 
+PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead 
+(Flat f) u t)) (THead (Flat f) (lift1 p u) (lift1 p t))))))).(\lambda (u: 
+T).(\lambda (t: T).(eq_ind_r T (THead (Flat f) (lift1 p u) (lift1 p t)) 
+(\lambda (t0: T).(eq T (lift n n0 t0) (THead (Flat f) (lift n n0 (lift1 p u)) 
+(lift n n0 (lift1 p t))))) (eq_ind_r T (THead (Flat f) (lift n n0 (lift1 p 
+u)) (lift n n0 (lift1 p t))) (\lambda (t0: T).(eq T t0 (THead (Flat f) (lift 
+n n0 (lift1 p u)) (lift n n0 (lift1 p t))))) (refl_equal T (THead (Flat f) 
+(lift n n0 (lift1 p u)) (lift n n0 (lift1 p t)))) (lift n n0 (THead (Flat f) 
+(lift1 p u) (lift1 p t))) (lift_flat f (lift1 p u) (lift1 p t) n n0)) (lift1 
+p (THead (Flat f) u t)) (H u t)))))))) hds)).
 
 theorem lift1_cons_tail:
  \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(eq 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift1/props.ma
index eeec1fc7d..5b92ff26a 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift1/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift1/props.ma
@@ -22,6 +22,23 @@ include "lift/props.ma".
 
 include "drop1/defs.ma".
 
+theorem lift1_lift1:
+ \forall (is1: PList).(\forall (is2: PList).(\forall (t: T).(eq T (lift1 is1 
+(lift1 is2 t)) (lift1 (papp is1 is2) t))))
+\def
+ \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2: 
+PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 (papp p is2) 
+t))))) (\lambda (is2: PList).(\lambda (t: T).(refl_equal T (lift1 is2 t)))) 
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: 
+((\forall (is2: PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 
+(papp p is2) t)))))).(\lambda (is2: PList).(\lambda (t: T).(sym_eq T (lift n 
+n0 (lift1 (papp p is2) t)) (lift n n0 (lift1 p (lift1 is2 t))) (sym_eq T 
+(lift n n0 (lift1 p (lift1 is2 t))) (lift n n0 (lift1 (papp p is2) t)) 
+(sym_eq T (lift n n0 (lift1 (papp p is2) t)) (lift n n0 (lift1 p (lift1 is2 
+t))) (f_equal3 nat nat T T lift n n n0 n0 (lift1 (papp p is2) t) (lift1 p 
+(lift1 is2 t)) (refl_equal nat n) (refl_equal nat n0) (sym_eq T (lift1 p 
+(lift1 is2 t)) (lift1 (papp p is2) t) (H is2 t)))))))))))) is1).
+
 theorem lift1_xhg:
  \forall (hds: PList).(\forall (t: T).(eq T (lift1 (Ss hds) (lift (S O) O t)) 
 (lift (S O) O (lift1 hds t))))
@@ -59,57 +76,60 @@ t0))))) (refl_equal TList (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O
 
 theorem lift1_free:
  \forall (hds: PList).(\forall (i: nat).(\forall (t: T).(eq T (lift1 hds 
-(lift (S i) O t)) (lift (S (trans hds i)) O (ctrans hds i t)))))
+(lift (S i) O t)) (lift (S (trans hds i)) O (lift1 (ptrans hds i) t)))))
 \def
  \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: 
 nat).(\forall (t: T).(eq T (lift1 p (lift (S i) O t)) (lift (S (trans p i)) O 
-(ctrans p i t)))))) (\lambda (i: nat).(\lambda (t: T).(refl_equal T (lift (S 
-i) O t)))) (\lambda (h: nat).(\lambda (d: nat).(\lambda (hds0: 
+(lift1 (ptrans p i) t)))))) (\lambda (i: nat).(\lambda (t: T).(refl_equal T 
+(lift (S i) O t)))) (\lambda (h: nat).(\lambda (d: nat).(\lambda (hds0: 
 PList).(\lambda (H: ((\forall (i: nat).(\forall (t: T).(eq T (lift1 hds0 
-(lift (S i) O t)) (lift (S (trans hds0 i)) O (ctrans hds0 i t))))))).(\lambda 
-(i: nat).(\lambda (t: T).(eq_ind_r T (lift (S (trans hds0 i)) O (ctrans hds0 
-i t)) (\lambda (t0: T).(eq T (lift h d t0) (lift (S (match (blt (trans hds0 
-i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans 
-hds0 i) h)])) O (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h 
-(minus d (S (trans hds0 i))) (ctrans hds0 i t)) | false \Rightarrow (ctrans 
-hds0 i t)])))) (xinduction bool (blt (trans hds0 i) d) (\lambda (b: bool).(eq 
-T (lift h d (lift (S (trans hds0 i)) O (ctrans hds0 i t))) (lift (S (match b 
-with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 
-i) h)])) O (match b with [true \Rightarrow (lift h (minus d (S (trans hds0 
-i))) (ctrans hds0 i t)) | false \Rightarrow (ctrans hds0 i t)])))) (\lambda 
-(x_x: bool).(bool_ind (\lambda (b: bool).((eq bool (blt (trans hds0 i) d) b) 
-\to (eq T (lift h d (lift (S (trans hds0 i)) O (ctrans hds0 i t))) (lift (S 
-(match b with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus 
-(trans hds0 i) h)])) O (match b with [true \Rightarrow (lift h (minus d (S 
-(trans hds0 i))) (ctrans hds0 i t)) | false \Rightarrow (ctrans hds0 i 
-t)]))))) (\lambda (H0: (eq bool (blt (trans hds0 i) d) true)).(eq_ind_r nat 
-(plus (S (trans hds0 i)) (minus d (S (trans hds0 i)))) (\lambda (n: nat).(eq 
-T (lift h n (lift (S (trans hds0 i)) O (ctrans hds0 i t))) (lift (S (trans 
-hds0 i)) O (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i t))))) 
+(lift (S i) O t)) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) 
+t))))))).(\lambda (i: nat).(\lambda (t: T).(eq_ind_r T (lift (S (trans hds0 
+i)) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T (lift h d t0) (lift 
+(S (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | 
+false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match (blt (trans hds0 
+i) d) with [true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans 
+hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t)))) (xinduction bool (blt 
+(trans hds0 i) d) (\lambda (b: bool).(eq T (lift h d (lift (S (trans hds0 i)) 
+O (lift1 (ptrans hds0 i) t))) (lift (S (match b with [true \Rightarrow (trans 
+hds0 i) | false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match b with 
+[true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | 
+false \Rightarrow (ptrans hds0 i)]) t)))) (\lambda (x_x: bool).(bool_ind 
+(\lambda (b: bool).((eq bool (blt (trans hds0 i) d) b) \to (eq T (lift h d 
+(lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (match b with 
+[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) 
+h)])) O (lift1 (match b with [true \Rightarrow (PCons h (minus d (S (trans 
+hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t))))) 
+(\lambda (H0: (eq bool (blt (trans hds0 i) d) true)).(eq_ind_r nat (plus (S 
+(trans hds0 i)) (minus d (S (trans hds0 i)))) (\lambda (n: nat).(eq T (lift h 
+n (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (trans hds0 
+i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t)))) 
 (eq_ind_r T (lift (S (trans hds0 i)) O (lift h (minus d (S (trans hds0 i))) 
-(ctrans hds0 i t))) (\lambda (t0: T).(eq T t0 (lift (S (trans hds0 i)) O 
-(lift h (minus d (S (trans hds0 i))) (ctrans hds0 i t))))) (refl_equal T 
-(lift (S (trans hds0 i)) O (lift h (minus d (S (trans hds0 i))) (ctrans hds0 
-i t)))) (lift h (plus (S (trans hds0 i)) (minus d (S (trans hds0 i)))) (lift 
-(S (trans hds0 i)) O (ctrans hds0 i t))) (lift_d (ctrans hds0 i t) h (S 
-(trans hds0 i)) (minus d (S (trans hds0 i))) O (le_O_n (minus d (S (trans 
-hds0 i)))))) d (le_plus_minus (S (trans hds0 i)) d (bge_le (S (trans hds0 i)) 
-d (le_bge (S (trans hds0 i)) d (lt_le_S (trans hds0 i) d (blt_lt d (trans 
-hds0 i) H0))))))) (\lambda (H0: (eq bool (blt (trans hds0 i) d) 
-false)).(eq_ind_r T (lift (plus h (S (trans hds0 i))) O (ctrans hds0 i t)) 
-(\lambda (t0: T).(eq T t0 (lift (S (plus (trans hds0 i) h)) O (ctrans hds0 i 
-t)))) (eq_ind nat (S (plus h (trans hds0 i))) (\lambda (n: nat).(eq T (lift n 
-O (ctrans hds0 i t)) (lift (S (plus (trans hds0 i) h)) O (ctrans hds0 i t)))) 
+(lift1 (ptrans hds0 i) t))) (\lambda (t0: T).(eq T t0 (lift (S (trans hds0 
+i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t)))) 
+(refl_equal T (lift (S (trans hds0 i)) O (lift1 (PCons h (minus d (S (trans 
+hds0 i))) (ptrans hds0 i)) t))) (lift h (plus (S (trans hds0 i)) (minus d (S 
+(trans hds0 i)))) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) 
+(lift_d (lift1 (ptrans hds0 i) t) h (S (trans hds0 i)) (minus d (S (trans 
+hds0 i))) O (le_O_n (minus d (S (trans hds0 i)))))) d (le_plus_minus (S 
+(trans hds0 i)) d (bge_le (S (trans hds0 i)) d (le_bge (S (trans hds0 i)) d 
+(lt_le_S (trans hds0 i) d (blt_lt d (trans hds0 i) H0))))))) (\lambda (H0: 
+(eq bool (blt (trans hds0 i) d) false)).(eq_ind_r T (lift (plus h (S (trans 
+hds0 i))) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T t0 (lift (S 
+(plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) (eq_ind nat (S (plus 
+h (trans hds0 i))) (\lambda (n: nat).(eq T (lift n O (lift1 (ptrans hds0 i) 
+t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) 
 (eq_ind_r nat (plus (trans hds0 i) h) (\lambda (n: nat).(eq T (lift (S n) O 
-(ctrans hds0 i t)) (lift (S (plus (trans hds0 i) h)) O (ctrans hds0 i t)))) 
-(refl_equal T (lift (S (plus (trans hds0 i) h)) O (ctrans hds0 i t))) (plus h 
-(trans hds0 i)) (plus_comm h (trans hds0 i))) (plus h (S (trans hds0 i))) 
-(plus_n_Sm h (trans hds0 i))) (lift h d (lift (S (trans hds0 i)) O (ctrans 
-hds0 i t))) (lift_free (ctrans hds0 i t) (S (trans hds0 i)) h O d (eq_ind nat 
-(S (plus O (trans hds0 i))) (\lambda (n: nat).(le d n)) (eq_ind_r nat (plus 
-(trans hds0 i) O) (\lambda (n: nat).(le d (S n))) (le_S d (plus (trans hds0 
-i) O) (le_plus_trans d (trans hds0 i) O (bge_le d (trans hds0 i) H0))) (plus 
-O (trans hds0 i)) (plus_comm O (trans hds0 i))) (plus O (S (trans hds0 i))) 
-(plus_n_Sm O (trans hds0 i))) (le_O_n d)))) x_x))) (lift1 hds0 (lift (S i) O 
-t)) (H i t)))))))) hds).
+(lift1 (ptrans hds0 i) t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans 
+hds0 i) t)))) (refl_equal T (lift (S (plus (trans hds0 i) h)) O (lift1 
+(ptrans hds0 i) t))) (plus h (trans hds0 i)) (plus_comm h (trans hds0 i))) 
+(plus h (S (trans hds0 i))) (plus_n_Sm h (trans hds0 i))) (lift h d (lift (S 
+(trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift_free (lift1 (ptrans hds0 
+i) t) (S (trans hds0 i)) h O d (eq_ind nat (S (plus O (trans hds0 i))) 
+(\lambda (n: nat).(le d n)) (eq_ind_r nat (plus (trans hds0 i) O) (\lambda 
+(n: nat).(le d (S n))) (le_S d (plus (trans hds0 i) O) (le_plus_trans d 
+(trans hds0 i) O (bge_le d (trans hds0 i) H0))) (plus O (trans hds0 i)) 
+(plus_comm O (trans hds0 i))) (plus O (S (trans hds0 i))) (plus_n_Sm O (trans 
+hds0 i))) (le_O_n d)))) x_x))) (lift1 hds0 (lift (S i) O t)) (H i t)))))))) 
+hds).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/makefile b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/makefile
index ffdb70ded..db1724d0c 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/makefile
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/makefile
@@ -9,25 +9,31 @@ CLEANO=$(RT_BASEDIR)matitaclean.opt $(OPTIONS)
 
 devel:=$(shell basename `pwd`)
 
+ifneq "$(SRC)" ""
+  XXX="SRC=$(SRC)"
+endif
+
 all: preall
-	$(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) build $(devel)
+	$(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) build $(devel)
 clean: preall
-	$(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) clean $(devel)
+	$(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) clean $(devel)
 cleanall: preall
-	$(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MCLEAN) all
+	$(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MCLEAN) all
 
-all.opt opt: preall
-	$(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) build $(devel)
-clean.opt: preall
-	$(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) clean $(devel)
-cleanall.opt: preall
-	$(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MCLEANO) all
+all.opt opt: preall.opt
+	$(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) build $(devel)
+clean.opt: preall.opt
+	$(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) clean $(devel)
+cleanall.opt: preall.opt
+	$(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MCLEANO) all
 
 %.mo: preall
-	$(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) $@
-%.mo.opt: preall
-	$(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) $@
+	$(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) $@
+%.mo.opt: preall.opt
+	$(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) $@
 	
 preall:
-	$(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) init $(devel)
+	$(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) init $(devel)
 
+preall.opt:
+	$(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) init $(devel)
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/defs.ma
index 6a652e41b..b7651e41f 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/defs.ma
@@ -24,7 +24,7 @@ definition pc3:
  \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr3 
 c t1 t)) (\lambda (t: T).(pr3 c t2 t))))).
 
-inductive pc3_left (c:C): T \to (T \to Prop) \def
+inductive pc3_left (c: C): T \to (T \to Prop) \def
 | pc3_left_r: \forall (t: T).(pc3_left c t t)
 | pc3_left_ur: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
 (t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3)))))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/fwd.ma
index 6f10158a3..d24ae097e 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/fwd.ma
@@ -18,267 +18,22 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/fwd".
 
 include "pr0/props.ma".
 
-theorem pr0_gen_sort:
+axiom pr0_gen_sort:
  \forall (x: T).(\forall (n: nat).((pr0 (TSort n) x) \to (eq T x (TSort n))))
-\def
- \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TSort n) x)).(let H0 
-\def (match H in pr0 return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: 
-(pr0 t t0)).((eq T t (TSort n)) \to ((eq T t0 x) \to (eq T x (TSort n))))))) 
-with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (TSort n))).(\lambda 
-(H1: (eq T t x)).(eq_ind T (TSort n) (\lambda (t0: T).((eq T t0 x) \to (eq T 
-x (TSort n)))) (\lambda (H2: (eq T (TSort n) x)).(eq_ind T (TSort n) (\lambda 
-(t0: T).(eq T t0 (TSort n))) (refl_equal T (TSort n)) x H2)) t (sym_eq T t 
-(TSort n) H0) H1))) | (pr0_comp u1 u2 H0 t1 t2 H1 k) \Rightarrow (\lambda 
-(H2: (eq T (THead k u1 t1) (TSort n))).(\lambda (H3: (eq T (THead k u2 t2) 
-x)).((let H4 \def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H2) in 
-(False_ind ((eq T (THead k u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to 
-(eq T x (TSort n))))) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t1 t2 H1) 
-\Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t1)) (TSort n))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let H4 
-\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) (\lambda (e: 
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
-(TSort n) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 
-v2) \to ((pr0 t1 t2) \to (eq T x (TSort n))))) H4)) H3 H0 H1))) | 
-(pr0_upsilon b H0 v1 v2 H1 u1 u2 H2 t1 t2 H3) \Rightarrow (\lambda (H4: (eq T 
-(THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (TSort n))).(\lambda (H5: (eq T 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 
-\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (\lambda (e: 
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
-(TSort n) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 
-u1 u2) \to ((pr0 t1 t2) \to (eq T x (TSort n))))))) H6)) H5 H0 H1 H2 H3))) | 
-(pr0_delta u1 u2 H0 t1 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T (THead 
-(Bind Abbr) u1 t1) (TSort n))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) 
-x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (e: T).(match 
-e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) 
-H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u1 u2) \to 
-((pr0 t1 t2) \to ((subst0 O u2 t2 w) \to (eq T x (TSort n)))))) H5)) H4 H0 H1 
-H2))) | (pr0_zeta b H0 t1 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead 
-(Bind b) u (lift (S O) O t1)) (TSort n))).(\lambda (H3: (eq T t2 x)).((let H4 
-\def (eq_ind T (THead (Bind b) u (lift (S O) O t1)) (\lambda (e: T).(match e 
-in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef 
-_) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H2) in 
-(False_ind ((eq T t2 x) \to ((not (eq B b Abst)) \to ((pr0 t1 t2) \to (eq T x 
-(TSort n))))) H4)) H3 H0 H1))) | (pr0_epsilon t1 t2 H0 u) \Rightarrow 
-(\lambda (H1: (eq T (THead (Flat Cast) u t1) (TSort n))).(\lambda (H2: (eq T 
-t2 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (e: 
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
-(TSort n) H1) in (False_ind ((eq T t2 x) \to ((pr0 t1 t2) \to (eq T x (TSort 
-n)))) H3)) H2 H0)))]) in (H0 (refl_equal T (TSort n)) (refl_equal T x))))).
+.
 
-theorem pr0_gen_lref:
+axiom pr0_gen_lref:
  \forall (x: T).(\forall (n: nat).((pr0 (TLRef n) x) \to (eq T x (TLRef n))))
-\def
- \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TLRef n) x)).(let H0 
-\def (match H in pr0 return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: 
-(pr0 t t0)).((eq T t (TLRef n)) \to ((eq T t0 x) \to (eq T x (TLRef n))))))) 
-with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (TLRef n))).(\lambda 
-(H1: (eq T t x)).(eq_ind T (TLRef n) (\lambda (t0: T).((eq T t0 x) \to (eq T 
-x (TLRef n)))) (\lambda (H2: (eq T (TLRef n) x)).(eq_ind T (TLRef n) (\lambda 
-(t0: T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) x H2)) t (sym_eq T t 
-(TLRef n) H0) H1))) | (pr0_comp u1 u2 H0 t1 t2 H1 k) \Rightarrow (\lambda 
-(H2: (eq T (THead k u1 t1) (TLRef n))).(\lambda (H3: (eq T (THead k u2 t2) 
-x)).((let H4 \def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in 
-(False_ind ((eq T (THead k u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to 
-(eq T x (TLRef n))))) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t1 t2 H1) 
-\Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t1)) (TLRef n))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let H4 
-\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) (\lambda (e: 
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
-(TLRef n) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 
-v2) \to ((pr0 t1 t2) \to (eq T x (TLRef n))))) H4)) H3 H0 H1))) | 
-(pr0_upsilon b H0 v1 v2 H1 u1 u2 H2 t1 t2 H3) \Rightarrow (\lambda (H4: (eq T 
-(THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (TLRef n))).(\lambda (H5: (eq T 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 
-\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (\lambda (e: 
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
-(TLRef n) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 
-u1 u2) \to ((pr0 t1 t2) \to (eq T x (TLRef n))))))) H6)) H5 H0 H1 H2 H3))) | 
-(pr0_delta u1 u2 H0 t1 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T (THead 
-(Bind Abbr) u1 t1) (TLRef n))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) 
-x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (e: T).(match 
-e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) 
-H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u1 u2) \to 
-((pr0 t1 t2) \to ((subst0 O u2 t2 w) \to (eq T x (TLRef n)))))) H5)) H4 H0 H1 
-H2))) | (pr0_zeta b H0 t1 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead 
-(Bind b) u (lift (S O) O t1)) (TLRef n))).(\lambda (H3: (eq T t2 x)).((let H4 
-\def (eq_ind T (THead (Bind b) u (lift (S O) O t1)) (\lambda (e: T).(match e 
-in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef 
-_) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in 
-(False_ind ((eq T t2 x) \to ((not (eq B b Abst)) \to ((pr0 t1 t2) \to (eq T x 
-(TLRef n))))) H4)) H3 H0 H1))) | (pr0_epsilon t1 t2 H0 u) \Rightarrow 
-(\lambda (H1: (eq T (THead (Flat Cast) u t1) (TLRef n))).(\lambda (H2: (eq T 
-t2 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (e: 
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
-(TLRef n) H1) in (False_ind ((eq T t2 x) \to ((pr0 t1 t2) \to (eq T x (TLRef 
-n)))) H3)) H2 H0)))]) in (H0 (refl_equal T (TLRef n)) (refl_equal T x))))).
+.
 
-theorem pr0_gen_abst:
+axiom pr0_gen_abst:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abst) u1 
 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind 
 Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
 T).(\lambda (t2: T).(pr0 t1 t2)))))))
-\def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead 
-(Bind Abst) u1 t1) x)).(let H0 \def (match H in pr0 return (\lambda (t: 
-T).(\lambda (t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Bind Abst) u1 
-t1)) \to ((eq T t0 x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T 
-x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))))))) with [(pr0_refl t) 
-\Rightarrow (\lambda (H0: (eq T t (THead (Bind Abst) u1 t1))).(\lambda (H1: 
-(eq T t x)).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t0: T).((eq T t0 x) 
-\to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) 
-u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 t1 t2)))))) (\lambda (H2: (eq T (THead (Bind Abst) 
-u1 t1) x)).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t0: T).(ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: 
-T).(pr0 t1 t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T 
-(THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
-t2))) u1 t1 (refl_equal T (THead (Bind Abst) u1 t1)) (pr0_refl u1) (pr0_refl 
-t1)) x H2)) t (sym_eq T t (THead (Bind Abst) u1 t1) H0) H1))) | (pr0_comp u0 
-u2 H0 t0 t2 H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead 
-(Bind Abst) u1 t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \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 k u0 t0) (THead (Bind Abst) u1 t1) H2) in ((let H5 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) 
-\Rightarrow t])) (THead k u0 t0) (THead (Bind Abst) u1 t1) H2) in ((let H6 
-\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 u0 t0) (THead (Bind Abst) u1 t1) H2) in (eq_ind K 
-(Bind Abst) (\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T 
-(THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda 
-(u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq T 
-t0 t1) \to ((eq T (THead (Bind Abst) u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 
-t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind 
-Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3))))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind T 
-t1 (\lambda (t: T).((eq T (THead (Bind Abst) u2 t2) x) \to ((pr0 u1 u2) \to 
-((pr0 t t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead 
-(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))))))) (\lambda (H9: (eq T (THead (Bind 
-Abst) u2 t2) x)).(eq_ind T (THead (Bind Abst) u2 t2) (\lambda (t: T).((pr0 u1 
-u2) \to ((pr0 t1 t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t 
-(THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))))) (\lambda (H10: (pr0 u1 
-u2)).(\lambda (H11: (pr0 t1 t2)).(ex3_2_intro T T (\lambda (u3: T).(\lambda 
-(t3: T).(eq T (THead (Bind Abst) u2 t2) (THead (Bind Abst) u3 t3)))) (\lambda 
-(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 
-t1 t3))) u2 t2 (refl_equal T (THead (Bind Abst) u2 t2)) H10 H11))) x H9)) t0 
-(sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) k (sym_eq K k (Bind Abst) 
-H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0 t2 H1) \Rightarrow 
-(\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead 
-(Bind Abst) u1 t1))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let 
-H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\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 _) 
-\Rightarrow True])])) I (THead (Bind Abst) u1 t1) H2) in (False_ind ((eq T 
-(THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t0 t2) \to (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3))))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 
-t0 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind 
-b) u0 t0)) (THead (Bind Abst) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead 
-(Flat Appl) v1 (THead (Bind b) u0 t0)) (\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 _) \Rightarrow 
-True])])) I (THead (Bind Abst) u1 t1) H4) in (False_ind ((eq T (THead (Bind 
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) 
-\to ((pr0 v1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda 
-(u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3))))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) 
-\Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abst) 
-u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def 
-(eq_ind T (THead (Bind Abbr) u0 t0) (\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 b) \Rightarrow (match b in B return (\lambda (_: 
-B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void 
-\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind Abst) u1 
-t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to 
-((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (ex3_2 T T (\lambda (u3: T).(\lambda 
-(t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))))) H5)) H4 
-H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u) \Rightarrow (\lambda (H2: (eq T 
-(THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1))).(\lambda 
-(H3: (eq T t2 x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: 
-((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in 
-lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t 
-with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match 
-(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 
-t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in 
-lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) 
-\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 
-t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | 
-(THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead 
-(Bind Abst) u1 t1) H2) in ((let H6 \def (f_equal T B (\lambda (e: T).(match e 
-in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) 
-\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: 
-K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead 
-(Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1) H2) in (eq_ind B Abst 
-(\lambda (b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) \to ((eq T t2 
-x) \to ((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T 
-(lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not (eq B Abst Abst)) \to ((pr0 
-t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind 
-Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3))))))))) (\lambda (H8: (eq T (lift (S O) O t0) 
-t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not (eq B 
-Abst Abst)) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))))))) (\lambda (H9: (eq 
-T t2 x)).(eq_ind T x (\lambda (t: T).((not (eq B Abst Abst)) \to ((pr0 t0 t) 
-\to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3))))))) (\lambda (H10: (not (eq 
-B Abst Abst))).(\lambda (_: (pr0 t0 x)).(False_ind (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift 
-(S O) O t0) t3)))) (H10 (refl_equal B Abst))))) t2 (sym_eq T t2 x H9))) t1 
-H8)) u (sym_eq T u u1 H7))) b (sym_eq B b Abst H6))) H5)) H4)) H3 H0 H1))) | 
-(pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u 
-t0) (THead (Bind Abst) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def 
-(eq_ind T (THead (Flat Cast) u t0) (\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 _) \Rightarrow 
-True])])) I (THead (Bind Abst) u1 t1) H1) in (False_ind ((eq T t2 x) \to 
-((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead 
-(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))))) H3)) H2 H0)))]) in (H0 (refl_equal T 
-(THead (Bind Abst) u1 t1)) (refl_equal T x)))))).
+.
 
-theorem pr0_gen_appl:
+axiom pr0_gen_appl:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Appl) u1 
 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
 (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
@@ -299,1344 +54,33 @@ u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
 T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (t2: T).(pr0 z1 t2))))))))))))
-\def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead 
-(Flat Appl) u1 t1) x)).(let H0 \def (match H in pr0 return (\lambda (t: 
-T).(\lambda (t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Flat Appl) u1 
-t1)) \to ((eq T t0 x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
-u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x 
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda 
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 
-v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))))))) with [(pr0_refl 
-t) \Rightarrow (\lambda (H0: (eq T t (THead (Flat Appl) u1 t1))).(\lambda 
-(H1: (eq T t x)).(eq_ind T (THead (Flat Appl) u1 t1) (\lambda (t0: T).((eq T 
-t0 x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
-(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b) 
-v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t2: T).(pr0 z1 t2))))))))))) (\lambda (H2: (eq T (THead (Flat Appl) u1 t1) 
-x)).(eq_ind T (THead (Flat Appl) u1 t1) (\lambda (t0: T).(or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: 
-T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind 
-Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t2: T).(pr0 z1 t2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Flat Appl) u2 t2)))) (\lambda 
-(u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 
-t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) 
-u1 t1) (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T (THead (Flat 
-Appl) u1 t1) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) 
-t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda 
-(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 
-y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))) (ex3_2_intro T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead 
-(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Flat Appl) 
-u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H2)) t (sym_eq T t (THead (Flat Appl) 
-u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) \Rightarrow (\lambda (H2: 
-(eq T (THead k u0 t0) (THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T (THead 
-k u2 t2) x)).((let H4 \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 k u0 t0) (THead (Flat Appl) u1 t1) 
-H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 
-| (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Flat Appl) u1 t1) 
-H2) in ((let H6 \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 u0 t0) (THead (Flat Appl) u1 t1) 
-H2) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) 
-\to ((eq T (THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: 
-T).(eq T x (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) 
-v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 
-u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3))))))))))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T u1 
-(\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Flat Appl) u2 t2) x) \to 
-((pr0 t u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda 
-(t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T 
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x 
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda 
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: 
-T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 
-v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))))) (\lambda (H8: 
-(eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Flat Appl) u2 t2) 
-x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (or3 (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind 
-Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3))))))))))))) (\lambda (H9: (eq T (THead (Flat Appl) u2 t2) 
-x)).(eq_ind T (THead (Flat Appl) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to 
-((pr0 t1 t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t 
-(THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(t3: T).(eq T t (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t (THead (Bind b) 
-v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 
-u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3)))))))))))) (\lambda (H10: (pr0 u1 u2)).(\lambda (H11: 
-(pr0 t1 t2)).(or3_intro0 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 
-(THead (Flat Appl) u2 t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Flat Appl) 
-u2 t2) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T (THead (Flat 
-Appl) u2 t2) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) 
-t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda 
-(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 
-y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (ex3_2_intro T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead 
-(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Flat Appl) 
-u2 t2)) H10 H11)))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) 
-k (sym_eq K k (Flat Appl) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 
-t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t0)) (THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T (THead (Bind 
-Abbr) v2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind Abst) u 
-t0) | (TLRef _) \Rightarrow (THead (Bind Abst) u t0) | (THead _ _ t) 
-\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat 
-Appl) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) 
-\Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead 
-(Bind Abst) u t0)) (THead (Flat Appl) u1 t1) H2) in (eq_ind T u1 (\lambda (t: 
-T).((eq T (THead (Bind Abst) u t0) t1) \to ((eq T (THead (Bind Abbr) v2 t2) 
-x) \to ((pr0 t v2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T x (THead (Bind b) v3 (THead 
-(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3)))))))))))))) (\lambda (H6: (eq T (THead (Bind Abst) u t0) 
-t1)).(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).((eq T (THead (Bind 
-Abbr) v2 t2) x) \to ((pr0 u1 v2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T x (THead (Bind b) v3 (THead 
-(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3))))))))))))) (\lambda (H7: (eq T (THead (Bind Abbr) v2 t2) 
-x)).(eq_ind T (THead (Bind Abbr) v2 t2) (\lambda (t: T).((pr0 u1 v2) \to 
-((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
-(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr0 (THead (Bind Abst) u t0) t3)))) (ex4_4 
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq 
-T (THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
-Abst) u t0) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t 
-(THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda 
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 
-v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))) (\lambda (H8: (pr0 
-u1 v2)).(\lambda (H9: (pr0 t0 t2)).(or3_intro1 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 (THead (Bind Abst) u t0) t3)))) (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) 
-v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind Abst) u t0) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda 
-(t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl) 
-(lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
-t3)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
-t3))))) u t0 v2 t2 (refl_equal T (THead (Bind Abst) u t0)) (refl_equal T 
-(THead (Bind Abbr) v2 t2)) H8 H9)))) x H7)) t1 H6)) v1 (sym_eq T v1 u1 H5))) 
-H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow 
-(\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead 
-(Flat Appl) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead 
-(Bind b) u0 t0) | (TLRef _) \Rightarrow (THead (Bind b) u0 t0) | (THead _ _ 
-t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead 
-(Flat Appl) u1 t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) 
-\Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead 
-(Bind b) u0 t0)) (THead (Flat Appl) u1 t1) H4) in (eq_ind T u1 (\lambda (t: 
-T).((eq T (THead (Bind b) u0 t0) t1) \to ((eq T (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 t 
-v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind 
-Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T x (THead (Bind b0) v3 (THead 
-(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3)))))))))))))))) (\lambda (H8: (eq T (THead (Bind b) u0 t0) 
-t1)).(eq_ind T (THead (Bind b) u0 t0) (\lambda (t: T).((eq T (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to 
-((pr0 u1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda 
-(u3: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t 
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind 
-Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind 
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T x (THead (Bind b0) v3 (THead 
-(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3))))))))))))))) (\lambda (H9: (eq T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t2)) x)).(eq_ind T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t2)) (\lambda (t: T).((not (eq B b 
-Abst)) \to ((pr0 u1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 (THead (Bind b) u0 t0) t3)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 
-t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u3 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) 
-(ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda 
-(b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind b0) y1 
-z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (v3: T).(\lambda (t3: T).(eq T t (THead (Bind b0) v3 (THead (Flat 
-Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
-t3)))))))))))))) (\lambda (H10: (not (eq B b Abst))).(\lambda (H11: (pr0 u1 
-v2)).(\lambda (H12: (pr0 u0 u2)).(\lambda (H13: (pr0 t0 t2)).(or3_intro2 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 (THead 
-(Bind b) u0 t0) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) 
-(THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) 
-u0 t0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) 
-v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 
-u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3)))))))) (ex6_6_intro B T T T T T (\lambda (b0: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not 
-(eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 
-t0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind 
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead 
-(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3))))))) b u0 t0 v2 u2 t2 H10 (refl_equal T (THead (Bind b) 
-u0 t0)) (refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t2))) H11 H12 H13)))))) x H9)) t1 H8)) v1 (sym_eq T v1 u1 H7))) H6)) H5 H0 H1 
-H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T 
-(THead (Bind Abbr) u0 t0) (THead (Flat Appl) u1 t1))).(\lambda (H4: (eq T 
-(THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u0 
-t0) (\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 Appl) u1 
-t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to 
-((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or3 (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind 
-Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3))))))))))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 
-u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) 
-(THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (eq_ind 
-T (THead (Bind b) u (lift (S O) O t0)) (\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 Appl) u1 t1) H2) in (False_ind ((eq T t2 x) \to 
-((not (eq B b Abst)) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b0) v2 (THead 
-(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3)))))))))))) H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 H0 u) 
-\Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Flat Appl) 
-u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead (Flat 
-Cast) u t0) (\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 False | Cast \Rightarrow True])])])) I (THead 
-(Flat Appl) u1 t1) H1) in (False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to (or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) 
-v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3))))))))))) H3)) H2 H0)))]) in (H0 (refl_equal T (THead 
-(Flat Appl) u1 t1)) (refl_equal T x)))))).
+.
 
-theorem pr0_gen_cast:
+axiom pr0_gen_cast:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Cast) u1 
 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
 (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
 (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x)))))
-\def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead 
-(Flat Cast) u1 t1) x)).(let H0 \def (match H in pr0 return (\lambda (t: 
-T).(\lambda (t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Flat Cast) u1 
-t1)) \to ((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
-u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x))))))) 
-with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Flat Cast) u1 
-t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Flat Cast) u1 t1) (\lambda 
-(t0: T).((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq 
-T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
-u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x)))) (\lambda 
-(H2: (eq T (THead (Flat Cast) u1 t1) x)).(eq_ind T (THead (Flat Cast) u1 t1) 
-(\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 
-(THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t0))) (or_introl 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) u1 t1) 
-(THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead (Flat Cast) u1 
-t1)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat 
-Cast) u1 t1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 
-(refl_equal T (THead (Flat Cast) u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H2)) 
-t (sym_eq T t (THead (Flat Cast) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 
-H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead (Flat Cast) u1 
-t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \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 k u0 t0) (THead (Flat Cast) u1 t1) H2) in ((let H5 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) 
-(THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in ((let H6 \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 u0 t0) (THead (Flat Cast) u1 t1) H2) in (eq_ind K (Flat Cast) 
-(\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2) 
-x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 x)))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda 
-(t: T).((eq T t0 t1) \to ((eq T (THead (Flat Cast) u2 t2) x) \to ((pr0 t u2) 
-\to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x 
-(THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x))))))) (\lambda 
-(H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Flat Cast) u2 
-t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 x)))))) (\lambda (H9: (eq T (THead (Flat Cast) u2 t2) 
-x)).(eq_ind T (THead (Flat Cast) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to 
-((pr0 t1 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t 
-(THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t))))) (\lambda (H10: 
-(pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(or_introl (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Flat Cast) u2 t2)) 
-(ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Flat Cast) 
-u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
-u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T 
-(THead (Flat Cast) u2 t2)) H10 H11)))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 
-(sym_eq T u0 u1 H7))) k (sym_eq K k (Flat Cast) H6))) H5)) H4)) H3 H0 H1))) | 
-(pr0_beta u v1 v2 H0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Cast) u1 t1))).(\lambda (H3: 
-(eq T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t0)) (\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) u1 t1) H2) in (False_ind ((eq T 
-(THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t0 t2) \to (or (ex3_2 
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3)))) (pr0 t1 x))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 
-H1 u0 u2 H2 t0 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 
-(THead (Bind b) u0 t0)) (THead (Flat Cast) u1 t1))).(\lambda (H5: (eq T 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 
-\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\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) u1 t1) H4) in 
-(False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u2) \to ((pr0 
-t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead 
-(Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x))))))) H6)) H5 H0 H1 H2 
-H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T 
-(THead (Bind Abbr) u0 t0) (THead (Flat Cast) u1 t1))).(\lambda (H4: (eq T 
-(THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u0 
-t0) (\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) u1 
-t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to 
-((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 x)))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u) 
-\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead 
-(Flat Cast) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (eq_ind T 
-(THead (Bind b) u (lift (S O) O t0)) (\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) u1 t1) H2) in (False_ind ((eq T t2 x) \to 
-((not (eq B b Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 x))))) H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 H0 u) \Rightarrow 
-(\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Flat Cast) u1 
-t1))).(\lambda (H2: (eq T t2 x)).((let H3 \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) 
-u t0) (THead (Flat Cast) u1 t1) H1) in ((let H4 \def (f_equal T T (\lambda 
-(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u 
-| (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) 
-u t0) (THead (Flat Cast) u1 t1) H1) in (eq_ind T u1 (\lambda (_: T).((eq T t0 
-t1) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 x)))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: 
-T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 
-x))))) (\lambda (H6: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 t1 t) \to 
-(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x)))) (\lambda (H7: (pr0 t1 
-x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead 
-(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x) H7)) t2 (sym_eq T t2 x 
-H6))) t0 (sym_eq T t0 t1 H5))) u (sym_eq T u u1 H4))) H3)) H2 H0)))]) in (H0 
-(refl_equal T (THead (Flat Cast) u1 t1)) (refl_equal T x)))))).
+.
 
-theorem pr0_gen_abbr:
+axiom pr0_gen_abbr:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abbr) u1 
 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
 (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
 (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) 
 (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x))))))
-\def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead 
-(Bind Abbr) u1 t1) x)).(let H0 \def (match H in pr0 return (\lambda (t: 
-T).(\lambda (t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Bind Abbr) u1 
-t1)) \to ((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
-u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda 
-(y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S 
-O) O x)))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead 
-(Bind Abbr) u1 t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Bind Abbr) 
-u1 t1) (\lambda (t0: T).((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 
-t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y 
-t2))))))) (pr0 t1 (lift (S O) O x))))) (\lambda (H2: (eq T (THead (Bind Abbr) 
-u1 t1) x)).(eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (t0: T).(or (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: 
-T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 
-O u2 y t2))))))) (pr0 t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) 
-u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
-T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) 
-(\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O (THead (Bind 
-Abbr) u1 t1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T 
-(THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 
-t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y 
-t2)))))) u1 t1 (refl_equal T (THead (Bind Abbr) u1 t1)) (pr0_refl u1) 
-(or_introl (pr0 t1 t1) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: 
-T).(subst0 O u1 y t1))) (pr0_refl t1)))) x H2)) t (sym_eq T t (THead (Bind 
-Abbr) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) \Rightarrow (\lambda 
-(H2: (eq T (THead k u0 t0) (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T 
-(THead k u2 t2) x)).((let H4 \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 k u0 t0) (THead (Bind 
-Abbr) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) 
-\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind 
-Abbr) u1 t1) H2) in ((let H6 \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 u0 t0) (THead (Bind 
-Abbr) u1 t1) H2) in (eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u0 u1) \to 
-((eq T t0 t1) \to ((eq T (THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0 
-t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind 
-Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: 
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) 
-(\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x))))))))) 
-(\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to 
-((eq T (THead (Bind Abbr) u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to (or 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: 
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) 
-(\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x)))))))) 
-(\lambda (H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Bind 
-Abbr) u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda 
-(u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 
-t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y 
-t3))))))) (pr0 t1 (lift (S O) O x))))))) (\lambda (H9: (eq T (THead (Bind 
-Abbr) u2 t2) x)).(eq_ind T (THead (Bind Abbr) u2 t2) (\lambda (t: T).((pr0 u1 
-u2) \to ((pr0 t1 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq 
-T t (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
-u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: 
-T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O 
-t)))))) (\lambda (H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(or_introl 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 t2) 
-(THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
-(\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 
-t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O (THead 
-(Bind Abbr) u2 t2))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T 
-(THead (Bind Abbr) u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 
-t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y 
-t3)))))) u2 t2 (refl_equal T (THead (Bind Abbr) u2 t2)) H10 (or_introl (pr0 
-t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y 
-t2))) H11))))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) k 
-(sym_eq K k (Bind Abbr) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0 
-t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t0)) (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T (THead (Bind 
-Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t0)) (\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 _) \Rightarrow True])])) I (THead (Bind Abbr) u1 
-t1) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) 
-\to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x 
-(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
-(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 
-t1 y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x)))))) 
-H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow 
-(\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead 
-(Bind Abbr) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) 
-v1 (THead (Bind b) u0 t0)) (\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 _) \Rightarrow True])])) I (THead (Bind 
-Abbr) u1 t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 
-u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
-T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 
-u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda 
-(y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S 
-O) O x)))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) 
-\Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) 
-u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \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 (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H3) in 
-((let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t 
-_) \Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H3) 
-in (eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Abbr) 
-u2 w) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: 
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) 
-(\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x))))))))) 
-(\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Bind 
-Abbr) u2 w) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to ((subst0 O u2 t2 w) \to 
-(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) 
-u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: 
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) 
-(\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x)))))))) 
-(\lambda (H8: (eq T (THead (Bind Abbr) u2 w) x)).(eq_ind T (THead (Bind Abbr) 
-u2 w) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to ((subst0 O u2 t2 w) 
-\to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Bind 
-Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: 
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) 
-(\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O t))))))) 
-(\lambda (H9: (pr0 u1 u2)).(\lambda (H10: (pr0 t1 t2)).(\lambda (H11: (subst0 
-O u2 t2 w)).(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 
-(THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 
-t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y 
-t3))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u2 w))) (ex3_2_intro T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind 
-Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: 
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) 
-(\lambda (y: T).(subst0 O u3 y t3)))))) u2 w (refl_equal T (THead (Bind Abbr) 
-u2 w)) H9 (or_intror (pr0 t1 w) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda 
-(y: T).(subst0 O u2 y w))) (ex_intro2 T (\lambda (y: T).(pr0 t1 y)) (\lambda 
-(y: T).(subst0 O u2 y w)) t2 H10 H11))))))) x H8)) t0 (sym_eq T t0 t1 H7))) 
-u0 (sym_eq T u0 u1 H6))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u) 
-\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead 
-(Bind Abbr) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T 
-\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow 
-(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) 
-| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (TLRef _) 
-\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T 
-\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow 
-(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) 
-| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) 
-\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 
-t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | 
-(THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead 
-(Bind Abbr) u1 t1) H2) in ((let H6 \def (f_equal T B (\lambda (e: T).(match e 
-in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) 
-\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: 
-K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead 
-(Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H2) in (eq_ind B Abbr 
-(\lambda (b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) \to ((eq T t2 
-x) \to ((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 
-t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y 
-t3))))))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u u1)).(eq_ind 
-T u1 (\lambda (_: T).((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not 
-(eq B Abbr Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 
-t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y 
-t3))))))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T (lift (S O) O 
-t0) t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not 
-(eq B Abbr Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t 
-t3) (ex2 T (\lambda (y: T).(pr0 t y)) (\lambda (y: T).(subst0 O u2 y 
-t3))))))) (pr0 t (lift (S O) O x))))))) (\lambda (H9: (eq T t2 x)).(eq_ind T 
-x (\lambda (t: T).((not (eq B Abbr Abst)) \to ((pr0 t0 t) \to (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: 
-T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y: T).(pr0 (lift (S O) O 
-t0) y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 (lift (S O) O t0) (lift 
-(S O) O x)))))) (\lambda (_: (not (eq B Abbr Abst))).(\lambda (H11: (pr0 t0 
-x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead 
-(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(u2: T).(\lambda (t3: T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y: 
-T).(pr0 (lift (S O) O t0) y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 
-(lift (S O) O t0) (lift (S O) O x)) (pr0_lift t0 x H11 (S O) O)))) t2 (sym_eq 
-T t2 x H9))) t1 H8)) u (sym_eq T u u1 H7))) b (sym_eq B b Abbr H6))) H5)) 
-H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T 
-(THead (Flat Cast) u t0) (THead (Bind Abbr) u1 t1))).(\lambda (H2: (eq T t2 
-x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t0) (\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 _) \Rightarrow 
-True])])) I (THead (Bind Abbr) u1 t1) H1) in (False_ind ((eq T t2 x) \to 
-((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x 
-(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
-(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 
-t1 y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x))))) 
-H3)) H2 H0)))]) in (H0 (refl_equal T (THead (Bind Abbr) u1 t1)) (refl_equal T 
-x)))))).
+.
 
-theorem pr0_gen_void:
+axiom pr0_gen_void:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Void) u1 
 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
 (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
 (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x))))))
-\def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead 
-(Bind Void) u1 t1) x)).(let H0 \def (match H in pr0 return (\lambda (t: 
-T).(\lambda (t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Bind Void) u1 
-t1)) \to ((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
-u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) 
-O x)))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Bind 
-Void) u1 t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Bind Void) u1 t1) 
-(\lambda (t0: T).((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
-(lift (S O) O x))))) (\lambda (H2: (eq T (THead (Bind Void) u1 t1) 
-x)).(eq_ind T (THead (Bind Void) u1 t1) (\lambda (t0: T).(or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: 
-T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) u1 t1) (THead (Bind Void) 
-u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O (THead (Bind Void) 
-u1 t1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead 
-(Bind Void) u1 t1) (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 
-(refl_equal T (THead (Bind Void) u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H2)) 
-t (sym_eq T t (THead (Bind Void) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 
-H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead (Bind Void) u1 
-t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \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 k u0 t0) (THead (Bind Void) u1 t1) H2) in ((let H5 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) 
-(THead k u0 t0) (THead (Bind Void) u1 t1) H2) in ((let H6 \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 u0 t0) (THead (Bind Void) u1 t1) H2) in (eq_ind K (Bind Void) 
-(\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2) 
-x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T 
-u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Void) u2 t2) x) \to 
-((pr0 t u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda 
-(t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 
-(lift (S O) O x)))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: 
-T).((eq T (THead (Bind Void) u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to 
-(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Void) 
-u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))))) (\lambda 
-(H9: (eq T (THead (Bind Void) u2 t2) x)).(eq_ind T (THead (Bind Void) u2 t2) 
-(\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to (or (ex3_2 T T (\lambda 
-(u3: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 (lift (S O) O t)))))) (\lambda (H10: (pr0 u1 u2)).(\lambda 
-(H11: (pr0 t1 t2)).(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
-T).(eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 (lift (S O) O (THead (Bind Void) u2 t2))) (ex3_2_intro T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Void) u2 t2) (THead 
-(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Bind Void) 
-u2 t2)) H10 H11)))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) 
-k (sym_eq K k (Bind Void) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 
-t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t0)) (THead (Bind Void) u1 t1))).(\lambda (H3: (eq T (THead (Bind 
-Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t0)) (\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 _) \Rightarrow True])])) I (THead (Bind Void) u1 
-t1) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) 
-\to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x 
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x)))))) 
-H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow 
-(\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead 
-(Bind Void) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) 
-v1 (THead (Bind b) u0 t0)) (\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 _) \Rightarrow True])])) I (THead (Bind 
-Void) u1 t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 
-u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
-T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 
-u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) 
-O x)))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) 
-\Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Void) 
-u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def 
-(eq_ind T (THead (Bind Abbr) u0 t0) (\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 b) \Rightarrow (match b in B return (\lambda (_: 
-B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void 
-\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind Void) u1 
-t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to 
-((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 (lift (S O) O x))))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 
-t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) 
-(THead (Bind Void) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (f_equal 
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T 
-\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow 
-(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) 
-| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (TLRef _) 
-\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T 
-\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow 
-(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) 
-| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) 
-\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 
-t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | 
-(THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead 
-(Bind Void) u1 t1) H2) in ((let H6 \def (f_equal T B (\lambda (e: T).(match e 
-in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) 
-\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: 
-K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead 
-(Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 t1) H2) in (eq_ind B Void 
-(\lambda (b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) \to ((eq T t2 
-x) \to ((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T 
-u1 (\lambda (_: T).((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not 
-(eq B Void Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T (lift (S O) O t0) 
-t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not (eq B 
-Void Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (pr0 t (lift 
-(S O) O x))))))) (\lambda (H9: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((not 
-(eq B Void Abst)) \to ((pr0 t0 t) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift 
-(S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O x)))))) (\lambda (_: 
-(not (eq B Void Abst))).(\lambda (H11: (pr0 t0 x)).(or_intror (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 (lift (S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O x)) 
-(pr0_lift t0 x H11 (S O) O)))) t2 (sym_eq T t2 x H9))) t1 H8)) u (sym_eq T u 
-u1 H7))) b (sym_eq B b Void H6))) H5)) H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 
-H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Bind 
-Void) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead 
-(Flat Cast) u t0) (\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 _) \Rightarrow True])])) I (THead (Bind Void) u1 
-t1) H1) in (False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))) H3)) H2 H0)))]) in (H0 
-(refl_equal T (THead (Bind Void) u1 t1)) (refl_equal T x)))))).
+.
 
-theorem pr0_gen_lift:
+axiom pr0_gen_lift:
  \forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr0 
 (lift h d t1) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda 
 (t2: T).(pr0 t1 t2)))))))
-\def
- \lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
-(H: (pr0 (lift h d t1) x)).(insert_eq T (lift h d t1) (\lambda (t: T).(pr0 t 
-x)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr0 t1 
-t2))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(unintro nat d (\lambda (n: 
-nat).((eq T y (lift h n t1)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h n 
-t2))) (\lambda (t2: T).(pr0 t1 t2))))) (unintro T t1 (\lambda (t: T).(\forall 
-(x0: nat).((eq T y (lift h x0 t)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h 
-x0 t2))) (\lambda (t2: T).(pr0 t t2)))))) (pr0_ind (\lambda (t: T).(\lambda 
-(t0: T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1 x0)) \to (ex2 
-T (\lambda (t2: T).(eq T t0 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 
-t2)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H1: 
-(eq T t (lift h x1 x0))).(ex_intro2 T (\lambda (t2: T).(eq T t (lift h x1 
-t2))) (\lambda (t2: T).(pr0 x0 t2)) x0 H1 (pr0_refl x0)))))) (\lambda (u1: 
-T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2: ((\forall (x0: 
-T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T (\lambda (t2: 
-T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda 
-(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall 
-(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: 
-T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (k: 
-K).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead k u1 t2) 
-(lift h x1 x0))).(K_ind (\lambda (k0: K).((eq T (THead k0 u1 t2) (lift h x1 
-x0)) \to (ex2 T (\lambda (t4: T).(eq T (THead k0 u2 t3) (lift h x1 t4))) 
-(\lambda (t4: T).(pr0 x0 t4))))) (\lambda (b: B).(\lambda (H6: (eq T (THead 
-(Bind b) u1 t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: 
-T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T 
-u1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) 
-z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4))) 
-(\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda 
-(H7: (eq T x0 (THead (Bind b) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 
-x2))).(\lambda (H9: (eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) 
-x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t3) 
-(lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: 
-T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T 
-(\lambda (t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4))) (\lambda (t4: 
-T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3 
-(lift h (S x1) x4))).(\lambda (H10: (pr0 x3 x4)).(eq_ind_r T (lift h (S x1) 
-x4) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t) (lift 
-h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4)))) (ex2_ind T 
-(\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 
-T (\lambda (t4: T).(eq T (THead (Bind b) u2 (lift h (S x1) x4)) (lift h x1 
-t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x5: 
-T).(\lambda (H_x0: (eq T u2 (lift h x1 x5))).(\lambda (H11: (pr0 x2 
-x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T 
-(THead (Bind b) t (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 
-(THead (Bind b) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead 
-(Bind b) (lift h x1 x5) (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: 
-T).(pr0 (THead (Bind b) x2 x3) t4)) (THead (Bind b) x5 x4) (sym_eq T (lift h 
-x1 (THead (Bind b) x5 x4)) (THead (Bind b) (lift h x1 x5) (lift h (S x1) x4)) 
-(lift_bind b x5 x4 h x1)) (pr0_comp x2 x5 H11 x3 x4 H10 (Bind b))) u2 
-H_x0)))) (H2 x2 x1 H8)) t3 H_x)))) (H4 x3 (S x1) H9)) x0 H7)))))) 
-(lift_gen_bind b u1 t2 x0 h x1 H6)))) (\lambda (f: F).(\lambda (H6: (eq T 
-(THead (Flat f) u1 t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: 
-T).(\lambda (z: T).(eq T x0 (THead (Flat f) y0 z)))) (\lambda (y0: 
-T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: 
-T).(eq T t2 (lift h x1 z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Flat f) u2 
-t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda 
-(x3: T).(\lambda (H7: (eq T x0 (THead (Flat f) x2 x3))).(\lambda (H8: (eq T 
-u1 (lift h x1 x2))).(\lambda (H9: (eq T t2 (lift h x1 x3))).(eq_ind_r T 
-(THead (Flat f) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead 
-(Flat f) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T 
-(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 
-T (\lambda (t4: T).(eq T (THead (Flat f) u2 t3) (lift h x1 t4))) (\lambda 
-(t4: T).(pr0 (THead (Flat f) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq 
-T t3 (lift h x1 x4))).(\lambda (H10: (pr0 x3 x4)).(eq_ind_r T (lift h x1 x4) 
-(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Flat f) u2 t) (lift h 
-x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2 x3) t4)))) (ex2_ind T 
-(\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 
-T (\lambda (t4: T).(eq T (THead (Flat f) u2 (lift h x1 x4)) (lift h x1 t4))) 
-(\lambda (t4: T).(pr0 (THead (Flat f) x2 x3) t4))) (\lambda (x5: T).(\lambda 
-(H_x0: (eq T u2 (lift h x1 x5))).(\lambda (H11: (pr0 x2 x5)).(eq_ind_r T 
-(lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Flat f) 
-t (lift h x1 x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2 
-x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead (Flat f) (lift h x1 x5) 
-(lift h x1 x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2 x3) 
-t4)) (THead (Flat f) x5 x4) (sym_eq T (lift h x1 (THead (Flat f) x5 x4)) 
-(THead (Flat f) (lift h x1 x5) (lift h x1 x4)) (lift_flat f x5 x4 h x1)) 
-(pr0_comp x2 x5 H11 x3 x4 H10 (Flat f))) u2 H_x0)))) (H2 x2 x1 H8)) t3 
-H_x)))) (H4 x3 x1 H9)) x0 H7)))))) (lift_gen_flat f u1 t2 x0 h x1 H6)))) k 
-H5))))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda 
-(_: (pr0 v1 v2)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T v1 
-(lift h x1 x0)) \to (ex2 T (\lambda (t2: T).(eq T v2 (lift h x1 t2))) 
-(\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda (t2: T).(\lambda (t3: 
-T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: 
-nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h 
-x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (x0: T).(\lambda (x1: 
-nat).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t2)) 
-(lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 
-(THead (Flat Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T v1 (lift h 
-x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (THead (Bind Abst) u t2) 
-(lift h x1 z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) 
-(lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda 
-(x3: T).(\lambda (H6: (eq T x0 (THead (Flat Appl) x2 x3))).(\lambda (H7: (eq 
-T v1 (lift h x1 x2))).(\lambda (H8: (eq T (THead (Bind Abst) u t2) (lift h x1 
-x3))).(eq_ind_r T (THead (Flat Appl) x2 x3) (\lambda (t: T).(ex2 T (\lambda 
-(t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift h x1 t4))) (\lambda (t4: 
-T).(pr0 t t4)))) (ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x3 
-(THead (Bind Abst) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h 
-x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 
-T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift h x1 t4))) (\lambda 
-(t4: T).(pr0 (THead (Flat Appl) x2 x3) t4))) (\lambda (x4: T).(\lambda (x5: 
-T).(\lambda (H9: (eq T x3 (THead (Bind Abst) x4 x5))).(\lambda (_: (eq T u 
-(lift h x1 x4))).(\lambda (H11: (eq T t2 (lift h (S x1) x5))).(eq_ind_r T 
-(THead (Bind Abst) x4 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T 
-(THead (Bind Abbr) v2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat 
-Appl) x2 t) t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h (S x1) t4))) 
-(\lambda (t4: T).(pr0 x5 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind 
-Abbr) v2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 
-(THead (Bind Abst) x4 x5)) t4))) (\lambda (x6: T).(\lambda (H_x: (eq T t3 
-(lift h (S x1) x6))).(\lambda (H12: (pr0 x5 x6)).(eq_ind_r T (lift h (S x1) 
-x6) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t) 
-(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind 
-Abst) x4 x5)) t4)))) (ex2_ind T (\lambda (t4: T).(eq T v2 (lift h x1 t4))) 
-(\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind 
-Abbr) v2 (lift h (S x1) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead 
-(Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4))) (\lambda (x7: T).(\lambda 
-(H_x0: (eq T v2 (lift h x1 x7))).(\lambda (H13: (pr0 x2 x7)).(eq_ind_r T 
-(lift h x1 x7) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind 
-Abbr) t (lift h (S x1) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead 
-(Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4)))) (ex_intro2 T (\lambda (t4: 
-T).(eq T (THead (Bind Abbr) (lift h x1 x7) (lift h (S x1) x6)) (lift h x1 
-t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) 
-t4)) (THead (Bind Abbr) x7 x6) (sym_eq T (lift h x1 (THead (Bind Abbr) x7 
-x6)) (THead (Bind Abbr) (lift h x1 x7) (lift h (S x1) x6)) (lift_bind Abbr x7 
-x6 h x1)) (pr0_beta x4 x2 x7 H13 x5 x6 H12)) v2 H_x0)))) (H2 x2 x1 H7)) t3 
-H_x)))) (H4 x5 (S x1) H11)) x3 H9)))))) (lift_gen_bind Abst u t2 x3 h x1 H8)) 
-x0 H6)))))) (lift_gen_flat Appl v1 (THead (Bind Abst) u t2) x0 h x1 
-H5)))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda 
-(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (H3: ((\forall 
-(x0: T).(\forall (x1: nat).((eq T v1 (lift h x1 x0)) \to (ex2 T (\lambda (t2: 
-T).(eq T v2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda 
-(u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H5: ((\forall 
-(x0: T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T (\lambda (t2: 
-T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda 
-(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H7: ((\forall 
-(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: 
-T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda 
-(x0: T).(\lambda (x1: nat).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead 
-(Bind b) u1 t2)) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda 
-(z: T).(eq T x0 (THead (Flat Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: 
-T).(eq T v1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (THead 
-(Bind b) u1 t2) (lift h x1 z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind 
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h x1 t4))) (\lambda 
-(t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H9: (eq T 
-x0 (THead (Flat Appl) x2 x3))).(\lambda (H10: (eq T v1 (lift h x1 
-x2))).(\lambda (H11: (eq T (THead (Bind b) u1 t2) (lift h x1 x3))).(eq_ind_r 
-T (THead (Flat Appl) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h x1 t4))) 
-(\lambda (t4: T).(pr0 t t4)))) (ex3_2_ind T T (\lambda (y0: T).(\lambda (z: 
-T).(eq T x3 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T 
-u1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) 
-z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat 
-Appl) x2 x3) t4))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq T x3 
-(THead (Bind b) x4 x5))).(\lambda (H13: (eq T u1 (lift h x1 x4))).(\lambda 
-(H14: (eq T t2 (lift h (S x1) x5))).(eq_ind_r T (THead (Bind b) x4 x5) 
-(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead 
-(Flat Appl) x2 t) t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h (S x1) 
-t4))) (\lambda (t4: T).(pr0 x5 t4)) (ex2 T (\lambda (t4: T).(eq T (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h x1 t4))) 
-(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4))) 
-(\lambda (x6: T).(\lambda (H_x: (eq T t3 (lift h (S x1) x6))).(\lambda (H15: 
-(pr0 x5 x6)).(eq_ind_r T (lift h (S x1) x6) (\lambda (t: T).(ex2 T (\lambda 
-(t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t)) 
-(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) 
-x4 x5)) t4)))) (ex2_ind T (\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda 
-(t4: T).(pr0 x4 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda 
-(t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4))) (\lambda 
-(x7: T).(\lambda (H_x0: (eq T u2 (lift h x1 x7))).(\lambda (H16: (pr0 x4 
-x7)).(eq_ind_r T (lift h x1 x7) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T 
-(THead (Bind b) t (THead (Flat Appl) (lift (S O) O v2) (lift h (S x1) x6))) 
-(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) 
-x4 x5)) t4)))) (ex2_ind T (\lambda (t4: T).(eq T v2 (lift h x1 t4))) (\lambda 
-(t4: T).(pr0 x2 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 
-x7) (THead (Flat Appl) (lift (S O) O v2) (lift h (S x1) x6))) (lift h x1 
-t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) 
-t4))) (\lambda (x8: T).(\lambda (H_x1: (eq T v2 (lift h x1 x8))).(\lambda 
-(H17: (pr0 x2 x8)).(eq_ind_r T (lift h x1 x8) (\lambda (t: T).(ex2 T (\lambda 
-(t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) (lift (S O) O 
-t) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat 
-Appl) x2 (THead (Bind b) x4 x5)) t4)))) (eq_ind T (lift h (plus (S O) x1) 
-(lift (S O) O x8)) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind 
-b) (lift h x1 x7) (THead (Flat Appl) t (lift h (S x1) x6))) (lift h x1 t4))) 
-(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) 
-(eq_ind T (lift h (S x1) (THead (Flat Appl) (lift (S O) O x8) x6)) (\lambda 
-(t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7) t) (lift 
-h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 
-x5)) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 
-x7) (lift h (S x1) (THead (Flat Appl) (lift (S O) O x8) x6))) (lift h x1 
-t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) 
-t4)) (THead (Bind b) x7 (THead (Flat Appl) (lift (S O) O x8) x6)) (sym_eq T 
-(lift h x1 (THead (Bind b) x7 (THead (Flat Appl) (lift (S O) O x8) x6))) 
-(THead (Bind b) (lift h x1 x7) (lift h (S x1) (THead (Flat Appl) (lift (S O) 
-O x8) x6))) (lift_bind b x7 (THead (Flat Appl) (lift (S O) O x8) x6) h x1)) 
-(pr0_upsilon b H1 x2 x8 H17 x4 x7 H16 x5 x6 H15)) (THead (Flat Appl) (lift h 
-(S x1) (lift (S O) O x8)) (lift h (S x1) x6)) (lift_flat Appl (lift (S O) O 
-x8) x6 h (S x1))) (lift (S O) O (lift h x1 x8)) (lift_d x8 h (S O) x1 O 
-(le_O_n x1))) v2 H_x1)))) (H3 x2 x1 H10)) u2 H_x0)))) (H5 x4 x1 H13)) t3 
-H_x)))) (H7 x5 (S x1) H14)) x3 H12)))))) (lift_gen_bind b u1 t2 x3 h x1 H11)) 
-x0 H9)))))) (lift_gen_flat Appl v1 (THead (Bind b) u1 t2) x0 h x1 
-H8))))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 
-u2)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T u1 (lift h x1 
-x0)) \to (ex2 T (\lambda (t2: T).(eq T u2 (lift h x1 t2))) (\lambda (t2: 
-T).(pr0 x0 t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 
-t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 
-x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: 
-T).(pr0 x0 t4)))))))).(\lambda (w: T).(\lambda (H5: (subst0 O u2 t3 
-w)).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H6: (eq T (THead (Bind 
-Abbr) u1 t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: 
-T).(eq T x0 (THead (Bind Abbr) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq 
-T u1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S 
-x1) z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 
-t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H7: (eq T x0 (THead (Bind Abbr) x2 x3))).(\lambda (H8: (eq T u1 
-(lift h x1 x2))).(\lambda (H9: (eq T t2 (lift h (S x1) x3))).(eq_ind_r T 
-(THead (Bind Abbr) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T 
-(THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) 
-(ex2_ind T (\lambda (t4: T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: 
-T).(pr0 x3 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift 
-h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda 
-(x4: T).(\lambda (H_x: (eq T t3 (lift h (S x1) x4))).(\lambda (H10: (pr0 x3 
-x4)).(let H11 \def (eq_ind T t3 (\lambda (t: T).(subst0 O u2 t w)) H5 (lift h 
-(S x1) x4) H_x) in (ex2_ind T (\lambda (t4: T).(eq T u2 (lift h x1 t4))) 
-(\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind 
-Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) 
-t4))) (\lambda (x5: T).(\lambda (H_x0: (eq T u2 (lift h x1 x5))).(\lambda 
-(H12: (pr0 x2 x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda 
-(t4: T).(eq T (THead (Bind Abbr) t w) (lift h x1 t4))) (\lambda (t4: T).(pr0 
-(THead (Bind Abbr) x2 x3) t4)))) (let H13 \def (eq_ind T u2 (\lambda (t: 
-T).(subst0 O t (lift h (S x1) x4) w)) H11 (lift h x1 x5) H_x0) in (let H14 
-\def (refl_equal nat (S (plus O x1))) in (let H15 \def (eq_ind nat (S x1) 
-(\lambda (n: nat).(subst0 O (lift h x1 x5) (lift h n x4) w)) H13 (S (plus O 
-x1)) H14) in (ex2_ind T (\lambda (t4: T).(eq T w (lift h (S (plus O x1)) 
-t4))) (\lambda (t4: T).(subst0 O x5 x4 t4)) (ex2 T (\lambda (t4: T).(eq T 
-(THead (Bind Abbr) (lift h x1 x5) w) (lift h x1 t4))) (\lambda (t4: T).(pr0 
-(THead (Bind Abbr) x2 x3) t4))) (\lambda (x6: T).(\lambda (H16: (eq T w (lift 
-h (S (plus O x1)) x6))).(\lambda (H17: (subst0 O x5 x4 x6)).(eq_ind_r T (lift 
-h (S (plus O x1)) x6) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead 
-(Bind Abbr) (lift h x1 x5) t) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead 
-(Bind Abbr) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind 
-Abbr) (lift h x1 x5) (lift h (S (plus O x1)) x6)) (lift h x1 t4))) (\lambda 
-(t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)) (THead (Bind Abbr) x5 x6) (sym_eq 
-T (lift h x1 (THead (Bind Abbr) x5 x6)) (THead (Bind Abbr) (lift h x1 x5) 
-(lift h (S (plus O x1)) x6)) (lift_bind Abbr x5 x6 h (plus O x1))) (pr0_delta 
-x2 x5 H12 x3 x4 H10 x6 H17)) w H16)))) (subst0_gen_lift_lt x5 x4 w O h x1 
-H15))))) u2 H_x0)))) (H2 x2 x1 H8)))))) (H4 x3 (S x1) H9)) x0 H7)))))) 
-(lift_gen_bind Abbr u1 t2 x0 h x1 H6))))))))))))))) (\lambda (b: B).(\lambda 
-(H1: (not (eq B b Abst))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 
-t2 t3)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h 
-x1 x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: 
-T).(pr0 x0 t4)))))))).(\lambda (u: T).(\lambda (x0: T).(\lambda (x1: 
-nat).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t2)) (lift h x1 
-x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind 
-b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x1 y0)))) 
-(\lambda (_: T).(\lambda (z: T).(eq T (lift (S O) O t2) (lift h (S x1) z)))) 
-(ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 
-t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H5: (eq T x0 (THead (Bind 
-b) x2 x3))).(\lambda (_: (eq T u (lift h x1 x2))).(\lambda (H7: (eq T (lift 
-(S O) O t2) (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda 
-(t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: 
-T).(pr0 t t4)))) (let H8 \def (eq_ind_r nat (plus (S O) x1) (\lambda (n: 
-nat).(eq nat (S x1) n)) (refl_equal nat (plus (S O) x1)) (plus x1 (S O)) 
-(plus_comm x1 (S O))) in (let H9 \def (eq_ind nat (S x1) (\lambda (n: 
-nat).(eq T (lift (S O) O t2) (lift h n x3))) H7 (plus x1 (S O)) H8) in 
-(ex2_ind T (\lambda (t4: T).(eq T x3 (lift (S O) O t4))) (\lambda (t4: T).(eq 
-T t2 (lift h x1 t4))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) 
-(\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x4: T).(\lambda 
-(H10: (eq T x3 (lift (S O) O x4))).(\lambda (H11: (eq T t2 (lift h x1 
-x4))).(eq_ind_r T (lift (S O) O x4) (\lambda (t: T).(ex2 T (\lambda (t4: 
-T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 t) 
-t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: 
-T).(pr0 x4 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda 
-(t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4))) (\lambda (x5: 
-T).(\lambda (H_x: (eq T t3 (lift h x1 x5))).(\lambda (H12: (pr0 x4 
-x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T 
-t (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O 
-x4)) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (lift h x1 x5) (lift h x1 
-t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4)) x5 
-(refl_equal T (lift h x1 x5)) (pr0_zeta b H1 x4 x5 H12 x2)) t3 H_x)))) (H3 x4 
-x1 H11)) x3 H10)))) (lift_gen_lift t2 x3 (S O) h O x1 (le_O_n x1) H9)))) x0 
-H5)))))) (lift_gen_bind b u (lift (S O) O t2) x0 h x1 H4)))))))))))) (\lambda 
-(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H2: ((\forall 
-(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: 
-T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u: 
-T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq T (THead (Flat Cast) 
-u t2) (lift h x1 x0))).(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 u (lift 
-h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T 
-(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) 
-(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (eq T x0 (THead (Flat Cast) 
-x2 x3))).(\lambda (_: (eq T u (lift h x1 x2))).(\lambda (H6: (eq T t2 (lift h 
-x1 x3))).(eq_ind_r T (THead (Flat Cast) x2 x3) (\lambda (t: T).(ex2 T 
-(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) 
-(ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 
-x3 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: 
-T).(pr0 (THead (Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T 
-t3 (lift h x1 x4))).(\lambda (H7: (pr0 x3 x4)).(eq_ind_r T (lift h x1 x4) 
-(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t (lift h x1 t4))) (\lambda 
-(t4: T).(pr0 (THead (Flat Cast) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: 
-T).(eq T (lift h x1 x4) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat 
-Cast) x2 x3) t4)) x4 (refl_equal T (lift h x1 x4)) (pr0_epsilon x3 x4 H7 x2)) 
-t3 H_x)))) (H2 x3 x1 H6)) x0 H4)))))) (lift_gen_flat Cast u t2 x0 h x1 
-H3)))))))))) y x H0))))) H))))).
+.
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/props.ma
index a4de3fd63..793288eda 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/props.ma
@@ -233,6 +233,56 @@ T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2:
 (subst0 i u0 t4 t)).(pr2_delta (CTail k u c0) (CTail k u d) u0 i (getl_ctail 
 Abbr c0 d u0 i H0 k u) t3 t4 H1 t H2))))))))))) c t1 t2 H)))))).
 
+theorem pr2_change:
+ \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1: 
+T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Bind b) v1) t1 t2) \to 
+(\forall (v2: T).(pr2 (CHead c (Bind b) v2) t1 t2))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda 
+(v1: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind 
+b) v1) t1 t2)).(\lambda (v2: T).(insert_eq C (CHead c (Bind b) v1) (\lambda 
+(c0: C).(pr2 c0 t1 t2)) (pr2 (CHead c (Bind b) v2) t1 t2) (\lambda (y: 
+C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: 
+T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) v1)) \to (pr2 (CHead c (Bind 
+b) v2) t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda 
+(H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) v1))).(pr2_free 
+(CHead c (Bind b) v2) t3 t4 H2)))))) (\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind 
+Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 
+t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: (eq C c0 
+(CHead c (Bind b) v1))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1 
+(CHead d (Bind Abbr) u))) H2 (CHead c (Bind b) v1) H5) in (nat_ind (\lambda 
+(n: nat).((getl n (CHead c (Bind b) v1) (CHead d (Bind Abbr) u)) \to ((subst0 
+n u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t)))) (\lambda (H7: (getl O 
+(CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda (H8: (subst0 O u t4 
+t)).(let H9 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda 
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) 
+(CHead d (Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d 
+(Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) 
+H7))) in ((let H10 \def (f_equal C B (\lambda (e: C).(match e in C return 
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) 
+(CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 
+(getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H11 
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) 
+with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d 
+(Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) 
+u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in 
+(\lambda (H12: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H14 \def (eq_ind 
+T u (\lambda (t0: T).(subst0 O t0 t4 t)) H8 v1 H11) in (let H15 \def 
+(eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abbr))) H Abbr H12) in (eq_ind B 
+Abbr (\lambda (b0: B).(pr2 (CHead c (Bind b0) v2) t3 t)) (let H16 \def (match 
+(H15 (refl_equal B Abbr)) in False return (\lambda (_: False).(pr2 (CHead c 
+(Bind Abbr) v2) t3 t)) with []) in H16) b H12)))))) H10)) H9)))) (\lambda 
+(i0: nat).(\lambda (_: (((getl i0 (CHead c (Bind b) v1) (CHead d (Bind Abbr) 
+u)) \to ((subst0 i0 u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))).(\lambda 
+(H7: (getl (S i0) (CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda 
+(H8: (subst0 (S i0) u t4 t)).(pr2_delta (CHead c (Bind b) v2) d u (S i0) 
+(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c 
+(CHead d (Bind Abbr) u) v1 i0 H7) v2) t3 t4 H3 t H8))))) i H6 H4))))))))))))) 
+y t1 t2 H1))) H0)))))))).
+
 theorem pr2_lift:
  \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h 
 d c e) \to (\forall (t1: T).(\forall (t2: T).((pr2 e t1 t2) \to (pr2 c (lift 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/defs.ma
index 69266bbb1..5e14a9fc6 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/defs.ma
@@ -18,7 +18,7 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/defs".
 
 include "pr2/defs.ma".
 
-inductive pr3 (c:C): T \to (T \to Prop) \def
+inductive pr3 (c: C): T \to (T \to Prop) \def
 | pr3_refl: \forall (t: T).(pr3 c t t)
 | pr3_sing: \forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall (t3: 
 T).((pr3 c t2 t3) \to (pr3 c t1 t3))))).
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/fwd.ma
index 6c0e84a66..b1705e99f 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/fwd.ma
@@ -1469,3 +1469,91 @@ y2) z1 z2))))))) x2 x3 x4 x5 x6 x7 H8 (pr3_refl c (THead (Bind x2) x3 x4))
 H15 (pr3_pr2 c x0 x6 H11) (pr3_pr2 c x3 x7 H12) (pr3_pr2 (CHead c (Bind x2) 
 x7) x4 x5 H13))))) x1 H9))))))))))))) H7)) H6)))))))))))) y x H0))))) H))))).
 
+theorem pr3_gen_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u1: 
+T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind b) u1 t1) x) \to (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind b) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 (CHead c (Bind b) u1) t1 t2)))) (pr3 (CHead c (Bind 
+b) u1) t1 (lift (S O) O x)))))))))
+\def
+ \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall 
+(c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind 
+b0) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
+(THead (Bind b0) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b0) u1) t1 t2)))) (pr3 
+(CHead c (Bind b0) u1) t1 (lift (S O) O x)))))))))) (\lambda (_: (not (eq B 
+Abbr Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: 
+T).(\lambda (H0: (pr3 c (THead (Bind Abbr) u1 t1) x)).(let H1 \def 
+(pr3_gen_abbr c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) 
+u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (or (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) 
+t1 (lift (S O) O x))) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind 
+Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) 
+t1 (lift (S O) O x))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T x 
+(THead (Bind Abbr) x0 x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: (pr3 
+(CHead c (Bind Abbr) u1) t1 x1)).(or_introl (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S 
+O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
+(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) x0 x1 
+H3 H4 H5))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S 
+O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
+(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 
+(CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) H2)) H1)))))))) (\lambda (H: 
+(not (eq B Abst Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: 
+T).(\lambda (x: T).(\lambda (_: (pr3 c (THead (Bind Abst) u1 t1) x)).(let H1 
+\def (match (H (refl_equal B Abst)) in False return (\lambda (_: False).(or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 (CHead c (Bind Abst) u1) t1 t2)))) (pr3 (CHead c 
+(Bind Abst) u1) t1 (lift (S O) O x)))) with []) in H1))))))) (\lambda (_: 
+(not (eq B Void Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: 
+T).(\lambda (x: T).(\lambda (H0: (pr3 c (THead (Bind Void) u1 t1) x)).(let H1 
+\def (pr3_gen_void c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
+(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t2)))))) (pr3 (CHead c 
+(Bind Void) u1) t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) 
+u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda 
+(H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) 
+u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) 
+u) t1 t2))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
+(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b0) u) t1 t2))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 
+t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H3: (eq T x (THead (Bind Void) x0 
+x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: ((\forall (b0: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(or_introl (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Void) u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S 
+O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
+(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 t2))) x0 x1 
+H3 H4 (H5 Void u1)))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Void) u1) t1 
+(lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 
+t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x)) H2)) H1)))))))) b).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/iso.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/iso.ma
index 4b3519cd9..99a06b1a6 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/iso.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/iso.ma
@@ -18,7 +18,189 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/iso".
 
 include "pr3/fwd.ma".
 
-include "iso/fwd.ma".
+include "iso/props.ma".
+
+include "tlist/props.ma".
+
+theorem pr3_iso_appls_abbr:
+ \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).(let u1 \def (THeads (Flat 
+Appl) vs (TLRef i)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
+(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) vs (lift (S i) O w)) 
+u2))))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda 
+(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind 
+(\lambda (t: TList).(let u1 \def (THeads (Flat Appl) t (TLRef i)) in (\forall 
+(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to 
+(pr3 c (THeads (Flat Appl) t (lift (S i) O w)) u2)))))) (\lambda (u2: 
+T).(\lambda (H0: (pr3 c (TLRef i) u2)).(\lambda (H1: (((iso (TLRef i) u2) \to 
+(\forall (P: Prop).P)))).(let H2 \def (pr3_gen_lref c u2 i H0) in (or_ind (eq 
+T u2 (TLRef i)) (ex3_3 C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_: 
+T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u: 
+T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(v: T).(eq T u2 (lift (S i) O v)))))) (pr3 c (lift (S i) O w) u2) (\lambda 
+(H3: (eq T u2 (TLRef i))).(let H4 \def (eq_ind T u2 (\lambda (t: T).((iso 
+(TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H3) in (eq_ind_r T 
+(TLRef i) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (H4 (iso_refl (TLRef 
+i)) (pr3 c (lift (S i) O w) (TLRef i))) u2 H3))) (\lambda (H3: (ex3_3 C T T 
+(\lambda (d0: C).(\lambda (u: T).(\lambda (_: T).(getl i c (CHead d0 (Bind 
+Abbr) u))))) (\lambda (d0: C).(\lambda (u: T).(\lambda (v: T).(pr3 d0 u v)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T u2 (lift (S i) O 
+v))))))).(ex3_3_ind C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_: 
+T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u: 
+T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(v: T).(eq T u2 (lift (S i) O v))))) (pr3 c (lift (S i) O w) u2) (\lambda 
+(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H4: (getl i c (CHead x0 
+(Bind Abbr) x1))).(\lambda (H5: (pr3 x0 x1 x2)).(\lambda (H6: (eq T u2 (lift 
+(S i) O x2))).(let H7 \def (eq_ind T u2 (\lambda (t: T).((iso (TLRef i) t) 
+\to (\forall (P: Prop).P))) H1 (lift (S i) O x2) H6) in (eq_ind_r T (lift (S 
+i) O x2) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (let H8 \def (eq_ind C 
+(CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H (CHead x0 (Bind 
+Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) 
+H4)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return 
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow 
+c0])) (CHead d (Bind Abbr) w) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d 
+(Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) H4)) in ((let H10 \def (f_equal 
+C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
+\Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) w) (CHead 
+x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind 
+Abbr) x1) H4)) in (\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 
+(\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 w H10) in (let H13 
+\def (eq_ind_r T x1 (\lambda (t: T).(pr3 x0 t x2)) H5 w H10) in (let H14 \def 
+(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) w))) H12 d 
+H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c0: C).(pr3 c0 w x2)) H13 d 
+H11) in (pr3_lift c d (S i) O (getl_drop Abbr c d w i H14) w x2 H15))))))) 
+H9))) u2 H6)))))))) H3)) H2))))) (\lambda (t: T).(\lambda (t0: 
+TList).(\lambda (H0: ((\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (TLRef 
+i)) u2) \to ((((iso (THeads (Flat Appl) t0 (TLRef i)) u2) \to (\forall (P: 
+Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (lift (S i) O w)) 
+u2)))))).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads 
+(Flat Appl) t0 (TLRef i))) u2)).(\lambda (H2: (((iso (THead (Flat Appl) t 
+(THeads (Flat Appl) t0 (TLRef i))) u2) \to (\forall (P: Prop).P)))).(let H3 
+\def (pr3_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) u2 H1) in (or3_ind 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 
+t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 
+c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t 
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t 
+(THeads (Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (H4: (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: 
+T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))))).(ex3_2_ind T T (\lambda 
+(u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
+(THeads (Flat Appl) t0 (TLRef i)) t2))) (pr3 c (THead (Flat Appl) t (THeads 
+(Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H5: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c t 
+x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(let H8 \def 
+(eq_ind T u2 (\lambda (t1: T).((iso (THead (Flat Appl) t (THeads (Flat Appl) 
+t0 (TLRef i))) t1) \to (\forall (P: Prop).P))) H2 (THead (Flat Appl) x0 x1) 
+H5) in (eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(pr3 c (THead 
+(Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) t1)) (H8 (iso_head t 
+x0 (THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) (pr3 c (THead (Flat 
+Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) (THead (Flat Appl) x0 x1))) 
+u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t 
+u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 
+c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O 
+w))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H6: (pr3 c t 
+x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
+Abst) x0 x1))).(\lambda (H8: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c 
+(Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) (THead (Flat Appl) t 
+(THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t 
+(THead (Bind Abst) x0 x1)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift 
+(S i) O w))) c (pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead 
+(Bind Abst) x0 x1) (H0 (THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso 
+(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (P: 
+Prop).(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H9 P)))) t Appl) (THead 
+(Bind Abbr) t x1) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) 
+(THead (Bind Abbr) t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) 
+x0 x1)) (THead (Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 
+(pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t 
+x1) c (pr3_head_12 c t x2 H6 (Bind Abbr) x1 x3 (H8 Abbr x2)) u2 H5)))))))))) 
+H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind 
+B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
+(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat 
+Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda 
+(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not 
+(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
+(THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c (THead (Bind x0) x5 (THead 
+(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H8: (pr3 c t x4)).(\lambda 
+(H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t 
+(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat 
+Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Bind x0) 
+x1 (THead (Flat Appl) (lift (S O) O t) x2)) (THead (Flat Appl) t (THeads 
+(Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t (THead (Bind 
+x0) x1 x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c 
+(pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead (Bind x0) x1 
+x2) (H0 (THead (Bind x0) x1 x2) H6 (\lambda (H11: (iso (THeads (Flat Appl) t0 
+(TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (P: 
+Prop).(iso_flats_lref_bind_false Appl x0 i x1 x2 t0 H11 P)))) t Appl) (THead 
+(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr3_pr2 c (THead (Flat 
+Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift 
+(S O) O t) x2)) (pr2_free c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) 
+(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr0_upsilon x0 
+H5 t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2))))) (THead (Bind 
+x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr3_head_12 c x1 x1 
+(pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift (S O) O t) x2) (THead 
+(Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead c (Bind x0) x1) (lift 
+(S O) O t) (lift (S O) O x4) (pr3_lift (CHead c (Bind x0) x1) c (S O) O 
+(drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H8) (Flat Appl) x2 x2 
+(pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) 
+u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) 
+(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c (pr3_head_12 
+c x1 x5 H9 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat 
+Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 
+(lift (S O) O x4) Appl)) u2 H7)))))))))))))) H4)) H3)))))))) vs)))))).
 
 theorem pr3_iso_appls_cast:
  \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(let u1 
@@ -180,3 +362,775 @@ O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c
 (THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) 
 x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))) vs)))).
 
+theorem pr3_iso_appl_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: 
+T).(\forall (t: T).(let u1 \def (THead (Flat Appl) v1 (THead (Bind b) v2 t)) 
+in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
+(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) v2 (THead (Flat Appl) 
+(lift (S O) O v1) t)) u2))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v1: T).(\lambda 
+(v2: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H0: (pr3 c 
+(THead (Flat Appl) v1 (THead (Bind b) v2 t)) u2)).(\lambda (H1: (((iso (THead 
+(Flat Appl) v1 (THead (Bind b) v2 t)) u2) \to (\forall (P: Prop).P)))).(let 
+H2 \def (pr3_gen_appl c v1 (THead (Bind b) v2 t) u2 H0) in (or3_ind (ex3_2 T 
+T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 c (THead (Bind b) v2 t) t2)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
+Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1 
+t2)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) 
+(\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 
+z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat 
+Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b0: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c (THead (Bind b) v2 
+(THead (Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (H3: (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 c (THead (Bind b) v2 t) t2))))).(ex3_2_ind T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
+(THead (Bind b) v2 t) t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) 
+(lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq 
+T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v1 x0)).(\lambda (_: 
+(pr3 c (THead (Bind b) v2 t) x1)).(let H7 \def (eq_ind T u2 (\lambda (t0: 
+T).((iso (THead (Flat Appl) v1 (THead (Bind b) v2 t)) t0) \to (\forall (P: 
+Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl) 
+x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S 
+O) O v1) t)) t0)) (H7 (iso_head v1 x0 (THead (Bind b) v2 t) x1 (Flat Appl)) 
+(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead 
+(Flat Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda 
+(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
+Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1 
+t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
+(THead (Bind b) v2 t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: 
+T).(pr3 (CHead c (Bind b0) u) z1 t2))))))) (pr3 c (THead (Bind b) v2 (THead 
+(Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c (THead (Bind Abbr) 
+x2 x3) u2)).(\lambda (H5: (pr3 c v1 x2)).(\lambda (H6: (pr3 c (THead (Bind b) 
+v2 t) (THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b0: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b0) u) x1 x3))))).(pr3_t (THead (Bind Abbr) x2 x3) 
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) c (let H_x \def 
+(pr3_gen_bind b H c v2 t (THead (Bind Abst) x0 x1) H6) in (let H8 \def H_x in 
+(or_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst) 
+x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 
+u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) 
+(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind Abst) x0 x1))) (pr3 c 
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind 
+Abbr) x2 x3)) (\lambda (H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
+T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: 
+T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) 
+(lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3)) (\lambda (x4: T).(\lambda 
+(x5: T).(\lambda (H10: (eq T (THead (Bind Abst) x0 x1) (THead (Bind b) x4 
+x5))).(\lambda (H11: (pr3 c v2 x4)).(\lambda (H12: (pr3 (CHead c (Bind b) v2) 
+t x5)).(let H13 \def (f_equal T B (\lambda (e: T).(match e in T return 
+(\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow 
+Abst | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with 
+[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (THead (Bind Abst) 
+x0 x1) (THead (Bind b) x4 x5) H10) in ((let H14 \def (f_equal T T (\lambda 
+(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 
+| (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind 
+Abst) x0 x1) (THead (Bind b) x4 x5) H10) in ((let H15 \def (f_equal T T 
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) \Rightarrow t0])) 
+(THead (Bind Abst) x0 x1) (THead (Bind b) x4 x5) H10) in (\lambda (H16: (eq T 
+x0 x4)).(\lambda (H17: (eq B Abst b)).(let H18 \def (eq_ind_r T x5 (\lambda 
+(t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H12 x1 H15) in (let H19 \def 
+(eq_ind_r T x4 (\lambda (t0: T).(pr3 c v2 t0)) H11 x0 H16) in (let H20 \def 
+(eq_ind_r B b (\lambda (b0: B).(pr3 (CHead c (Bind b0) v2) t x1)) H18 Abst 
+H17) in (let H21 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H 
+Abst H17) in (eq_ind B Abst (\lambda (b0: B).(pr3 c (THead (Bind b0) v2 
+(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3))) (let H22 
+\def (match (H21 (refl_equal B Abst)) in False return (\lambda (_: 
+False).(pr3 c (THead (Bind Abst) v2 (THead (Flat Appl) (lift (S O) O v1) t)) 
+(THead (Bind Abbr) x2 x3))) with []) in H22) b H17)))))))) H14)) H13))))))) 
+H9)) (\lambda (H9: (pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind 
+Abst) x0 x1)))).(pr3_t (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O 
+x2) (lift (S O) O (THead (Bind Abst) x0 x1)))) (THead (Bind b) v2 (THead 
+(Flat Appl) (lift (S O) O v1) t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift 
+(S O) O v1) t) (THead (Flat Appl) (lift (S O) O x2) (lift (S O) O (THead 
+(Bind Abst) x0 x1))) (Bind b) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O 
+v1) (lift (S O) O x2) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop 
+(Bind b) O c c (drop_refl c) v2) v1 x2 H5) t (lift (S O) O (THead (Bind Abst) 
+x0 x1)) H9 Appl)) (THead (Bind Abbr) x2 x3) (eq_ind T (lift (S O) O (THead 
+(Flat Appl) x2 (THead (Bind Abst) x0 x1))) (\lambda (t0: T).(pr3 c (THead 
+(Bind b) v2 t0) (THead (Bind Abbr) x2 x3))) (pr3_sing c (THead (Bind Abbr) x2 
+x1) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst) 
+x0 x1)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 
+(THead (Bind Abst) x0 x1)))) (THead (Bind Abbr) x2 x1) (pr0_zeta b H (THead 
+(Flat Appl) x2 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x1) (pr0_beta 
+x0 x2 x2 (pr0_refl x2) x1 x1 (pr0_refl x1)) v2)) (THead (Bind Abbr) x2 x3) 
+(pr3_head_12 c x2 x2 (pr3_refl c x2) (Bind Abbr) x1 x3 (H7 Abbr x2))) (THead 
+(Flat Appl) (lift (S O) O x2) (lift (S O) O (THead (Bind Abst) x0 x1))) 
+(lift_flat Appl x2 (THead (Bind Abst) x0 x1) (S O) O)))) H8))) u2 H4))))))))) 
+H3)) (\lambda (H3: (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
+b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead 
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) 
+y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr3 c v1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind 
+B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 z1)))))))) (\lambda 
+(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: 
+T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) 
+O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) 
+y2) z1 z2))))))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O 
+v1) t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0 
+Abst))).(\lambda (H5: (pr3 c (THead (Bind b) v2 t) (THead (Bind x0) x1 
+x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) 
+O x4) x3)) u2)).(\lambda (H7: (pr3 c v1 x4)).(\lambda (H8: (pr3 c x1 
+x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t (THead (Bind 
+x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind b) v2 (THead 
+(Flat Appl) (lift (S O) O v1) t)) c (let H_x \def (pr3_gen_bind b H c v2 t 
+(THead (Bind x0) x1 x2) H5) in (let H10 \def H_x in (or_ind (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) (pr3 (CHead c (Bind 
+b) v2) t (lift (S O) O (THead (Bind x0) x1 x2))) (pr3 c (THead (Bind b) v2 
+(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat 
+Appl) (lift (S O) O x4) x3))) (\lambda (H11: (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 (CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 (CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead 
+(Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat Appl) 
+(lift (S O) O x4) x3))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H12: (eq 
+T (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7))).(\lambda (H13: (pr3 c v2 
+x6)).(\lambda (H14: (pr3 (CHead c (Bind b) v2) t x7)).(let H15 \def (f_equal 
+T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) 
+\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) \Rightarrow (match 
+k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
+\Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in 
+((let H16 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
+T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ t0 
+_) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in 
+((let H17 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
+T).T) with [(TSort _) \Rightarrow x2 | (TLRef _) \Rightarrow x2 | (THead _ _ 
+t0) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in 
+(\lambda (H18: (eq T x1 x6)).(\lambda (H19: (eq B x0 b)).(let H20 \def 
+(eq_ind_r T x7 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H14 x2 H17) 
+in (let H21 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c v2 t0)) H13 x1 H18) 
+in (let H22 \def (eq_ind B x0 (\lambda (b0: B).(pr3 (CHead c (Bind b0) x5) x2 
+x3)) H9 b H19) in (let H23 \def (eq_ind B x0 (\lambda (b0: B).(not (eq B b0 
+Abst))) H4 b H19) in (eq_ind_r B b (\lambda (b0: B).(pr3 c (THead (Bind b) v2 
+(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind b0) x5 (THead (Flat 
+Appl) (lift (S O) O x4) x3)))) (pr3_head_21 c v2 x5 (pr3_t x1 v2 c H21 x5 H8) 
+(Bind b) (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat Appl) (lift (S 
+O) O x4) x3) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O 
+x4) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c 
+(drop_refl c) v2) v1 x4 H7) t x3 (pr3_t x2 t (CHead c (Bind b) v2) H20 x3 
+(pr3_pr3_pr3_t c v2 x1 H21 x2 x3 (Bind b) (pr3_pr3_pr3_t c x1 x5 H8 x2 x3 
+(Bind b) H22))) Appl)) x0 H19)))))))) H16)) H15))))))) H11)) (\lambda (H11: 
+(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind x0) x1 x2)))).(pr3_t 
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead 
+(Bind x0) x1 x2)))) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) 
+t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat 
+Appl) (lift (S O) O x4) (lift (S O) O (THead (Bind x0) x1 x2))) (Bind b) 
+(pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O x4) (pr3_lift 
+(CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) v2) 
+v1 x4 H7) t (lift (S O) O (THead (Bind x0) x1 x2)) H11 Appl)) (THead (Bind 
+x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (eq_ind T (lift (S O) O 
+(THead (Flat Appl) x4 (THead (Bind x0) x1 x2))) (\lambda (t0: T).(pr3 c 
+(THead (Bind b) v2 t0) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O 
+x4) x3)))) (pr3_sing c (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O 
+x4) x2)) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x4 (THead (Bind 
+x0) x1 x2)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) 
+x4 (THead (Bind x0) x1 x2)))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S 
+O) O x4) x2)) (pr0_zeta b H (THead (Flat Appl) x4 (THead (Bind x0) x1 x2)) 
+(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr0_upsilon x0 
+H4 x4 x4 (pr0_refl x4) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2)) v2)) (THead 
+(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (pr3_head_12 c x1 x5 
+H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat Appl) 
+(lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H9 (lift (S 
+O) O x4) Appl))) (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead 
+(Bind x0) x1 x2))) (lift_flat Appl x4 (THead (Bind x0) x1 x2) (S O) O)))) 
+H10))) u2 H6))))))))))))) H3)) H2)))))))))).
+
+theorem pr3_iso_appls_appl_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (v: T).(\forall (u: 
+T).(\forall (t: T).(\forall (vs: TList).(let u1 \def (THeads (Flat Appl) vs 
+(THead (Flat Appl) v (THead (Bind b) u t))) in (\forall (c: C).(\forall (u2: 
+T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c 
+(THeads (Flat Appl) vs (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) 
+t))) u2)))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v: T).(\lambda 
+(u: T).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: 
+TList).(let u1 \def (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind 
+b) u t))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 
+u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead 
+(Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))) (\lambda (c: 
+C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) v (THead (Bind b) 
+u t)) u2)).(\lambda (H1: (((iso (THead (Flat Appl) v (THead (Bind b) u t)) 
+u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b H v u t c u2 H0 H1))))) 
+(\lambda (t0: T).(\lambda (t1: TList).(\lambda (H0: ((\forall (c: C).(\forall 
+(u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u 
+t))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind 
+b) u t))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1 
+(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))).(\lambda 
+(c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t0 (THeads 
+(Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t)))) u2)).(\lambda 
+(H2: (((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v 
+(THead (Bind b) u t)))) u2) \to (\forall (P: Prop).P)))).(let H3 \def 
+(pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind 
+b) u t))) u2 H1) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
+T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 
+u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead 
+(Flat Appl) v (THead (Bind b) u t))) t2)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
+Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
+Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: 
+B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2)))))))) (ex6_6 B T T T 
+T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead 
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) 
+y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c 
+(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat 
+Appl) (lift (S O) O v) t)))) u2) (\lambda (H4: (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
+(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) 
+t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
+(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
+Appl) v (THead (Bind b) u t))) t2))) (pr3 c (THead (Flat Appl) t0 (THeads 
+(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) 
+u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T u2 (THead (Flat 
+Appl) x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat 
+Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) x1)).(let H8 \def 
+(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl) 
+t1 (THead (Flat Appl) v (THead (Bind b) u t)))) t2) \to (\forall (P: 
+Prop).P))) H2 (THead (Flat Appl) x0 x1) H5) in (eq_ind_r T (THead (Flat Appl) 
+x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 
+(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) t2)) (H8 
+(iso_head t0 x0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u 
+t))) x1 (Flat Appl)) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 
+(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) (THead (Flat 
+Appl) x0 x1))) u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
+Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
+Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: 
+B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))))).(ex4_4_ind T T 
+T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c 
+(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u3: T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
+Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: 
+B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))) (pr3 c (THead 
+(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) 
+(lift (S O) O v) t)))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) 
+u2)).(\lambda (H6: (pr3 c t0 x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t1 
+(THead (Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 
+x1))).(\lambda (H8: ((\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind 
+b0) u0) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0 
+(THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) 
+t)))) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Flat 
+Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S 
+O) O v) t)))) c (pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u 
+(THead (Flat Appl) (lift (S O) O v) t))) (THead (Bind Abst) x0 x1) (H0 c 
+(THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso (THeads (Flat Appl) t1 (THead 
+(Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 x1))).(\lambda (P: 
+Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 (THead (Bind b) u t) 
+t1 H9 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c (THead (Flat Appl) 
+t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr2_free c (THead 
+(Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr0_beta 
+x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) 
+x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c t0 x2 H6 (Bind Abbr) x1 x3 
+(H8 Abbr x2)) u2 H5)))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda 
+(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
+(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead 
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) 
+y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind 
+B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u 
+t))) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b0) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))) (pr3 c 
+(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat 
+Appl) (lift (S O) O v) t)))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda 
+(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not 
+(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t1 (THead (Flat 
+Appl) v (THead (Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c 
+(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda 
+(H8: (pr3 c t0 x4)).(\lambda (H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c 
+(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S 
+O) O x4) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u 
+(THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t (THead (Bind x0) x1 (THead 
+(Flat Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) 
+t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t 
+(THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0 (THeads 
+(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c 
+(pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) 
+(lift (S O) O v) t))) (THead (Bind x0) x1 x2) (H0 c (THead (Bind x0) x1 x2) 
+H6 (\lambda (H11: (iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead 
+(Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (P: 
+Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind b) u t) t1 
+H11 P)))) t0 Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) 
+x2)) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind 
+x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat 
+Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) 
+(lift (S O) O t0) x2)) (pr0_upsilon x0 H5 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl 
+x1) x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S 
+O) O x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat 
+Appl) (lift (S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2) 
+(pr3_head_12 (CHead c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4) 
+(pr3_lift (CHead c (Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c 
+(drop_refl c) x1) t0 x4 H8) (Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind 
+x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 
+(THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat 
+Appl) (lift (S O) O x4) x2)) c (pr3_head_12 c x1 x5 H9 (Bind x0) (THead (Flat 
+Appl) (lift (S O) O x4) x2) (THead (Flat Appl) (lift (S O) O x4) x3) 
+(pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 (lift (S O) O x4) Appl)) u2 
+H7)))))))))))))) H4)) H3))))))))) vs)))))).
+
+theorem pr3_iso_appls_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (vs: TList).(\forall (u: 
+T).(\forall (t: T).(let u1 \def (THeads (Flat Appl) vs (THead (Bind b) u t)) 
+in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
+(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) 
+(lifts (S O) O vs) t)) u2))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (vs: 
+TList).(tlist_ind_rew (\lambda (t: TList).(\forall (u: T).(\forall (t0: 
+T).(let u1 \def (THeads (Flat Appl) t (THead (Bind b) u t0)) in (\forall (c: 
+C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: 
+Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t) 
+t0)) u2))))))))) (\lambda (u: T).(\lambda (t: T).(\lambda (c: C).(\lambda 
+(u2: T).(\lambda (H0: (pr3 c (THead (Bind b) u t) u2)).(\lambda (_: (((iso 
+(THead (Bind b) u t) u2) \to (\forall (P: Prop).P)))).H0)))))) (\lambda (ts: 
+TList).(\lambda (t: T).(\lambda (H0: ((\forall (u: T).(\forall (t0: 
+T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) ts (THead 
+(Bind b) u t0)) u2) \to ((((iso (THeads (Flat Appl) ts (THead (Bind b) u t0)) 
+u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat 
+Appl) (lifts (S O) O ts) t0)) u2))))))))).(\lambda (u: T).(\lambda (t0: 
+T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THeads (Flat Appl) 
+(TApp ts t) (THead (Bind b) u t0)) u2)).(\lambda (H2: (((iso (THeads (Flat 
+Appl) (TApp ts t) (THead (Bind b) u t0)) u2) \to (\forall (P: 
+Prop).P)))).(eq_ind_r TList (TApp (lifts (S O) O ts) (lift (S O) O t)) 
+(\lambda (t1: TList).(pr3 c (THead (Bind b) u (THeads (Flat Appl) t1 t0)) 
+u2)) (eq_ind_r T (THeads (Flat Appl) (lifts (S O) O ts) (THead (Flat Appl) 
+(lift (S O) O t) t0)) (\lambda (t1: T).(pr3 c (THead (Bind b) u t1) u2)) (let 
+H3 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind b) u t0)) 
+(\lambda (t1: T).(pr3 c t1 u2)) H1 (THeads (Flat Appl) ts (THead (Flat Appl) 
+t (THead (Bind b) u t0))) (theads_tapp (Flat Appl) ts t (THead (Bind b) u 
+t0))) in (let H4 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind 
+b) u t0)) (\lambda (t1: T).((iso t1 u2) \to (\forall (P: Prop).P))) H2 
+(THeads (Flat Appl) ts (THead (Flat Appl) t (THead (Bind b) u t0))) 
+(theads_tapp (Flat Appl) ts t (THead (Bind b) u t0))) in (TList_ind (\lambda 
+(t1: TList).(((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall 
+(u3: T).((pr3 c0 (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to 
+((((iso (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to (\forall (P: 
+Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O 
+t1) t2)) u3)))))))) \to ((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) t 
+(THead (Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat 
+Appl) t (THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c 
+(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t1) (THead (Flat Appl) 
+(lift (S O) O t) t0))) u2))))) (\lambda (_: ((\forall (u0: T).(\forall (t1: 
+T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) TNil (THead 
+(Bind b) u0 t1)) u3) \to ((((iso (THeads (Flat Appl) TNil (THead (Bind b) u0 
+t1)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads 
+(Flat Appl) (lifts (S O) O TNil) t1)) u3))))))))).(\lambda (H6: (pr3 c 
+(THeads (Flat Appl) TNil (THead (Flat Appl) t (THead (Bind b) u t0))) 
+u2)).(\lambda (H7: (((iso (THeads (Flat Appl) TNil (THead (Flat Appl) t 
+(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b 
+H t u t0 c u2 H6 H7)))) (\lambda (t1: T).(\lambda (ts0: TList).(\lambda (_: 
+((((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall (u3: T).((pr3 
+c0 (THeads (Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads 
+(Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to 
+(pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O ts0) t2)) 
+u3)))))))) \to ((pr3 c (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead 
+(Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) ts0 (THead (Flat Appl) t 
+(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead 
+(Bind b) u (THeads (Flat Appl) (lifts (S O) O ts0) (THead (Flat Appl) (lift 
+(S O) O t) t0))) u2)))))).(\lambda (H5: ((\forall (u0: T).(\forall (t2: 
+T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) (TCons t1 
+ts0) (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads (Flat Appl) (TCons t1 
+ts0) (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 
+(THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O (TCons t1 ts0)) t2)) 
+u3))))))))).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t1 ts0) (THead 
+(Flat Appl) t (THead (Bind b) u t0))) u2)).(\lambda (H7: (((iso (THeads (Flat 
+Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) u2) \to 
+(\forall (P: Prop).P)))).(H5 u (THead (Flat Appl) (lift (S O) O t) t0) c u2 
+(pr3_iso_appls_appl_bind b H t u t0 (TCons t1 ts0) c u2 H6 H7) (\lambda (H8: 
+(iso (THeads (Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl) 
+(lift (S O) O t) t0))) u2)).(\lambda (P: Prop).(H7 (iso_trans (THeads (Flat 
+Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) (THeads 
+(Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl) (lift (S O) O 
+t) t0))) (iso_head t1 t1 (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead 
+(Bind b) u t0))) (THeads (Flat Appl) ts0 (THead (Bind b) u (THead (Flat Appl) 
+(lift (S O) O t) t0))) (Flat Appl)) u2 H8) P)))))))))) ts H0 H3 H4))) (THeads 
+(Flat Appl) (TApp (lifts (S O) O ts) (lift (S O) O t)) t0) (theads_tapp (Flat 
+Appl) (lifts (S O) O ts) (lift (S O) O t) t0)) (lifts (S O) O (TApp ts t)) 
+(lifts_tapp (S O) O t ts))))))))))) vs))).
+
+theorem pr3_iso_beta:
+ \forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 \def (THead (Flat 
+Appl) v (THead (Bind Abst) w t)) in (\forall (c: C).(\forall (u2: T).((pr3 c 
+u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind 
+Abbr) v t) u2))))))))
+\def
+ \lambda (v: T).(\lambda (w: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: 
+T).(\lambda (H: (pr3 c (THead (Flat Appl) v (THead (Bind Abst) w t)) 
+u2)).(\lambda (H0: (((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) u2) 
+\to (\forall (P: Prop).P)))).(let H1 \def (pr3_gen_appl c v (THead (Bind 
+Abst) w t) u2 H) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
+T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v 
+u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) w t) t2)))) 
+(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: 
+T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) 
+w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind 
+b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr3 c v u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c 
+(THead (Bind Abbr) v t) u2) (\lambda (H2: (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
+(THead (Bind Abst) w t) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: 
+T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) 
+w t) t2))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H3: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v 
+x0)).(\lambda (_: (pr3 c (THead (Bind Abst) w t) x1)).(let H6 \def (eq_ind T 
+u2 (\lambda (t0: T).((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) t0) 
+\to (\forall (P: Prop).P))) H0 (THead (Flat Appl) x0 x1) H3) in (eq_ind_r T 
+(THead (Flat Appl) x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind Abbr) v t) 
+t0)) (H6 (iso_head v x0 (THead (Bind Abst) w t) x1 (Flat Appl)) (pr3 c (THead 
+(Bind Abbr) v t) (THead (Flat Appl) x0 x1))) u2 H3))))))) H2)) (\lambda (H2: 
+(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: 
+T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) 
+w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind 
+b) u) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v 
+u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Bind Abbr) v t) 
+u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H3: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H4: (pr3 c v 
+x2)).(\lambda (H5: (pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) x0 
+x1))).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) 
+u) x1 x3))))).(let H7 \def (pr3_gen_abst c w t (THead (Bind Abst) x0 x1) H5) 
+in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst) 
+x0 x1) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w 
+u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda 
+(x4: T).(\lambda (x5: T).(\lambda (H8: (eq T (THead (Bind Abst) x0 x1) (THead 
+(Bind Abst) x4 x5))).(\lambda (H9: (pr3 c w x4)).(\lambda (H10: ((\forall (b: 
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x5))))).(let H11 \def (f_equal 
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) 
+(THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in ((let H12 \def 
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
+[(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) 
+\Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in 
+(\lambda (H13: (eq T x0 x4)).(let H14 \def (eq_ind_r T x5 (\lambda (t0: 
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) H10 x1 
+H12) in (let H15 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c w t0)) H9 x0 
+H13) in (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) v t) c 
+(pr3_head_12 c v x2 H4 (Bind Abbr) t x3 (pr3_t x1 t (CHead c (Bind Abbr) x2) 
+(H14 Abbr x2) x3 (H6 Abbr x2))) u2 H3))))) H11))))))) H7)))))))))) H2)) 
+(\lambda (H2: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead (Bind b) y1 
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat 
+Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v 
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
+(THead (Bind Abst) w t) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) 
+u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2))))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: B).(\lambda 
+(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: 
+T).(\lambda (H3: (not (eq B x0 Abst))).(\lambda (H4: (pr3 c (THead (Bind 
+Abst) w t) (THead (Bind x0) x1 x2))).(\lambda (H5: (pr3 c (THead (Bind x0) x5 
+(THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (_: (pr3 c v 
+x4)).(\lambda (_: (pr3 c x1 x5)).(\lambda (H8: (pr3 (CHead c (Bind x0) x5) x2 
+x3)).(let H9 \def (pr3_gen_abst c w t (THead (Bind x0) x1 x2) H4) in 
+(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w 
+u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda 
+(x6: T).(\lambda (x7: T).(\lambda (H10: (eq T (THead (Bind x0) x1 x2) (THead 
+(Bind Abst) x6 x7))).(\lambda (H11: (pr3 c w x6)).(\lambda (H12: ((\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x7))))).(let H13 \def 
+(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with 
+[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) 
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) 
+\Rightarrow b | (Flat _) \Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead 
+(Bind Abst) x6 x7) H10) in ((let H14 \def (f_equal T T (\lambda (e: T).(match 
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) 
+\Rightarrow x1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind x0) x1 x2) 
+(THead (Bind Abst) x6 x7) H10) in ((let H15 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 | 
+(TLRef _) \Rightarrow x2 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind x0) 
+x1 x2) (THead (Bind Abst) x6 x7) H10) in (\lambda (H16: (eq T x1 
+x6)).(\lambda (H17: (eq B x0 Abst)).(let H18 \def (eq_ind_r T x7 (\lambda 
+(t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) 
+H12 x2 H15) in (let H19 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c w t0)) 
+H11 x1 H16) in (let H20 \def (eq_ind B x0 (\lambda (b: B).(pr3 (CHead c (Bind 
+b) x5) x2 x3)) H8 Abst H17) in (let H21 \def (eq_ind B x0 (\lambda (b: 
+B).(pr3 c (THead (Bind b) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)) 
+H5 Abst H17) in (let H22 \def (eq_ind B x0 (\lambda (b: B).(not (eq B b 
+Abst))) H3 Abst H17) in (let H23 \def (match (H22 (refl_equal B Abst)) in 
+False return (\lambda (_: False).(pr3 c (THead (Bind Abbr) v t) u2)) with []) 
+in H23))))))))) H14)) H13))))))) H9)))))))))))))) H2)) H1)))))))).
+
+theorem pr3_iso_appls_beta:
+ \forall (us: TList).(\forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 
+\def (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) w t))) in 
+(\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
+(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) us (THead (Bind Abbr) 
+v t)) u2)))))))))
+\def
+ \lambda (us: TList).(TList_ind (\lambda (t: TList).(\forall (v: T).(\forall 
+(w: T).(\forall (t0: T).(let u1 \def (THeads (Flat Appl) t (THead (Flat Appl) 
+v (THead (Bind Abst) w t0))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 
+u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat 
+Appl) t (THead (Bind Abbr) v t0)) u2)))))))))) (\lambda (v: T).(\lambda (w: 
+T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H: (pr3 c 
+(THead (Flat Appl) v (THead (Bind Abst) w t)) u2)).(\lambda (H0: (((iso 
+(THead (Flat Appl) v (THead (Bind Abst) w t)) u2) \to (\forall (P: 
+Prop).P)))).(pr3_iso_beta v w t c u2 H H0)))))))) (\lambda (t: T).(\lambda 
+(t0: TList).(\lambda (H: ((\forall (v: T).(\forall (w: T).(\forall (t1: 
+T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (THead 
+(Flat Appl) v (THead (Bind Abst) w t1))) u2) \to ((((iso (THeads (Flat Appl) 
+t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) u2) \to (\forall (P: 
+Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1)) 
+u2)))))))))).(\lambda (v: T).(\lambda (w: T).(\lambda (t1: T).(\lambda (c: 
+C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) t (THeads (Flat 
+Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) u2)).(\lambda (H1: 
+(((iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Flat Appl) v 
+(THead (Bind Abst) w t1)))) u2) \to (\forall (P: Prop).P)))).(let H2 \def 
+(pr3_gen_appl c t (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind 
+Abst) w t1))) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
+T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) 
+t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2)))) (ex4_4 T T T T 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c 
+(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
+Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: 
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
+(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c 
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) 
+(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
+(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
+Appl) v (THead (Bind Abst) w t1))) t2))))).(ex3_2_ind T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
+(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2))) 
+(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) 
+u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat 
+Appl) x0 x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat 
+Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) x1)).(let H7 \def 
+(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t (THeads (Flat Appl) 
+t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) t2) \to (\forall (P: 
+Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl) 
+x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 
+(THead (Bind Abbr) v t1))) t2)) (H7 (iso_head t x0 (THeads (Flat Appl) t0 
+(THead (Flat Appl) v (THead (Bind Abst) w t1))) x1 (Flat Appl)) (pr3 c (THead 
+(Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) (THead (Flat 
+Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
+Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
+Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: 
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T T T 
+T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c 
+(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
+Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: 
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Flat 
+Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c 
+(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t x2)).(\lambda (H6: (pr3 
+c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) 
+(THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b: B).(\forall (u: 
+T).(pr3 (CHead c (Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) 
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c 
+(pr3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Flat Appl) t 
+(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads 
+(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind Abst) x0 x1) (H v w t1 
+c (THead (Bind Abst) x0 x1) H6 (\lambda (H8: (iso (THeads (Flat Appl) t0 
+(THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) x0 
+x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 
+(THead (Bind Abst) w t1) t0 H8 P)))) t Appl) (THead (Bind Abbr) t x1) 
+(pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) 
+t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead 
+(Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 (pr0_refl x1))))) u2 
+(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t x1) c (pr3_head_12 c t 
+x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3: 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) 
+w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c 
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
+Appl) v (THead (Bind Abst) w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) 
+u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead 
+(Bind Abbr) v t1))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq 
+B x0 Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v 
+(THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c 
+(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda 
+(H7: (pr3 c t x4)).(\lambda (H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c 
+(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S 
+O) O x4) x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) 
+v t1))) c (pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) 
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c 
+(pr3_t (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Flat Appl) t 
+(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads 
+(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind x0) x1 x2) (H v w t1 c 
+(THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads (Flat Appl) t0 (THead 
+(Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda 
+(P: Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind Abst) 
+w t1) t0 H10 P)))) t Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) 
+O t) x2)) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead 
+(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr2_free c (THead 
+(Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) 
+(lift (S O) O t) x2)) (pr0_upsilon x0 H4 t t (pr0_refl t) x1 x1 (pr0_refl x1) 
+x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O 
+x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl) 
+(lift (S O) O t) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 
+(CHead c (Bind x0) x1) (lift (S O) O t) (lift (S O) O x4) (pr3_lift (CHead c 
+(Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H7) 
+(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift 
+(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S 
+O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c 
+(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) 
+(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) 
+x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))))))) us).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/arity.ma
index 800e6a00e..e8c5f04ae 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/arity.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/arity.ma
@@ -16,67 +16,286 @@
 
 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/arity".
 
-include "ceqc/props.ma".
+include "csubc/props.ma".
+
+theorem sc3_arity_csubc:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 
+t a) \to (\forall (d1: C).(\forall (is: PList).((drop1 is d1 c1) \to (\forall 
+(c2: C).((csubc g d1 c2) \to (sc3 g a c2 (lift1 is t)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
+(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: 
+A).(\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: 
+C).((csubc g d1 c2) \to (sc3 g a0 c2 (lift1 is t0)))))))))) (\lambda (c: 
+C).(\lambda (n: nat).(\lambda (d1: C).(\lambda (is: PList).(\lambda (_: 
+(drop1 is d1 c)).(\lambda (c2: C).(\lambda (_: (csubc g d1 c2)).(eq_ind_r T 
+(TSort n) (\lambda (t0: T).(land (arity g c2 t0 (ASort O n)) (sn3 c2 t0))) 
+(conj (arity g c2 (TSort n) (ASort O n)) (sn3 c2 (TSort n)) (arity_sort g c2 
+n) (sn3_nf2 c2 (TSort n) (nf2_sort c2 n))) (lift1 is (TSort n)) (lift1_sort n 
+is))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
+nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: 
+A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (d1: C).(\forall 
+(is: PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g 
+a0 c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda 
+(H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(let 
+H_x \def (drop1_getl_trans is c d1 H3 Abbr d u i H0) in (let H5 \def H_x in 
+(ex2_ind C (\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2: 
+C).(getl (trans is i) d1 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) u)))) 
+(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x: C).(\lambda (_: (drop1 
+(ptrans is i) x d)).(\lambda (H7: (getl (trans is i) d1 (CHead x (Bind Abbr) 
+(lift1 (ptrans is i) u)))).(let H_x0 \def (csubc_getl_conf g d1 (CHead x 
+(Bind Abbr) (lift1 (ptrans is i) u)) (trans is i) H7 c2 H4) in (let H8 \def 
+H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans is i) c2 e2)) (\lambda (e2: 
+C).(csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) e2)) (sc3 g a0 c2 
+(lift1 is (TLRef i))) (\lambda (x0: C).(\lambda (H9: (getl (trans is i) c2 
+x0)).(\lambda (H10: (csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) 
+x0)).(let H11 \def (match H10 in csubc return (\lambda (c0: C).(\lambda (c3: 
+C).(\lambda (_: (csubc ? c0 c3)).((eq C c0 (CHead x (Bind Abbr) (lift1 
+(ptrans is i) u))) \to ((eq C c3 x0) \to (sc3 g a0 c2 (lift1 is (TLRef 
+i)))))))) with [(csubc_sort n) \Rightarrow (\lambda (H11: (eq C (CSort n) 
+(CHead x (Bind Abbr) (lift1 (ptrans is i) u)))).(\lambda (H12: (eq C (CSort 
+n) x0)).((let H13 \def (eq_ind C (CSort n) (\lambda (e: C).(match e in C 
+return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) 
+\Rightarrow False])) I (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) H11) in 
+(False_ind ((eq C (CSort n) x0) \to (sc3 g a0 c2 (lift1 is (TLRef i)))) H13)) 
+H12))) | (csubc_head c0 c3 H11 k v) \Rightarrow (\lambda (H12: (eq C (CHead 
+c0 k v) (CHead x (Bind Abbr) (lift1 (ptrans is i) u)))).(\lambda (H13: (eq C 
+(CHead c3 k v) x0)).((let H14 \def (f_equal C T (\lambda (e: C).(match e in C 
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t0) 
+\Rightarrow t0])) (CHead c0 k v) (CHead x (Bind Abbr) (lift1 (ptrans is i) 
+u)) H12) in ((let H15 \def (f_equal C K (\lambda (e: C).(match e in C return 
+(\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow 
+k0])) (CHead c0 k v) (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) H12) in 
+((let H16 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: 
+C).C) with [(CSort _) \Rightarrow c0 | (CHead c4 _ _) \Rightarrow c4])) 
+(CHead c0 k v) (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) H12) in (eq_ind 
+C x (\lambda (c4: C).((eq K k (Bind Abbr)) \to ((eq T v (lift1 (ptrans is i) 
+u)) \to ((eq C (CHead c3 k v) x0) \to ((csubc g c4 c3) \to (sc3 g a0 c2 
+(lift1 is (TLRef i)))))))) (\lambda (H17: (eq K k (Bind Abbr))).(eq_ind K 
+(Bind Abbr) (\lambda (k0: K).((eq T v (lift1 (ptrans is i) u)) \to ((eq C 
+(CHead c3 k0 v) x0) \to ((csubc g x c3) \to (sc3 g a0 c2 (lift1 is (TLRef 
+i))))))) (\lambda (H18: (eq T v (lift1 (ptrans is i) u))).(eq_ind T (lift1 
+(ptrans is i) u) (\lambda (t0: T).((eq C (CHead c3 (Bind Abbr) t0) x0) \to 
+((csubc g x c3) \to (sc3 g a0 c2 (lift1 is (TLRef i)))))) (\lambda (H19: (eq 
+C (CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)) x0)).(eq_ind C (CHead c3 
+(Bind Abbr) (lift1 (ptrans is i) u)) (\lambda (_: C).((csubc g x c3) \to (sc3 
+g a0 c2 (lift1 is (TLRef i))))) (\lambda (_: (csubc g x c3)).(let H21 \def 
+(eq_ind_r C x0 (\lambda (c4: C).(getl (trans is i) c2 c4)) H9 (CHead c3 (Bind 
+Abbr) (lift1 (ptrans is i) u)) H19) in (let H_y \def (sc3_abbr g a0 TNil) in 
+(eq_ind_r T (TLRef (trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y 
+(trans is i) c3 (lift1 (ptrans is i) u) c2 (eq_ind T (lift1 is (lift (S i) O 
+u)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (eq_ind T (lift1 (PConsTail is (S i) 
+O) u) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H2 d1 (PConsTail is (S i) O) 
+(drop1_cons_tail c d (S i) O (getl_drop Abbr c d u i H0) is d1 H3) c2 H4) 
+(lift1 is (lift (S i) O u)) (lift1_cons_tail u (S i) O is)) (lift (S (trans 
+is i)) O (lift1 (ptrans is i) u)) (lift1_free is i u)) H21) (lift1 is (TLRef 
+i)) (lift1_lref is i))))) x0 H19)) v (sym_eq T v (lift1 (ptrans is i) u) 
+H18))) k (sym_eq K k (Bind Abbr) H17))) c0 (sym_eq C c0 x H16))) H15)) H14)) 
+H13 H11))) | (csubc_abst c0 c3 H11 v a1 H12 w H13) \Rightarrow (\lambda (H14: 
+(eq C (CHead c0 (Bind Abst) v) (CHead x (Bind Abbr) (lift1 (ptrans is i) 
+u)))).(\lambda (H15: (eq C (CHead c3 (Bind Abbr) w) x0)).((let H16 \def 
+(eq_ind C (CHead c0 (Bind Abst) v) (\lambda (e: C).(match e in C return 
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) 
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow 
+False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) 
+\Rightarrow False])])) I (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) H14) 
+in (False_ind ((eq C (CHead c3 (Bind Abbr) w) x0) \to ((csubc g c0 c3) \to 
+((sc3 g (asucc g a1) c0 v) \to ((sc3 g a1 c3 w) \to (sc3 g a0 c2 (lift1 is 
+(TLRef i))))))) H16)) H15 H11 H12 H13)))]) in (H11 (refl_equal C (CHead x 
+(Bind Abbr) (lift1 (ptrans is i) u))) (refl_equal C x0)))))) H8)))))) 
+H5)))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
+(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0: 
+A).(\lambda (H1: (arity g d u (asucc g a0))).(\lambda (_: ((\forall (d1: 
+C).(\forall (is: PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 
+c2) \to (sc3 g (asucc g a0) c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda 
+(is: PList).(\lambda (H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: 
+(csubc g d1 c2)).(let H5 \def H0 in (let H_x \def (drop1_getl_trans is c d1 
+H3 Abst d u i H5) in (let H6 \def H_x in (ex2_ind C (\lambda (e2: C).(drop1 
+(ptrans is i) e2 d)) (\lambda (e2: C).(getl (trans is i) d1 (CHead e2 (Bind 
+Abst) (lift1 (ptrans is i) u)))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda 
+(x: C).(\lambda (H7: (drop1 (ptrans is i) x d)).(\lambda (H8: (getl (trans is 
+i) d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)))).(let H_x0 \def 
+(csubc_getl_conf g d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)) (trans is 
+i) H8 c2 H4) in (let H9 \def H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans 
+is i) c2 e2)) (\lambda (e2: C).(csubc g (CHead x (Bind Abst) (lift1 (ptrans 
+is i) u)) e2)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x0: C).(\lambda 
+(H10: (getl (trans is i) c2 x0)).(\lambda (H11: (csubc g (CHead x (Bind Abst) 
+(lift1 (ptrans is i) u)) x0)).(let H12 \def (match H11 in csubc return 
+(\lambda (c0: C).(\lambda (c3: C).(\lambda (_: (csubc ? c0 c3)).((eq C c0 
+(CHead x (Bind Abst) (lift1 (ptrans is i) u))) \to ((eq C c3 x0) \to (sc3 g 
+a0 c2 (lift1 is (TLRef i)))))))) with [(csubc_sort n) \Rightarrow (\lambda 
+(H12: (eq C (CSort n) (CHead x (Bind Abst) (lift1 (ptrans is i) 
+u)))).(\lambda (H13: (eq C (CSort n) x0)).((let H14 \def (eq_ind C (CSort n) 
+(\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) 
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead x (Bind Abst) 
+(lift1 (ptrans is i) u)) H12) in (False_ind ((eq C (CSort n) x0) \to (sc3 g 
+a0 c2 (lift1 is (TLRef i)))) H14)) H13))) | (csubc_head c0 c3 H12 k v) 
+\Rightarrow (\lambda (H13: (eq C (CHead c0 k v) (CHead x (Bind Abst) (lift1 
+(ptrans is i) u)))).(\lambda (H14: (eq C (CHead c3 k v) x0)).((let H15 \def 
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
+[(CSort _) \Rightarrow v | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k v) 
+(CHead x (Bind Abst) (lift1 (ptrans is i) u)) H13) in ((let H16 \def (f_equal 
+C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) 
+\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k v) (CHead x 
+(Bind Abst) (lift1 (ptrans is i) u)) H13) in ((let H17 \def (f_equal C C 
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
+\Rightarrow c0 | (CHead c4 _ _) \Rightarrow c4])) (CHead c0 k v) (CHead x 
+(Bind Abst) (lift1 (ptrans is i) u)) H13) in (eq_ind C x (\lambda (c4: 
+C).((eq K k (Bind Abst)) \to ((eq T v (lift1 (ptrans is i) u)) \to ((eq C 
+(CHead c3 k v) x0) \to ((csubc g c4 c3) \to (sc3 g a0 c2 (lift1 is (TLRef 
+i)))))))) (\lambda (H18: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda 
+(k0: K).((eq T v (lift1 (ptrans is i) u)) \to ((eq C (CHead c3 k0 v) x0) \to 
+((csubc g x c3) \to (sc3 g a0 c2 (lift1 is (TLRef i))))))) (\lambda (H19: (eq 
+T v (lift1 (ptrans is i) u))).(eq_ind T (lift1 (ptrans is i) u) (\lambda (t0: 
+T).((eq C (CHead c3 (Bind Abst) t0) x0) \to ((csubc g x c3) \to (sc3 g a0 c2 
+(lift1 is (TLRef i)))))) (\lambda (H20: (eq C (CHead c3 (Bind Abst) (lift1 
+(ptrans is i) u)) x0)).(eq_ind C (CHead c3 (Bind Abst) (lift1 (ptrans is i) 
+u)) (\lambda (_: C).((csubc g x c3) \to (sc3 g a0 c2 (lift1 is (TLRef i))))) 
+(\lambda (_: (csubc g x c3)).(let H22 \def (eq_ind_r C x0 (\lambda (c4: 
+C).(getl (trans is i) c2 c4)) H10 (CHead c3 (Bind Abst) (lift1 (ptrans is i) 
+u)) H20) in (let H_y \def (sc3_abst g a0 TNil) in (eq_ind_r T (TLRef (trans 
+is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y c2 (trans is i) 
+(csubc_arity_conf g d1 c2 H4 (TLRef (trans is i)) a0 (eq_ind T (lift1 is 
+(TLRef i)) (\lambda (t0: T).(arity g d1 t0 a0)) (arity_lift1 g a0 c is d1 
+(TLRef i) H3 (arity_abst g c d u i H0 a0 H1)) (TLRef (trans is i)) 
+(lift1_lref is i))) (nf2_lref_abst c2 c3 (lift1 (ptrans is i) u) (trans is i) 
+H22) I) (lift1 is (TLRef i)) (lift1_lref is i))))) x0 H20)) v (sym_eq T v 
+(lift1 (ptrans is i) u) H19))) k (sym_eq K k (Bind Abst) H18))) c0 (sym_eq C 
+c0 x H17))) H16)) H15)) H14 H12))) | (csubc_abst c0 c3 H12 v a1 H13 w H14) 
+\Rightarrow (\lambda (H15: (eq C (CHead c0 (Bind Abst) v) (CHead x (Bind 
+Abst) (lift1 (ptrans is i) u)))).(\lambda (H16: (eq C (CHead c3 (Bind Abbr) 
+w) x0)).((let H17 \def (f_equal C T (\lambda (e: C).(match e in C return 
+(\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t0) \Rightarrow 
+t0])) (CHead c0 (Bind Abst) v) (CHead x (Bind Abst) (lift1 (ptrans is i) u)) 
+H15) in ((let H18 \def (f_equal C C (\lambda (e: C).(match e in C return 
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c4 _ _) 
+\Rightarrow c4])) (CHead c0 (Bind Abst) v) (CHead x (Bind Abst) (lift1 
+(ptrans is i) u)) H15) in (eq_ind C x (\lambda (c4: C).((eq T v (lift1 
+(ptrans is i) u)) \to ((eq C (CHead c3 (Bind Abbr) w) x0) \to ((csubc g c4 
+c3) \to ((sc3 g (asucc g a1) c4 v) \to ((sc3 g a1 c3 w) \to (sc3 g a0 c2 
+(lift1 is (TLRef i))))))))) (\lambda (H19: (eq T v (lift1 (ptrans is i) 
+u))).(eq_ind T (lift1 (ptrans is i) u) (\lambda (t0: T).((eq C (CHead c3 
+(Bind Abbr) w) x0) \to ((csubc g x c3) \to ((sc3 g (asucc g a1) x t0) \to 
+((sc3 g a1 c3 w) \to (sc3 g a0 c2 (lift1 is (TLRef i)))))))) (\lambda (H20: 
+(eq C (CHead c3 (Bind Abbr) w) x0)).(eq_ind C (CHead c3 (Bind Abbr) w) 
+(\lambda (_: C).((csubc g x c3) \to ((sc3 g (asucc g a1) x (lift1 (ptrans is 
+i) u)) \to ((sc3 g a1 c3 w) \to (sc3 g a0 c2 (lift1 is (TLRef i))))))) 
+(\lambda (_: (csubc g x c3)).(\lambda (H22: (sc3 g (asucc g a1) x (lift1 
+(ptrans is i) u))).(\lambda (H23: (sc3 g a1 c3 w)).(let H24 \def (eq_ind_r C 
+x0 (\lambda (c4: C).(getl (trans is i) c2 c4)) H10 (CHead c3 (Bind Abbr) w) 
+H20) in (let H_y \def (sc3_abbr g a0 TNil) in (eq_ind_r T (TLRef (trans is 
+i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y (trans is i) c3 w c2 (let H_y0 
+\def (arity_lift1 g (asucc g a0) d (ptrans is i) x u H7 H1) in (let H_y1 \def 
+(sc3_arity_gen g x (lift1 (ptrans is i) u) (asucc g a1) H22) in (sc3_repl g 
+a1 c2 (lift (S (trans is i)) O w) (sc3_lift g a1 c3 w H23 c2 (S (trans is i)) 
+O (getl_drop Abbr c2 c3 w (trans is i) H24)) a0 (asucc_inj g a1 a0 
+(arity_mono g x (lift1 (ptrans is i) u) (asucc g a1) H_y1 (asucc g a0) 
+H_y0))))) H24) (lift1 is (TLRef i)) (lift1_lref is i))))))) x0 H20)) v 
+(sym_eq T v (lift1 (ptrans is i) u) H19))) c0 (sym_eq C c0 x H18))) H17)) H16 
+H12 H13 H14)))]) in (H12 (refl_equal C (CHead x (Bind Abst) (lift1 (ptrans is 
+i) u))) (refl_equal C x0)))))) H9)))))) H6))))))))))))))))) (\lambda (b: 
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: 
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall 
+(d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g 
+d1 c2) \to (sc3 g a1 c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda (a2: 
+A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: ((\forall 
+(d1: C).(\forall (is: PList).((drop1 is d1 (CHead c (Bind b) u)) \to (\forall 
+(c2: C).((csubc g d1 c2) \to (sc3 g a2 c2 (lift1 is t0))))))))).(\lambda (d1: 
+C).(\lambda (is: PList).(\lambda (H5: (drop1 is d1 c)).(\lambda (c2: 
+C).(\lambda (H6: (csubc g d1 c2)).(let H_y \def (sc3_bind g b H0 a1 a2 TNil) 
+in (eq_ind_r T (THead (Bind b) (lift1 is u) (lift1 (Ss is) t0)) (\lambda (t1: 
+T).(sc3 g a2 c2 t1)) (H_y c2 (lift1 is u) (lift1 (Ss is) t0) (H4 (CHead d1 
+(Bind b) (lift1 is u)) (Ss is) (drop1_skip_bind b c is d1 u H5) (CHead c2 
+(Bind b) (lift1 is u)) (csubc_head g d1 c2 H6 (Bind b) (lift1 is u))) (H2 d1 
+is H5 c2 H6)) (lift1 is (THead (Bind b) u t0)) (lift1_bind b is u 
+t0))))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: 
+A).(\lambda (H0: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (d1: 
+C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 
+c2) \to (sc3 g (asucc g a1) c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda 
+(a2: A).(\lambda (H2: (arity g (CHead c (Bind Abst) u) t0 a2)).(\lambda (H3: 
+((\forall (d1: C).(\forall (is: PList).((drop1 is d1 (CHead c (Bind Abst) u)) 
+\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a2 c2 (lift1 is 
+t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H4: (drop1 is d1 
+c)).(\lambda (c2: C).(\lambda (H5: (csubc g d1 c2)).(eq_ind_r T (THead (Bind 
+Abst) (lift1 is u) (lift1 (Ss is) t0)) (\lambda (t1: T).(land (arity g c2 t1 
+(AHead a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall 
+(is0: PList).((drop1 is0 d c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 
+is0 t1)))))))))) (conj (arity g c2 (THead (Bind Abst) (lift1 is u) (lift1 (Ss 
+is) t0)) (AHead a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to 
+(\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g a2 d (THead (Flat Appl) w 
+(lift1 is0 (THead (Bind Abst) (lift1 is u) (lift1 (Ss is) t0)))))))))) 
+(csubc_arity_conf g d1 c2 H5 (THead (Bind Abst) (lift1 is u) (lift1 (Ss is) 
+t0)) (AHead a1 a2) (arity_head g d1 (lift1 is u) a1 (arity_lift1 g (asucc g 
+a1) c is d1 u H4 H0) (lift1 (Ss is) t0) a2 (arity_lift1 g a2 (CHead c (Bind 
+Abst) u) (Ss is) (CHead d1 (Bind Abst) (lift1 is u)) t0 (drop1_skip_bind Abst 
+c is d1 u H4) H2))) (\lambda (d: C).(\lambda (w: T).(\lambda (H6: (sc3 g a1 d 
+w)).(\lambda (is0: PList).(\lambda (H7: (drop1 is0 d c2)).(eq_ind_r T (THead 
+(Bind Abst) (lift1 is0 (lift1 is u)) (lift1 (Ss is0) (lift1 (Ss is) t0))) 
+(\lambda (t1: T).(sc3 g a2 d (THead (Flat Appl) w t1))) (let H8 \def 
+(sc3_appl g a1 a2 TNil) in (H8 d w (lift1 (Ss is0) (lift1 (Ss is) t0)) (let 
+H_y \def (sc3_bind g Abbr (\lambda (H9: (eq B Abbr Abst)).(not_abbr_abst H9)) 
+a1 a2 TNil) in (H_y d w (lift1 (Ss is0) (lift1 (Ss is) t0)) (let H_x \def 
+(csubc_drop1_conf_rev g is0 d c2 H7 d1 H5) in (let H9 \def H_x in (ex2_ind C 
+(\lambda (c3: C).(drop1 is0 c3 d1)) (\lambda (c3: C).(csubc g c3 d)) (sc3 g 
+a2 (CHead d (Bind Abbr) w) (lift1 (Ss is0) (lift1 (Ss is) t0))) (\lambda (x: 
+C).(\lambda (H10: (drop1 is0 x d1)).(\lambda (H11: (csubc g x d)).(eq_ind_r T 
+(lift1 (papp (Ss is0) (Ss is)) t0) (\lambda (t1: T).(sc3 g a2 (CHead d (Bind 
+Abbr) w) t1)) (eq_ind_r PList (Ss (papp is0 is)) (\lambda (p: PList).(sc3 g 
+a2 (CHead d (Bind Abbr) w) (lift1 p t0))) (H3 (CHead x (Bind Abst) (lift1 
+(papp is0 is) u)) (Ss (papp is0 is)) (drop1_skip_bind Abst c (papp is0 is) x 
+u (drop1_trans is0 x d1 H10 is c H4)) (CHead d (Bind Abbr) w) (csubc_abst g x 
+d H11 (lift1 (papp is0 is) u) a1 (H1 x (papp is0 is) (drop1_trans is0 x d1 
+H10 is c H4) x (csubc_refl g x)) w H6)) (papp (Ss is0) (Ss is)) (papp_ss is0 
+is)) (lift1 (Ss is0) (lift1 (Ss is) t0)) (lift1_lift1 (Ss is0) (Ss is) 
+t0))))) H9))) H6)) H6 (lift1 is0 (lift1 is u)) (sc3_lift1 g c2 (asucc g a1) 
+is0 d (lift1 is u) (H1 d1 is H4 c2 H5) H7))) (lift1 is0 (THead (Bind Abst) 
+(lift1 is u) (lift1 (Ss is) t0))) (lift1_bind Abst is0 (lift1 is u) (lift1 
+(Ss is) t0))))))))) (lift1 is (THead (Bind Abst) u t0)) (lift1_bind Abst is u 
+t0)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda 
+(_: (arity g c u a1)).(\lambda (H1: ((\forall (d1: C).(\forall (is: 
+PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a1 
+c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity 
+g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (d1: C).(\forall (is: 
+PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g 
+(AHead a1 a2) c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: 
+PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g 
+d1 c2)).(let H_y \def (H1 d1 is H4 c2 H5) in (let H_y0 \def (H3 d1 is H4 c2 
+H5) in (let H6 \def H_y0 in (and_ind (arity g c2 (lift1 is t0) (AHead a1 a2)) 
+(\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: 
+PList).((drop1 is0 d c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 
+(lift1 is t0))))))))) (sc3 g a2 c2 (lift1 is (THead (Flat Appl) u t0))) 
+(\lambda (_: (arity g c2 (lift1 is t0) (AHead a1 a2))).(\lambda (H8: 
+((\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: 
+PList).((drop1 is0 d c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 
+(lift1 is t0))))))))))).(let H_y1 \def (H8 c2 (lift1 is u) H_y PNil) in 
+(eq_ind_r T (THead (Flat Appl) (lift1 is u) (lift1 is t0)) (\lambda (t1: 
+T).(sc3 g a2 c2 t1)) (H_y1 (drop1_nil c2)) (lift1 is (THead (Flat Appl) u 
+t0)) (lift1_flat Appl is u t0))))) H6)))))))))))))))))) (\lambda (c: 
+C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g 
+a0))).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) 
+\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (asucc g a0) c2 (lift1 is 
+u))))))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3: 
+((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: 
+C).((csubc g d1 c2) \to (sc3 g a0 c2 (lift1 is t0))))))))).(\lambda (d1: 
+C).(\lambda (is: PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2: 
+C).(\lambda (H5: (csubc g d1 c2)).(let H_y \def (sc3_cast g a0 TNil) in 
+(eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1 is t0)) (\lambda (t1: 
+T).(sc3 g a0 c2 t1)) (H_y c2 (lift1 is u) (H1 d1 is H4 c2 H5) (lift1 is t0) 
+(H3 d1 is H4 c2 H5)) (lift1 is (THead (Flat Cast) u t0)) (lift1_flat Cast is 
+u t0)))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: 
+A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (d1: C).(\forall 
+(is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g 
+a1 c2 (lift1 is t0))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 
+a2)).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H3: (drop1 is d1 
+c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(sc3_repl g a1 c2 (lift1 
+is t0) (H1 d1 is H3 c2 H4) a2 H2))))))))))))) c1 t a H))))).
 
 theorem sc3_arity:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
 a) \to (sc3 g a c t)))))
 \def
  \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
-(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0: 
-A).(sc3 g a0 c0 t0)))) (\lambda (c0: C).(\lambda (n: nat).(conj (arity g c0 
-(TSort n) (ASort O n)) (sn3 c0 (TSort n)) (arity_sort g c0 n) (sn3_nf2 c0 
-(TSort n) (nf2_sort c0 n))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
-T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) 
-u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (H2: (sc3 g a0 
-d u)).(let H_y \def (sc3_abbr g a0 TNil) in (H_y i d u c0 (sc3_lift g a0 d u 
-H2 c0 (S i) O (getl_drop Abbr c0 d u i H0)) H0)))))))))) (\lambda (c0: 
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 
-(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc 
-g a0))).(\lambda (_: (sc3 g (asucc g a0) d u)).(let H3 \def (sc3_abst g a0 
-TNil) in (H3 c0 i (arity_abst g c0 d u i H0 a0 H1) (nf2_lref_abst c0 d u i 
-H0) I)))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda 
-(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u 
-a1)).(\lambda (H2: (sc3 g a1 c0 u)).(\lambda (t0: T).(\lambda (a2: 
-A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: (sc3 g 
-a2 (CHead c0 (Bind b) u) t0)).(let H_y \def (sc3_bind g b H0 a1 a2 TNil) in 
-(H_y c0 u t0 H4 H2))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda 
-(a1: A).(\lambda (H0: (arity g c0 u (asucc g a1))).(\lambda (H1: (sc3 g 
-(asucc g a1) c0 u)).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H2: (arity g 
-(CHead c0 (Bind Abst) u) t0 a2)).(\lambda (H3: (sc3 g a2 (CHead c0 (Bind 
-Abst) u) t0)).(conj (arity g c0 (THead (Bind Abst) u t0) (AHead a1 a2)) 
-(\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is: 
-PList).((drop1 is d c0) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is (THead 
-(Bind Abst) u t0))))))))) (arity_head g c0 u a1 H0 t0 a2 H2) (\lambda (d: 
-C).(\lambda (w: T).(\lambda (H4: (sc3 g a1 d w)).(\lambda (is: 
-PList).(\lambda (H5: (drop1 is d c0)).(let H6 \def (sc3_appl g a1 a2 TNil) in 
-(eq_ind_r T (THead (Bind Abst) (lift1 is u) (lift1 (Ss is) t0)) (\lambda (t1: 
-T).(sc3 g a2 d (THead (Flat Appl) w t1))) (H6 d w (lift1 (Ss is) t0) (let H_y 
-\def (sc3_bind g Abbr (\lambda (H7: (eq B Abbr Abst)).(not_abbr_abst H7)) a1 
-a2 TNil) in (H_y d w (lift1 (Ss is) t0) (let H7 \def (sc3_ceqc_trans g a2 
-TNil) in (H7 (CHead d (Bind Abst) (lift1 is u)) (lift1 (Ss is) t0) (sc3_lift1 
-g (CHead c0 (Bind Abst) u) a2 (Ss is) (CHead d (Bind Abst) (lift1 is u)) t0 
-H3 (drop1_skip_bind Abst c0 is d u H5)) (CHead d (Bind Abbr) w) (or_intror 
-(csubc g (CHead d (Bind Abbr) w) (CHead d (Bind Abst) (lift1 is u))) (csubc g 
-(CHead d (Bind Abst) (lift1 is u)) (CHead d (Bind Abbr) w)) (csubc_abst g d d 
-(csubc_refl g d) (lift1 is u) a1 (sc3_lift1 g c0 (asucc g a1) is d u H1 H5) w 
-H4)))) H4)) H4 (lift1 is u) (sc3_lift1 g c0 (asucc g a1) is d u H1 H5)) 
-(lift1 is (THead (Bind Abst) u t0)) (lift1_bind Abst is u 
-t0)))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: 
-A).(\lambda (_: (arity g c0 u a1)).(\lambda (H1: (sc3 g a1 c0 u)).(\lambda 
-(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda 
-(H3: (sc3 g (AHead a1 a2) c0 t0)).(let H4 \def H3 in (and_ind (arity g c0 t0 
-(AHead a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall 
-(is: PList).((drop1 is d c0) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is 
-t0)))))))) (sc3 g a2 c0 (THead (Flat Appl) u t0)) (\lambda (_: (arity g c0 t0 
-(AHead a1 a2))).(\lambda (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a1 d 
-w) \to (\forall (is: PList).((drop1 is d c0) \to (sc3 g a2 d (THead (Flat 
-Appl) w (lift1 is t0)))))))))).(let H_y \def (H6 c0 u H1 PNil) in (H_y 
-(drop1_nil c0))))) H4))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda 
-(a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda (H1: (sc3 g (asucc 
-g a0) c0 u)).(\lambda (t0: T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: 
-(sc3 g a0 c0 t0)).(let H_y \def (sc3_cast g a0 TNil) in (H_y c0 u H1 t0 
-H3)))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: 
-(arity g c0 t0 a1)).(\lambda (H1: (sc3 g a1 c0 t0)).(\lambda (a2: A).(\lambda 
-(H2: (leq g a1 a2)).(sc3_repl g a1 c0 t0 H1 a2 H2)))))))) c t a H))))).
+(arity g c t a)).(let H_y \def (sc3_arity_csubc g c t a H c PNil) in (H_y 
+(drop1_nil c) c (csubc_refl g c))))))).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/props.ma
index 3965938ed..aa28b068b 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/props.ma
@@ -223,12 +223,77 @@ c2 (lift1 p t) (H c2 t H0 H15) c n n0 H14))) c3 (sym_eq C c3 e H13))) c1
 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal 
 PList (PCons n n0 p)) (refl_equal C c) (refl_equal C e))))))))))) hds)))).
 
-axiom sc3_abbr:
+theorem sc3_abbr:
  \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (i: 
 nat).(\forall (d: C).(\forall (v: T).(\forall (c: C).((sc3 g a c (THeads 
 (Flat Appl) vs (lift (S i) O v))) \to ((getl i c (CHead d (Bind Abbr) v)) \to 
 (sc3 g a c (THeads (Flat Appl) vs (TLRef i)))))))))))
-.
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs: 
+TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c: 
+C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c 
+(CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs (TLRef 
+i))))))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (vs: 
+TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(\lambda (c: 
+C).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs (lift (S i) O v)) 
+(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (lift (S i) O v))))).(\lambda 
+(H0: (getl i c (CHead d (Bind Abbr) v))).(let H1 \def H in (and_ind (arity g 
+c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0)) (sn3 c (THeads (Flat 
+Appl) vs (lift (S i) O v))) (land (arity g c (THeads (Flat Appl) vs (TLRef 
+i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))) (\lambda (H2: 
+(arity g c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0))).(\lambda 
+(H3: (sn3 c (THeads (Flat Appl) vs (lift (S i) O v)))).(conj (arity g c 
+(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs 
+(TLRef i))) (arity_appls_abbr g c d v i H0 vs (ASort n n0) H2) 
+(sn3_appls_abbr c d v i H0 vs H3)))) H1))))))))))) (\lambda (a0: A).(\lambda 
+(_: ((\forall (vs: TList).(\forall (i: nat).(\forall (d: C).(\forall (v: 
+T).(\forall (c: C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to 
+((getl i c (CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs 
+(TLRef i)))))))))))).(\lambda (a1: A).(\lambda (H0: ((\forall (vs: 
+TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c: 
+C).((sc3 g a1 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c 
+(CHead d (Bind Abbr) v)) \to (sc3 g a1 c (THeads (Flat Appl) vs (TLRef 
+i)))))))))))).(\lambda (vs: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda 
+(v: T).(\lambda (c: C).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs 
+(lift (S i) O v)) (AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 
+d0 w) \to (\forall (is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat 
+Appl) w (lift1 is (THeads (Flat Appl) vs (lift (S i) O v)))))))))))).(\lambda 
+(H2: (getl i c (CHead d (Bind Abbr) v))).(let H3 \def H1 in (and_ind (arity g 
+c (THeads (Flat Appl) vs (lift (S i) O v)) (AHead a0 a1)) (\forall (d0: 
+C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) 
+\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift 
+(S i) O v)))))))))) (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead 
+a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: 
+PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is 
+(THeads (Flat Appl) vs (TLRef i))))))))))) (\lambda (H4: (arity g c (THeads 
+(Flat Appl) vs (lift (S i) O v)) (AHead a0 a1))).(\lambda (H5: ((\forall (d0: 
+C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) 
+\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift 
+(S i) O v)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (TLRef i)) 
+(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall 
+(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is 
+(THeads (Flat Appl) vs (TLRef i)))))))))) (arity_appls_abbr g c d v i H2 vs 
+(AHead a0 a1) H4) (\lambda (d0: C).(\lambda (w: T).(\lambda (H6: (sc3 g a0 d0 
+w)).(\lambda (is: PList).(\lambda (H7: (drop1 is d0 c)).(let H_x \def 
+(drop1_getl_trans is c d0 H7 Abbr d v i H2) in (let H8 \def H_x in (ex2_ind C 
+(\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2: C).(getl (trans is 
+i) d0 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) v)))) (sc3 g a1 d0 (THead 
+(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (x: 
+C).(\lambda (_: (drop1 (ptrans is i) x d)).(\lambda (H10: (getl (trans is i) 
+d0 (CHead x (Bind Abbr) (lift1 (ptrans is i) v)))).(let H_y \def (H0 (TCons w 
+(lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is 
+(TLRef i))) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w t))) (eq_ind_r 
+T (TLRef (trans is i)) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w 
+(THeads (Flat Appl) (lifts1 is vs) t)))) (H_y (trans is i) x (lift1 (ptrans 
+is i) v) d0 (eq_ind T (lift1 is (lift (S i) O v)) (\lambda (t: T).(sc3 g a1 
+d0 (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 is vs) t)))) (eq_ind T 
+(lift1 is (THeads (Flat Appl) vs (lift (S i) O v))) (\lambda (t: T).(sc3 g a1 
+d0 (THead (Flat Appl) w t))) (H5 d0 w H6 is H7) (THeads (Flat Appl) (lifts1 
+is vs) (lift1 is (lift (S i) O v))) (lifts1_flat Appl is (lift (S i) O v) 
+vs)) (lift (S (trans is i)) O (lift1 (ptrans is i) v)) (lift1_free is i v)) 
+H10) (lift1 is (TLRef i)) (lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs 
+(TLRef i))) (lifts1_flat Appl is (TLRef i) vs)))))) H8))))))))))) 
+H3))))))))))))) a)).
 
 theorem sc3_cast:
  \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall 
@@ -506,19 +571,163 @@ t)))))).(\lambda (H4: ((\forall (vs0: TList).(\forall (i0: nat).(let t \def
 ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a c0 t)))))))))).(H4 vs i 
 c H H0 H1))) H2)))))))))).
 
-axiom sc3_bind:
+theorem sc3_bind:
  \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (a1: 
 A).(\forall (a2: A).(\forall (vs: TList).(\forall (c: C).(\forall (v: 
 T).(\forall (t: T).((sc3 g a2 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts 
 (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a2 c (THeads (Flat Appl) vs 
 (THead (Bind b) v t)))))))))))))
-.
+\def
+ \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda 
+(a1: A).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (vs: TList).(\forall 
+(c: C).(\forall (v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads 
+(Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads 
+(Flat Appl) vs (THead (Bind b) v t)))))))))) (\lambda (n: nat).(\lambda (n0: 
+nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: 
+T).(\lambda (H0: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl) 
+(lifts (S O) O vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat 
+Appl) (lifts (S O) O vs) t)))).(\lambda (H1: (sc3 g a1 c v)).(let H2 \def H0 
+in (and_ind (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O 
+vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S 
+O) O vs) t)) (land (arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) 
+(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t)))) (\lambda 
+(H3: (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) 
+(ASort n n0))).(\lambda (H4: (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) 
+(lifts (S O) O vs) t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Bind 
+b) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t))) 
+(arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 H1) t vs (ASort n n0) 
+H3) (sn3_appls_bind b H c v (sc3_sn3 g a1 c v H1) vs t H4)))) H2)))))))))) 
+(\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall (c: C).(\forall 
+(v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads (Flat Appl) 
+(lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads (Flat Appl) 
+vs (THead (Bind b) v t))))))))))).(\lambda (a0: A).(\lambda (H1: ((\forall 
+(vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 (CHead 
+c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) 
+\to (sc3 g a0 c (THeads (Flat Appl) vs (THead (Bind b) v 
+t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda 
+(t: T).(\lambda (H2: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl) 
+(lifts (S O) O vs) t) (AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a 
+d w) \to (\forall (is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g 
+a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) 
+t))))))))))).(\lambda (H3: (sc3 g a1 c v)).(let H4 \def H2 in (and_ind (arity 
+g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) (AHead a 
+a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: 
+PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat Appl) 
+w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))) (land (arity g 
+c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) (\forall (d: 
+C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) 
+\to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead 
+(Bind b) v t))))))))))) (\lambda (H5: (arity g (CHead c (Bind b) v) (THeads 
+(Flat Appl) (lifts (S O) O vs) t) (AHead a a0))).(\lambda (H6: ((\forall (d: 
+C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d 
+(CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads 
+(Flat Appl) (lifts (S O) O vs) t))))))))))).(conj (arity g c (THeads (Flat 
+Appl) vs (THead (Bind b) v t)) (AHead a a0)) (\forall (d: C).(\forall (w: 
+T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d 
+(THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Bind b) v 
+t)))))))))) (arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 H3) t vs 
+(AHead a a0) H5) (\lambda (d: C).(\lambda (w: T).(\lambda (H7: (sc3 g a d 
+w)).(\lambda (is: PList).(\lambda (H8: (drop1 is d c)).(let H_y \def (H1 
+(TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) 
+(lift1 is (THead (Bind b) v t))) (\lambda (t0: T).(sc3 g a0 d (THead (Flat 
+Appl) w t0))) (eq_ind_r T (THead (Bind b) (lift1 is v) (lift1 (Ss is) t)) 
+(\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 
+is vs) t0)))) (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind TList (lifts1 (Ss 
+is) (lifts (S O) O vs)) (\lambda (t0: TList).(sc3 g a0 (CHead d (Bind b) 
+(lift1 is v)) (THead (Flat Appl) (lift (S O) O w) (THeads (Flat Appl) t0 
+(lift1 (Ss is) t))))) (eq_ind T (lift1 (Ss is) (THeads (Flat Appl) (lifts (S 
+O) O vs) t)) (\lambda (t0: T).(sc3 g a0 (CHead d (Bind b) (lift1 is v)) 
+(THead (Flat Appl) (lift (S O) O w) t0))) (H6 (CHead d (Bind b) (lift1 is v)) 
+(lift (S O) O w) (sc3_lift g a d w H7 (CHead d (Bind b) (lift1 is v)) (S O) O 
+(drop_drop (Bind b) O d d (drop_refl d) (lift1 is v))) (Ss is) 
+(drop1_skip_bind b c is d v H8)) (THeads (Flat Appl) (lifts1 (Ss is) (lifts 
+(S O) O vs)) (lift1 (Ss is) t)) (lifts1_flat Appl (Ss is) t (lifts (S O) O 
+vs))) (lifts (S O) O (lifts1 is vs)) (lifts1_xhg is vs)) (sc3_lift1 g c a1 is 
+d v H3 H8)) (lift1 is (THead (Bind b) v t)) (lift1_bind b is v t)) (lift1 is 
+(THeads (Flat Appl) vs (THead (Bind b) v t))) (lifts1_flat Appl is (THead 
+(Bind b) v t) vs))))))))))) H4)))))))))))) a2))))).
 
-axiom sc3_appl:
+theorem sc3_appl:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (vs: 
 TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 c (THeads 
 (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: 
 T).((sc3 g (asucc g a1) c w) \to (sc3 g a2 c (THeads (Flat Appl) vs (THead 
 (Flat Appl) v (THead (Bind Abst) w t))))))))))))))
-.
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(A_ind (\lambda (a: 
+A).(\forall (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 
+g a c (THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) 
+\to (\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a c (THeads (Flat 
+Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))))))))))) (\lambda 
+(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: 
+T).(\lambda (t: T).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs 
+(THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead 
+(Bind Abbr) v t))))).(\lambda (H0: (sc3 g a1 c v)).(\lambda (w: T).(\lambda 
+(H1: (sc3 g (asucc g a1) c w)).(let H2 \def H in (and_ind (arity g c (THeads 
+(Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat 
+Appl) vs (THead (Bind Abbr) v t))) (land (arity g c (THeads (Flat Appl) vs 
+(THead (Flat Appl) v (THead (Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads 
+(Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H3: 
+(arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n 
+n0))).(\lambda (H4: (sn3 c (THeads (Flat Appl) vs (THead (Bind Abbr) v 
+t)))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v (THead 
+(Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat 
+Appl) v (THead (Bind Abst) w t)))) (arity_appls_appl g c v a1 (sc3_arity_gen 
+g c v a1 H0) w (sc3_arity_gen g c w (asucc g a1) H1) t vs (ASort n n0) H3) 
+(sn3_appls_beta c v t vs H4 w (sc3_sn3 g (asucc g a1) c w H1))))) 
+H2)))))))))))) (\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall 
+(c: C).(\forall (v: T).(\forall (t: T).((sc3 g a c (THeads (Flat Appl) vs 
+(THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: T).((sc3 g 
+(asucc g a1) c w) \to (sc3 g a c (THeads (Flat Appl) vs (THead (Flat Appl) v 
+(THead (Bind Abst) w t)))))))))))))).(\lambda (a0: A).(\lambda (H0: ((\forall 
+(vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 c 
+(THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to 
+(\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a0 c (THeads (Flat Appl) 
+vs (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))))))).(\lambda (vs: 
+TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H1: (land 
+(arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (AHead a a0)) 
+(\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: 
+PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads 
+(Flat Appl) vs (THead (Bind Abbr) v t)))))))))))).(\lambda (H2: (sc3 g a1 c 
+v)).(\lambda (w: T).(\lambda (H3: (sc3 g (asucc g a1) c w)).(let H4 \def H1 
+in (and_ind (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (AHead 
+a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall (is: 
+PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is 
+(THeads (Flat Appl) vs (THead (Bind Abbr) v t)))))))))) (land (arity g c 
+(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))) (AHead 
+a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall (is: 
+PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is 
+(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w 
+t)))))))))))) (\lambda (H5: (arity g c (THeads (Flat Appl) vs (THead (Bind 
+Abbr) v t)) (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w0: 
+T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d 
+(THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Bind Abbr) v 
+t)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v 
+(THead (Bind Abst) w t))) (AHead a a0)) (\forall (d: C).(\forall (w0: 
+T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d 
+(THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Flat Appl) v 
+(THead (Bind Abst) w t))))))))))) (arity_appls_appl g c v a1 (sc3_arity_gen g 
+c v a1 H2) w (sc3_arity_gen g c w (asucc g a1) H3) t vs (AHead a a0) H5) 
+(\lambda (d: C).(\lambda (w0: T).(\lambda (H7: (sc3 g a d w0)).(\lambda (is: 
+PList).(\lambda (H8: (drop1 is d c)).(eq_ind_r T (THeads (Flat Appl) (lifts1 
+is vs) (lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t)))) (\lambda 
+(t0: T).(sc3 g a0 d (THead (Flat Appl) w0 t0))) (eq_ind_r T (THead (Flat 
+Appl) (lift1 is v) (lift1 is (THead (Bind Abst) w t))) (\lambda (t0: T).(sc3 
+g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) t0)))) 
+(eq_ind_r T (THead (Bind Abst) (lift1 is w) (lift1 (Ss is) t)) (\lambda (t0: 
+T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) 
+(THead (Flat Appl) (lift1 is v) t0))))) (let H_y \def (H0 (TCons w0 (lifts1 
+is vs))) in (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind T (lift1 is (THead 
+(Bind Abbr) v t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads 
+(Flat Appl) (lifts1 is vs) t0)))) (eq_ind T (lift1 is (THeads (Flat Appl) vs 
+(THead (Bind Abbr) v t))) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 
+t0))) (H6 d w0 H7 is H8) (THeads (Flat Appl) (lifts1 is vs) (lift1 is (THead 
+(Bind Abbr) v t))) (lifts1_flat Appl is (THead (Bind Abbr) v t) vs)) (THead 
+(Bind Abbr) (lift1 is v) (lift1 (Ss is) t)) (lift1_bind Abbr is v t)) 
+(sc3_lift1 g c a1 is d v H2 H8) (lift1 is w) (sc3_lift1 g c (asucc g a1) is d 
+w H3 H8))) (lift1 is (THead (Bind Abst) w t)) (lift1_bind Abst is w t)) 
+(lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t))) (lift1_flat Appl is 
+v (THead (Bind Abst) w t))) (lift1 is (THeads (Flat Appl) vs (THead (Flat 
+Appl) v (THead (Bind Abst) w t)))) (lifts1_flat Appl is (THead (Flat Appl) v 
+(THead (Bind Abst) w t)) vs)))))))))) H4)))))))))))))) a2))).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/defs.ma
index 46e2fda80..03b7e952a 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/defs.ma
@@ -18,7 +18,7 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/defs".
 
 include "pr3/defs.ma".
 
-inductive sn3 (c:C): T \to Prop \def
+inductive sn3 (c: C): T \to Prop \def
 | sn3_sing: \forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to (\forall 
 (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/fwd.ma
index 9b751cedf..32cb16386 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/fwd.ma
@@ -20,6 +20,59 @@ include "sn3/defs.ma".
 
 include "pr3/props.ma".
 
+theorem sn3_gen_bind:
+ \forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
+(THead (Bind b) u t)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: 
+(sn3 c (THead (Bind b) u t))).(insert_eq T (THead (Bind b) u t) (\lambda (t0: 
+T).(sn3 c t0)) (land (sn3 c u) (sn3 (CHead c (Bind b) u) t)) (\lambda (y: 
+T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T y (THead 
+(Bind b) u t0)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t0)))) (unintro 
+T u (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind b) t0 x)) \to 
+(land (sn3 c t0) (sn3 (CHead c (Bind b) t0) x))))) (sn3_ind c (\lambda (t0: 
+T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Bind b) x x0)) \to 
+(land (sn3 c x) (sn3 (CHead c (Bind b) x) x0)))))) (\lambda (t1: T).(\lambda 
+(H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 
+c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) 
+\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall 
+(x0: T).((eq T t2 (THead (Bind b) x x0)) \to (land (sn3 c x) (sn3 (CHead c 
+(Bind b) x) x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T 
+t1 (THead (Bind b) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0: 
+T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c 
+t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Bind b) x1 
+x2)) \to (land (sn3 c x1) (sn3 (CHead c (Bind b) x1) x2))))))))) H2 (THead 
+(Bind b) x x0) H3) in (let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall 
+(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to 
+(sn3 c t2))))) H1 (THead (Bind b) x x0) H3) in (conj (sn3 c x) (sn3 (CHead c 
+(Bind b) x) x0) (sn3_sing c x (\lambda (t2: T).(\lambda (H6: (((eq T x t2) 
+\to (\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x t2)).(let H8 \def (H4 
+(THead (Bind b) t2 x0) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind 
+b) t2 x0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | 
+(TLRef _) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind b) x 
+x0) (THead (Bind b) t2 x0) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: 
+T).(pr3 c x t0)) H7 x H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: 
+T).((eq T x t0) \to (\forall (P0: Prop).P0))) H6 x H9) in (H11 (refl_equal T 
+x) P)))))) (pr3_head_12 c x t2 H7 (Bind b) x0 x0 (pr3_refl (CHead c (Bind b) 
+t2) x0)) t2 x0 (refl_equal T (THead (Bind b) t2 x0))) in (and_ind (sn3 c t2) 
+(sn3 (CHead c (Bind b) t2) x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda 
+(_: (sn3 (CHead c (Bind b) t2) x0)).H9)) H8)))))) (sn3_sing (CHead c (Bind b) 
+x) x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P: 
+Prop).P)))).(\lambda (H7: (pr3 (CHead c (Bind b) x) x0 t2)).(let H8 \def (H4 
+(THead (Bind b) x t2) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind 
+b) x t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind b) x 
+x0) (THead (Bind b) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: 
+T).(pr3 (CHead c (Bind b) x) x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T 
+t2 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in 
+(H11 (refl_equal T x0) P)))))) (pr3_head_12 c x x (pr3_refl c x) (Bind b) x0 
+t2 H7) x t2 (refl_equal T (THead (Bind b) x t2))) in (and_ind (sn3 c x) (sn3 
+(CHead c (Bind b) x) t2) (sn3 (CHead c (Bind b) x) t2) (\lambda (_: (sn3 c 
+x)).(\lambda (H10: (sn3 (CHead c (Bind b) x) t2)).H10)) H8))))))))))))))) y 
+H0))))) H))))).
+
 theorem sn3_gen_flat:
  \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
 (THead (Flat f) u t)) \to (land (sn3 c u) (sn3 c t))))))
@@ -68,6 +121,35 @@ T x0) P)))))) (pr3_thin_dx c x0 t2 H7 x f) x t2 (refl_equal T (THead (Flat f)
 x t2))) in (and_ind (sn3 c x) (sn3 c t2) (sn3 c t2) (\lambda (_: (sn3 c 
 x)).(\lambda (H10: (sn3 c t2)).H10)) H8))))))))))))))) y H0))))) H))))).
 
+theorem sn3_gen_head:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
+(THead k u t)) \to (sn3 c u)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (c: C).(\forall (u: 
+T).(\forall (t: T).((sn3 c (THead k0 u t)) \to (sn3 c u)))))) (\lambda (b: 
+B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead 
+(Bind b) u t))).(let H_x \def (sn3_gen_bind b c u t H) in (let H0 \def H_x in 
+(and_ind (sn3 c u) (sn3 (CHead c (Bind b) u) t) (sn3 c u) (\lambda (H1: (sn3 
+c u)).(\lambda (_: (sn3 (CHead c (Bind b) u) t)).H1)) H0)))))))) (\lambda (f: 
+F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead 
+(Flat f) u t))).(let H_x \def (sn3_gen_flat f c u t H) in (let H0 \def H_x in 
+(and_ind (sn3 c u) (sn3 c t) (sn3 c u) (\lambda (H1: (sn3 c u)).(\lambda (_: 
+(sn3 c t)).H1)) H0)))))))) k).
+
+theorem sn3_gen_cflat:
+ \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 (CHead 
+c (Flat f) u) t) \to (sn3 c t)))))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: 
+(sn3 (CHead c (Flat f) u) t)).(sn3_ind (CHead c (Flat f) u) (\lambda (t0: 
+T).(sn3 c t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 
+t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to 
+(sn3 (CHead c (Flat f) u) t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T 
+t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to 
+(sn3 c t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) 
+\to (\forall (P: Prop).P)))).(\lambda (H3: (pr3 c t1 t2)).(H1 t2 H2 
+(pr3_cflat c t1 t2 H3 f u))))))))) t H))))).
+
 theorem sn3_gen_lift:
  \forall (c1: C).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((sn3 c1 
 (lift h d t)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t)))))))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/props.ma
index 6a21455d2..c72798ba0 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/props.ma
@@ -141,6 +141,477 @@ H11))) H15))))) H13))) t2 H9))))))) H8)) (\lambda (H8: (pr2 c t0
 t2)).(sn3_pr3_trans c t0 (sn3_sing c t0 H3) t2 (pr3_pr2 c t0 t2 H8))) 
 H7))))))))) t H2)))))) u H))).
 
+theorem sn3_cflat:
+ \forall (c: C).(\forall (t: T).((sn3 c t) \to (\forall (f: F).(\forall (u: 
+T).(sn3 (CHead c (Flat f) u) t)))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(\lambda (f: 
+F).(\lambda (u: T).(sn3_ind c (\lambda (t0: T).(sn3 (CHead c (Flat f) u) t0)) 
+(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall 
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
+(sn3 (CHead c (Flat f) u) t2)))))).(sn3_pr2_intro (CHead c (Flat f) u) t1 
+(\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) \to (\forall (P: 
+Prop).P)))).(\lambda (H3: (pr2 (CHead c (Flat f) u) t1 t2)).(H1 t2 H2 
+(pr3_pr2 c t1 t2 (pr2_gen_cflat f c u t1 t2 H3)))))))))) t H))))).
+
+theorem sn3_shift:
+ \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c 
+(THead (Bind b) v t)) \to (sn3 (CHead c (Bind b) v) t)))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: 
+(sn3 c (THead (Bind b) v t))).(let H_x \def (sn3_gen_bind b c v t H) in (let 
+H0 \def H_x in (and_ind (sn3 c v) (sn3 (CHead c (Bind b) v) t) (sn3 (CHead c 
+(Bind b) v) t) (\lambda (_: (sn3 c v)).(\lambda (H2: (sn3 (CHead c (Bind b) 
+v) t)).H2)) H0))))))).
+
+theorem sn3_change:
+ \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1: 
+T).(\forall (t: T).((sn3 (CHead c (Bind b) v1) t) \to (\forall (v2: T).(sn3 
+(CHead c (Bind b) v2) t)))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda 
+(v1: T).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c (Bind b) v1) t)).(\lambda 
+(v2: T).(sn3_ind (CHead c (Bind b) v1) (\lambda (t0: T).(sn3 (CHead c (Bind 
+b) v2) t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) 
+\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to (sn3 
+(CHead c (Bind b) v1) t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 
+t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to 
+(sn3 (CHead c (Bind b) v2) t2)))))).(sn3_pr2_intro (CHead c (Bind b) v2) t1 
+(\lambda (t2: T).(\lambda (H3: (((eq T t1 t2) \to (\forall (P: 
+Prop).P)))).(\lambda (H4: (pr2 (CHead c (Bind b) v2) t1 t2)).(H2 t2 H3 
+(pr3_pr2 (CHead c (Bind b) v1) t1 t2 (pr2_change b H c v2 t1 t2 H4 
+v1)))))))))) t H0))))))).
+
+theorem sn3_cpr3_trans:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
+(k: K).(\forall (t: T).((sn3 (CHead c k u1) t) \to (sn3 (CHead c k u2) 
+t)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 
+u2)).(\lambda (k: K).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c k u1) 
+t)).(sn3_ind (CHead c k u1) (\lambda (t0: T).(sn3 (CHead c k u2) t0)) 
+(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
+(P: Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u1) 
+t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
+Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u2) 
+t2)))))).(sn3_sing (CHead c k u2) t1 (\lambda (t2: T).(\lambda (H3: (((eq T 
+t1 t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 (CHead c k u2) t1 
+t2)).(H2 t2 H3 (pr3_pr3_pr3_t c u1 u2 H t1 t2 k H4))))))))) t H0))))))).
+
+theorem sn3_bind:
+ \forall (b: B).(\forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: 
+T).((sn3 (CHead c (Bind b) u) t) \to (sn3 c (THead (Bind b) u t)))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c 
+u)).(sn3_ind c (\lambda (t: T).(\forall (t0: T).((sn3 (CHead c (Bind b) t) 
+t0) \to (sn3 c (THead (Bind b) t t0))))) (\lambda (t1: T).(\lambda (_: 
+((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 
+t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to 
+(\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (t: T).((sn3 (CHead c 
+(Bind b) t2) t) \to (sn3 c (THead (Bind b) t2 t))))))))).(\lambda (t: 
+T).(\lambda (H2: (sn3 (CHead c (Bind b) t1) t)).(sn3_ind (CHead c (Bind b) 
+t1) (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (\lambda (t2: 
+T).(\lambda (H3: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: 
+Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind b) 
+t1) t3)))))).(\lambda (H4: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: 
+Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 c (THead (Bind b) 
+t1 t3))))))).(sn3_sing c (THead (Bind b) t1 t2) (\lambda (t3: T).(\lambda 
+(H5: (((eq T (THead (Bind b) t1 t2) t3) \to (\forall (P: Prop).P)))).(\lambda 
+(H6: (pr3 c (THead (Bind b) t1 t2) t3)).(let H_x \def (bind_dec_not b Abst) 
+in (let H7 \def H_x in (or_ind (eq B b Abst) (not (eq B b Abst)) (sn3 c t3) 
+(\lambda (H8: (eq B b Abst)).(let H9 \def (eq_ind B b (\lambda (b0: B).(pr3 c 
+(THead (Bind b0) t1 t2) t3)) H6 Abst H8) in (let H10 \def (eq_ind B b 
+(\lambda (b0: B).((eq T (THead (Bind b0) t1 t2) t3) \to (\forall (P: 
+Prop).P))) H5 Abst H8) in (let H11 \def (eq_ind B b (\lambda (b0: B).(\forall 
+(t4: T).((((eq T t2 t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind 
+b0) t1) t2 t4) \to (sn3 c (THead (Bind b0) t1 t4)))))) H4 Abst H8) in (let 
+H12 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T t2 t4) \to 
+(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) t2 t4) \to (sn3 
+(CHead c (Bind b0) t1) t4))))) H3 Abst H8) in (let H13 \def (eq_ind B b 
+(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P))) 
+\to ((pr3 c t1 t4) \to (\forall (t0: T).((sn3 (CHead c (Bind b0) t4) t0) \to 
+(sn3 c (THead (Bind b0) t4 t0)))))))) H1 Abst H8) in (let H14 \def 
+(pr3_gen_abst c t1 t2 t3 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: 
+T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall 
+(u0: T).(pr3 (CHead c (Bind b0) u0) t2 t4))))) (sn3 c t3) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H15: (eq T t3 (THead (Bind Abst) x0 
+x1))).(\lambda (H16: (pr3 c t1 x0)).(\lambda (H17: ((\forall (b0: B).(\forall 
+(u0: T).(pr3 (CHead c (Bind b0) u0) t2 x1))))).(let H18 \def (eq_ind T t3 
+(\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) t0) \to (\forall (P: 
+Prop).P))) H10 (THead (Bind Abst) x0 x1) H15) in (eq_ind_r T (THead (Bind 
+Abst) x0 x1) (\lambda (t0: T).(sn3 c t0)) (let H_x0 \def (term_dec t1 x0) in 
+(let H19 \def H_x0 in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: 
+Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H20: (eq T t1 x0)).(let 
+H21 \def (eq_ind_r T x0 (\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) 
+(THead (Bind Abst) t0 x1)) \to (\forall (P: Prop).P))) H18 t1 H20) in (let 
+H22 \def (eq_ind_r T x0 (\lambda (t0: T).(pr3 c t1 t0)) H16 t1 H20) in 
+(eq_ind T t1 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t0 x1))) (let H_x1 
+\def (term_dec t2 x1) in (let H23 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 
+x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abst) t1 x1)) (\lambda 
+(H24: (eq T t2 x1)).(let H25 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T 
+(THead (Bind Abst) t1 t2) (THead (Bind Abst) t1 t0)) \to (\forall (P: 
+Prop).P))) H21 t2 H24) in (let H26 \def (eq_ind_r T x1 (\lambda (t0: 
+T).(\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) 
+H17 t2 H24) in (eq_ind T t2 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t1 
+t0))) (H25 (refl_equal T (THead (Bind Abst) t1 t2)) (sn3 c (THead (Bind Abst) 
+t1 t2))) x1 H24)))) (\lambda (H24: (((eq T t2 x1) \to (\forall (P: 
+Prop).P)))).(H11 x1 H24 (H17 Abst t1))) H23))) x0 H20)))) (\lambda (H20: 
+(((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x1 \def (term_dec t2 x1) 
+in (let H21 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: 
+Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H22: (eq T t2 x1)).(let 
+H23 \def (eq_ind_r T x1 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: 
+T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) H17 t2 H22) in (eq_ind T t2 (\lambda 
+(t0: T).(sn3 c (THead (Bind Abst) x0 t0))) (H13 x0 H20 H16 t2 (sn3_cpr3_trans 
+c t1 x0 H16 (Bind Abst) t2 (sn3_sing (CHead c (Bind Abst) t1) t2 H12))) x1 
+H22))) (\lambda (H22: (((eq T t2 x1) \to (\forall (P: Prop).P)))).(H13 x0 H20 
+H16 x1 (sn3_cpr3_trans c t1 x0 H16 (Bind Abst) x1 (H12 x1 H22 (H17 Abst 
+t1))))) H21)))) H19))) t3 H15))))))) H14)))))))) (\lambda (H8: (not (eq B b 
+Abst))).(let H_x0 \def (pr3_gen_bind b H8 c t1 t2 t3 H6) in (let H9 \def H_x0 
+in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
+b) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: 
+T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) t2 t4)))) (pr3 (CHead c (Bind 
+b) t1) t2 (lift (S O) O t3)) (sn3 c t3) (\lambda (H10: (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 
+(CHead c (Bind b) t1) t2 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda 
+(t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 (CHead c (Bind b) 
+t1) t2 t4))) (sn3 c t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq 
+T t3 (THead (Bind b) x0 x1))).(\lambda (H12: (pr3 c t1 x0)).(\lambda (H13: 
+(pr3 (CHead c (Bind b) t1) t2 x1)).(let H14 \def (eq_ind T t3 (\lambda (t0: 
+T).((eq T (THead (Bind b) t1 t2) t0) \to (\forall (P: Prop).P))) H5 (THead 
+(Bind b) x0 x1) H11) in (eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: 
+T).(sn3 c t0)) (let H_x1 \def (term_dec t1 x0) in (let H15 \def H_x1 in 
+(or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: Prop).P)) (sn3 c (THead 
+(Bind b) x0 x1)) (\lambda (H16: (eq T t1 x0)).(let H17 \def (eq_ind_r T x0 
+(\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t0 x1)) \to 
+(\forall (P: Prop).P))) H14 t1 H16) in (let H18 \def (eq_ind_r T x0 (\lambda 
+(t0: T).(pr3 c t1 t0)) H12 t1 H16) in (eq_ind T t1 (\lambda (t0: T).(sn3 c 
+(THead (Bind b) t0 x1))) (let H_x2 \def (term_dec t2 x1) in (let H19 \def 
+H_x2 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: Prop).P)) (sn3 c 
+(THead (Bind b) t1 x1)) (\lambda (H20: (eq T t2 x1)).(let H21 \def (eq_ind_r 
+T x1 (\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t1 t0)) 
+\to (\forall (P: Prop).P))) H17 t2 H20) in (let H22 \def (eq_ind_r T x1 
+(\lambda (t0: T).(pr3 (CHead c (Bind b) t1) t2 t0)) H13 t2 H20) in (eq_ind T 
+t2 (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (H21 (refl_equal T (THead 
+(Bind b) t1 t2)) (sn3 c (THead (Bind b) t1 t2))) x1 H20)))) (\lambda (H20: 
+(((eq T t2 x1) \to (\forall (P: Prop).P)))).(H4 x1 H20 H13)) H19))) x0 
+H16)))) (\lambda (H16: (((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x2 
+\def (term_dec t2 x1) in (let H17 \def H_x2 in (or_ind (eq T t2 x1) ((eq T t2 
+x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind b) x0 x1)) (\lambda (H18: 
+(eq T t2 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t0: T).(pr3 (CHead c 
+(Bind b) t1) t2 t0)) H13 t2 H18) in (eq_ind T t2 (\lambda (t0: T).(sn3 c 
+(THead (Bind b) x0 t0))) (H1 x0 H16 H12 t2 (sn3_cpr3_trans c t1 x0 H12 (Bind 
+b) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3))) x1 H18))) (\lambda (H18: (((eq 
+T t2 x1) \to (\forall (P: Prop).P)))).(H1 x0 H16 H12 x1 (sn3_cpr3_trans c t1 
+x0 H12 (Bind b) x1 (H3 x1 H18 H13)))) H17)))) H15))) t3 H11))))))) H10)) 
+(\lambda (H10: (pr3 (CHead c (Bind b) t1) t2 (lift (S O) O 
+t3))).(sn3_gen_lift (CHead c (Bind b) t1) t3 (S O) O (sn3_pr3_trans (CHead c 
+(Bind b) t1) t2 (sn3_pr2_intro (CHead c (Bind b) t1) t2 (\lambda (t0: 
+T).(\lambda (H11: (((eq T t2 t0) \to (\forall (P: Prop).P)))).(\lambda (H12: 
+(pr2 (CHead c (Bind b) t1) t2 t0)).(H3 t0 H11 (pr3_pr2 (CHead c (Bind b) t1) 
+t2 t0 H12)))))) (lift (S O) O t3) H10) c (drop_drop (Bind b) O c c (drop_refl 
+c) t1))) H9)))) H7)))))))))) t H2)))))) u H)))).
+
+theorem sn3_beta:
+ \forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c (THead (Bind Abbr) v 
+t)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead 
+(Bind Abst) w t))))))))
+\def
+ \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead 
+(Bind Abbr) v t))).(insert_eq T (THead (Bind Abbr) v t) (\lambda (t0: T).(sn3 
+c t0)) (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead 
+(Bind Abst) w t))))) (\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t 
+(\lambda (t0: T).((eq T y (THead (Bind Abbr) v t0)) \to (\forall (w: T).((sn3 
+c w) \to (sn3 c (THead (Flat Appl) v (THead (Bind Abst) w t0))))))) (unintro 
+T v (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind Abbr) t0 x)) \to 
+(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) t0 (THead (Bind 
+Abst) w x)))))))) (sn3_ind c (\lambda (t0: T).(\forall (x: T).(\forall (x0: 
+T).((eq T t0 (THead (Bind Abbr) x x0)) \to (\forall (w: T).((sn3 c w) \to 
+(sn3 c (THead (Flat Appl) x (THead (Bind Abst) w x0))))))))) (\lambda (t1: 
+T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
+Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall 
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
+(\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Bind Abbr) x x0)) \to 
+(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) 
+w x0))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1 
+(THead (Bind Abbr) x x0))).(\lambda (w: T).(\lambda (H4: (sn3 c w)).(let H5 
+\def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to 
+(\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (\forall (x1: T).(\forall (x2: 
+T).((eq T t2 (THead (Bind Abbr) x1 x2)) \to (\forall (w0: T).((sn3 c w0) \to 
+(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) w0 x2)))))))))))) H2 (THead 
+(Bind Abbr) x x0) H3) in (let H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall 
+(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to 
+(sn3 c t2))))) H1 (THead (Bind Abbr) x x0) H3) in (sn3_ind c (\lambda (t0: 
+T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t0 x0)))) (\lambda (t2: 
+T).(\lambda (H7: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: 
+Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H8: ((\forall 
+(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to 
+(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t3 x0)))))))).(sn3_pr2_intro c 
+(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (\lambda (t3: T).(\lambda 
+(H9: (((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t3) \to (\forall 
+(P: Prop).P)))).(\lambda (H10: (pr2 c (THead (Flat Appl) x (THead (Bind Abst) 
+t2 x0)) t3)).(let H11 \def (pr2_gen_appl c x (THead (Bind Abst) t2 x0) t3 
+H10) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
+(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
+(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) t2 x0) t4)))) 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T (THead (Bind Abst) t2 x0) (THead (Bind Abst) y1 z1)))))) (\lambda 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
+(Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind 
+b) u) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c t3) 
+(\lambda (H12: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
+(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
+(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) t2 x0) 
+t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
+(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
+(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) t2 x0) t4))) (sn3 
+c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H13: (eq T t3 (THead (Flat 
+Appl) x1 x2))).(\lambda (H14: (pr2 c x x1)).(\lambda (H15: (pr2 c (THead 
+(Bind Abst) t2 x0) x2)).(let H16 \def (eq_ind T t3 (\lambda (t0: T).((eq T 
+(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P: 
+Prop).P))) H9 (THead (Flat Appl) x1 x2) H13) in (eq_ind_r T (THead (Flat 
+Appl) x1 x2) (\lambda (t0: T).(sn3 c t0)) (let H17 \def (pr2_gen_abst c t2 x0 
+x2 H15) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead 
+(Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t2 u2))) 
+(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead 
+c (Bind b) u) x0 t4))))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3: 
+T).(\lambda (x4: T).(\lambda (H18: (eq T x2 (THead (Bind Abst) x3 
+x4))).(\lambda (H19: (pr2 c t2 x3)).(\lambda (H20: ((\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c (Bind b) u) x0 x4))))).(let H21 \def (eq_ind T x2 
+(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) 
+(THead (Flat Appl) x1 t0)) \to (\forall (P: Prop).P))) H16 (THead (Bind Abst) 
+x3 x4) H18) in (eq_ind_r T (THead (Bind Abst) x3 x4) (\lambda (t0: T).(sn3 c 
+(THead (Flat Appl) x1 t0))) (let H_x \def (term_dec t2 x3) in (let H22 \def 
+H_x in (or_ind (eq T t2 x3) ((eq T t2 x3) \to (\forall (P: Prop).P)) (sn3 c 
+(THead (Flat Appl) x1 (THead (Bind Abst) x3 x4))) (\lambda (H23: (eq T t2 
+x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: T).((eq T (THead (Flat Appl) 
+x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) x1 (THead (Bind Abst) t0 
+x4))) \to (\forall (P: Prop).P))) H21 t2 H23) in (let H25 \def (eq_ind_r T x3 
+(\lambda (t0: T).(pr2 c t2 t0)) H19 t2 H23) in (eq_ind T t2 (\lambda (t0: 
+T).(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t0 x4)))) (let H_x0 \def 
+(term_dec x x1) in (let H26 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to 
+(\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t2 
+x4))) (\lambda (H27: (eq T x x1)).(let H28 \def (eq_ind_r T x1 (\lambda (t0: 
+T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) 
+t0 (THead (Bind Abst) t2 x4))) \to (\forall (P: Prop).P))) H24 x H27) in (let 
+H29 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H27) in (eq_ind 
+T x (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) t2 
+x4)))) (let H_x1 \def (term_dec x0 x4) in (let H30 \def H_x1 in (or_ind (eq T 
+x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x 
+(THead (Bind Abst) t2 x4))) (\lambda (H31: (eq T x0 x4)).(let H32 \def 
+(eq_ind_r T x4 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind 
+Abst) t2 x0)) (THead (Flat Appl) x (THead (Bind Abst) t2 t0))) \to (\forall 
+(P: Prop).P))) H28 x0 H31) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: 
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 
+H31) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead 
+(Bind Abst) t2 t0)))) (H32 (refl_equal T (THead (Flat Appl) x (THead (Bind 
+Abst) t2 x0))) (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)))) x4 
+H31)))) (\lambda (H31: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead 
+(Bind Abbr) x x4) (\lambda (H32: (eq T (THead (Bind Abbr) x x0) (THead (Bind 
+Abbr) x x4))).(\lambda (P: Prop).(let H33 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind 
+Abbr) x x0) (THead (Bind Abbr) x x4) H32) in (let H34 \def (eq_ind_r T x4 
+(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H31 x0 H33) in 
+(let H35 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: 
+T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H33) in (H34 (refl_equal T x0) 
+P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) 
+(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead 
+(Bind Abbr) x x4)) t2 (sn3_sing c t2 H7))) H30))) x1 H27)))) (\lambda (H27: 
+(((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind Abbr) x1 x4) 
+(\lambda (H28: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x1 
+x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e 
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) 
+\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
+(THead (Bind Abbr) x1 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind 
+Abbr) x x0) (THead (Bind Abbr) x1 x4) H28) in (\lambda (H31: (eq T x 
+x1)).(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (let H33 \def 
+(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) 
+H27 x H31) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 
+x H31) in (H33 (refl_equal T x) P)))))) H29)))) (pr3_head_12 c x x1 (pr3_pr2 
+c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20 
+Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) t2 (sn3_sing c t2 
+H7))) H26))) x3 H23)))) (\lambda (H23: (((eq T t2 x3) \to (\forall (P: 
+Prop).P)))).(let H_x0 \def (term_dec x x1) in (let H24 \def H_x0 in (or_ind 
+(eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) 
+x1 (THead (Bind Abst) x3 x4))) (\lambda (H25: (eq T x x1)).(let H26 \def 
+(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H25) in (eq_ind T x 
+(\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4)))) 
+(let H_x1 \def (term_dec x0 x4) in (let H27 \def H_x1 in (or_ind (eq T x0 x4) 
+((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x (THead 
+(Bind Abst) x3 x4))) (\lambda (H28: (eq T x0 x4)).(let H29 \def (eq_ind_r T 
+x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+x0 t0)))) H20 x0 H28) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat 
+Appl) x (THead (Bind Abst) x3 t0)))) (H8 x3 H23 (pr3_pr2 c t2 x3 H19)) x4 
+H28))) (\lambda (H28: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead 
+(Bind Abbr) x x4) (\lambda (H29: (eq T (THead (Bind Abbr) x x0) (THead (Bind 
+Abbr) x x4))).(\lambda (P: Prop).(let H30 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind 
+Abbr) x x0) (THead (Bind Abbr) x x4) H29) in (let H31 \def (eq_ind_r T x4 
+(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H28 x0 H30) in 
+(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: 
+T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (H31 (refl_equal T x0) 
+P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) 
+(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead 
+(Bind Abbr) x x4)) x3 (H7 x3 H23 (pr3_pr2 c t2 x3 H19)))) H27))) x1 H25))) 
+(\lambda (H25: (((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind 
+Abbr) x1 x4) (\lambda (H26: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
+x1 x4))).(\lambda (P: Prop).(let H27 \def (f_equal T T (\lambda (e: T).(match 
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) 
+\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
+(THead (Bind Abbr) x1 x4) H26) in ((let H28 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind 
+Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in (\lambda (H29: (eq T x 
+x1)).(let H30 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (let H31 \def 
+(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) 
+H25 x H29) in (let H32 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 
+x H29) in (H31 (refl_equal T x) P)))))) H27)))) (pr3_head_12 c x x1 (pr3_pr2 
+c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20 
+Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) x3 (H7 x3 H23 
+(pr3_pr2 c t2 x3 H19)))) H24)))) H22))) x2 H18))))))) H17)) t3 H13))))))) 
+H12)) (\lambda (H12: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T 
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind Abst) t2 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
+Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+z1 t4))))))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (x4: T).(\lambda (H13: (eq T (THead (Bind Abst) t2 x0) (THead 
+(Bind Abst) x1 x2))).(\lambda (H14: (eq T t3 (THead (Bind Abbr) x3 
+x4))).(\lambda (H15: (pr2 c x x3)).(\lambda (H16: ((\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H17 \def (eq_ind T t3 
+(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) 
+\to (\forall (P: Prop).P))) H9 (THead (Bind Abbr) x3 x4) H14) in (eq_ind_r T 
+(THead (Bind Abbr) x3 x4) (\lambda (t0: T).(sn3 c t0)) (let H18 \def (f_equal 
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) \Rightarrow t0])) 
+(THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in ((let H19 \def 
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
+[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) 
+\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in 
+(\lambda (_: (eq T t2 x1)).(let H21 \def (eq_ind_r T x2 (\lambda (t0: 
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t0 x4)))) H16 x0 
+H19) in (let H_x \def (term_dec x x3) in (let H22 \def H_x in (or_ind (eq T x 
+x3) ((eq T x x3) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abbr) x3 x4)) 
+(\lambda (H23: (eq T x x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: 
+T).(pr2 c x t0)) H15 x H23) in (eq_ind T x (\lambda (t0: T).(sn3 c (THead 
+(Bind Abbr) t0 x4))) (let H_x0 \def (term_dec x0 x4) in (let H25 \def H_x0 in 
+(or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead 
+(Bind Abbr) x x4)) (\lambda (H26: (eq T x0 x4)).(let H27 \def (eq_ind_r T x4 
+(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 
+t0)))) H21 x0 H26) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Bind Abbr) 
+x t0))) (sn3_sing c (THead (Bind Abbr) x x0) H6) x4 H26))) (\lambda (H26: 
+(((eq T x0 x4) \to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x x4) 
+(\lambda (H27: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x 
+x4))).(\lambda (P: Prop).(let H28 \def (f_equal T T (\lambda (e: T).(match e 
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) 
+\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
+(THead (Bind Abbr) x x4) H27) in (let H29 \def (eq_ind_r T x4 (\lambda (t0: 
+T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H26 x0 H28) in (let H30 \def 
+(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c 
+(Bind b) u) x0 t0)))) H21 x0 H28) in (H29 (refl_equal T x0) P)))))) (pr3_pr2 
+c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4 
+(Bind Abbr) (H21 Abbr x))))) H25))) x3 H23))) (\lambda (H23: (((eq T x x3) 
+\to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x3 x4) (\lambda (H24: (eq 
+T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x3 x4))).(\lambda (P: 
+Prop).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T return 
+(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | 
+(THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
+x3 x4) H24) in ((let H26 \def (f_equal T T (\lambda (e: T).(match e in T 
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) 
+\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
+(THead (Bind Abbr) x3 x4) H24) in (\lambda (H27: (eq T x x3)).(let H28 \def 
+(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c 
+(Bind b) u) x0 t0)))) H21 x0 H26) in (let H29 \def (eq_ind_r T x3 (\lambda 
+(t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H23 x H27) in (let H30 
+\def (eq_ind_r T x3 (\lambda (t0: T).(pr2 c x t0)) H15 x H27) in (H29 
+(refl_equal T x) P)))))) H25)))) (pr3_head_12 c x x3 (pr3_pr2 c x x3 H15) 
+(Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x3) x0 x4 (H21 Abbr x3))))) 
+H22)))))) H18)) t3 H14)))))))))) H12)) (\lambda (H12: (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) 
+y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3) 
+(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda 
+(x5: T).(\lambda (x6: T).(\lambda (H13: (not (eq B x1 Abst))).(\lambda (H14: 
+(eq T (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3))).(\lambda (H15: (eq 
+T t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda 
+(_: (pr2 c x x5)).(\lambda (H17: (pr2 c x2 x6)).(\lambda (H18: (pr2 (CHead c 
+(Bind x1) x6) x3 x4)).(let H19 \def (eq_ind T t3 (\lambda (t0: T).((eq T 
+(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P: 
+Prop).P))) H9 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) 
+H15) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) 
+x4)) (\lambda (t0: T).(sn3 c t0)) (let H20 \def (f_equal T B (\lambda (e: 
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | 
+(TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return 
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
+Abst])])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in ((let H21 
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
+with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) 
+\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in 
+((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
+T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ 
+t0) \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) 
+in (\lambda (H23: (eq T t2 x2)).(\lambda (H24: (eq B Abst x1)).(let H25 \def 
+(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H18 x0 
+H22) in (let H26 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H17 t2 
+H23) in (let H27 \def (eq_ind_r B x1 (\lambda (b: B).(pr2 (CHead c (Bind b) 
+x6) x0 x4)) H25 Abst H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b: 
+B).(not (eq B b Abst))) H13 Abst H24) in (eq_ind B Abst (\lambda (b: B).(sn3 
+c (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (let H29 
+\def (match (H28 (refl_equal B Abst)) in False return (\lambda (_: 
+False).(sn3 c (THead (Bind Abst) x6 (THead (Flat Appl) (lift (S O) O x5) 
+x4)))) with []) in H29) x1 H24)))))))) H21)) H20)) t3 H15)))))))))))))) H12)) 
+H11))))))))) w H4))))))))))) y H0))))) H)))).
+
 theorem sn3_appl_lref:
  \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (v: 
 T).((sn3 c v) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))
@@ -261,6 +732,195 @@ False])) I (THead (Bind x0) x1 x2) H8) in (False_ind (sn3 c (THead (Bind x0)
 x5 (THead (Flat Appl) (lift (S O) O x4) x3))) H14)) t2 H9)))))))))))))) H6)) 
 H5))))))))) v H0))))).
 
+theorem sn3_appl_abbr:
+ \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abbr) w)) \to (\forall (v: T).((sn3 c (THead (Flat Appl) v 
+(lift (S i) O w))) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda 
+(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (v: T).(\lambda (H0: (sn3 c 
+(THead (Flat Appl) v (lift (S i) O w)))).(insert_eq T (THead (Flat Appl) v 
+(lift (S i) O w)) (\lambda (t: T).(sn3 c t)) (sn3 c (THead (Flat Appl) v 
+(TLRef i))) (\lambda (y: T).(\lambda (H1: (sn3 c y)).(unintro T v (\lambda 
+(t: T).((eq T y (THead (Flat Appl) t (lift (S i) O w))) \to (sn3 c (THead 
+(Flat Appl) t (TLRef i))))) (sn3_ind c (\lambda (t: T).(\forall (x: T).((eq T 
+t (THead (Flat Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x 
+(TLRef i)))))) (\lambda (t1: T).(\lambda (H2: ((\forall (t2: T).((((eq T t1 
+t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c 
+t2)))))).(\lambda (H3: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
+Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).((eq T t2 (THead (Flat 
+Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x (TLRef 
+i)))))))))).(\lambda (x: T).(\lambda (H4: (eq T t1 (THead (Flat Appl) x (lift 
+(S i) O w)))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: 
+T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall 
+(x0: T).((eq T t2 (THead (Flat Appl) x0 (lift (S i) O w))) \to (sn3 c (THead 
+(Flat Appl) x0 (TLRef i))))))))) H3 (THead (Flat Appl) x (lift (S i) O w)) 
+H4) in (let H6 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: T).((((eq T t 
+t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (sn3 c t2))))) H2 
+(THead (Flat Appl) x (lift (S i) O w)) H4) in (sn3_pr2_intro c (THead (Flat 
+Appl) x (TLRef i)) (\lambda (t2: T).(\lambda (H7: (((eq T (THead (Flat Appl) 
+x (TLRef i)) t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr2 c (THead 
+(Flat Appl) x (TLRef i)) t2)).(let H9 \def (pr2_gen_appl c x (TLRef i) t2 H8) 
+in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x 
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq 
+T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) 
+(sn3 c t2) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T 
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: 
+T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H12: (pr2 c 
+x x0)).(\lambda (H13: (pr2 c (TLRef i) x1)).(let H14 \def (eq_ind T t2 
+(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: 
+Prop).P))) H7 (THead (Flat Appl) x0 x1) H11) in (eq_ind_r T (THead (Flat 
+Appl) x0 x1) (\lambda (t: T).(sn3 c t)) (let H15 \def (pr2_gen_lref c x1 i 
+H13) in (or_ind (eq T x1 (TLRef i)) (ex2_2 C T (\lambda (d0: C).(\lambda (u: 
+T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq 
+T x1 (lift (S i) O u))))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda (H16: 
+(eq T x1 (TLRef i))).(let H17 \def (eq_ind T x1 (\lambda (t: T).((eq T (THead 
+(Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P: 
+Prop).P))) H14 (TLRef i) H16) in (eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c 
+(THead (Flat Appl) x0 t))) (let H_x \def (term_dec x x0) in (let H18 \def H_x 
+in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c (THead 
+(Flat Appl) x0 (TLRef i))) (\lambda (H19: (eq T x x0)).(let H20 \def 
+(eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead 
+(Flat Appl) t (TLRef i))) \to (\forall (P: Prop).P))) H17 x H19) in (let H21 
+\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H19) in (eq_ind T x 
+(\lambda (t: T).(sn3 c (THead (Flat Appl) t (TLRef i)))) (H20 (refl_equal T 
+(THead (Flat Appl) x (TLRef i))) (sn3 c (THead (Flat Appl) x (TLRef i)))) x0 
+H19)))) (\lambda (H19: (((eq T x x0) \to (\forall (P: Prop).P)))).(H5 (THead 
+(Flat Appl) x0 (lift (S i) O w)) (\lambda (H20: (eq T (THead (Flat Appl) x 
+(lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)))).(\lambda (P: 
+Prop).(let H21 \def (f_equal T T (\lambda (e: T).(match e in T return 
+(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | 
+(THead _ t _) \Rightarrow t])) (THead (Flat Appl) x (lift (S i) O w)) (THead 
+(Flat Appl) x0 (lift (S i) O w)) H20) in (let H22 \def (eq_ind_r T x0 
+(\lambda (t: T).((eq T x t) \to (\forall (P0: Prop).P0))) H19 x H21) in (let 
+H23 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H21) in (H22 
+(refl_equal T x) P)))))) (pr3_pr2 c (THead (Flat Appl) x (lift (S i) O w)) 
+(THead (Flat Appl) x0 (lift (S i) O w)) (pr2_head_1 c x x0 H12 (Flat Appl) 
+(lift (S i) O w))) x0 (refl_equal T (THead (Flat Appl) x0 (lift (S i) O 
+w))))) H18))) x1 H16))) (\lambda (H16: (ex2_2 C T (\lambda (d0: C).(\lambda 
+(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
+T).(eq T x1 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda 
+(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
+T).(eq T x1 (lift (S i) O u)))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda 
+(x2: C).(\lambda (x3: T).(\lambda (H17: (getl i c (CHead x2 (Bind Abbr) 
+x3))).(\lambda (H18: (eq T x1 (lift (S i) O x3))).(let H19 \def (eq_ind T x1 
+(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 
+t)) \to (\forall (P: Prop).P))) H14 (lift (S i) O x3) H18) in (eq_ind_r T 
+(lift (S i) O x3) (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H20 
+\def (eq_ind C (CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H 
+(CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 
+(Bind Abbr) x3) H17)) in (let H21 \def (f_equal C C (\lambda (e: C).(match e 
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) 
+\Rightarrow c0])) (CHead d (Bind Abbr) w) (CHead x2 (Bind Abbr) x3) 
+(getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 (Bind Abbr) x3) H17)) in 
+((let H22 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: 
+C).T) with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d 
+(Bind Abbr) w) (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) 
+i H (CHead x2 (Bind Abbr) x3) H17)) in (\lambda (H23: (eq C d x2)).(let H24 
+\def (eq_ind_r T x3 (\lambda (t: T).(getl i c (CHead x2 (Bind Abbr) t))) H20 
+w H22) in (eq_ind T w (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 (lift (S 
+i) O t)))) (let H25 \def (eq_ind_r C x2 (\lambda (c0: C).(getl i c (CHead c0 
+(Bind Abbr) w))) H24 d H23) in (let H_x \def (term_dec x x0) in (let H26 \def 
+H_x in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c 
+(THead (Flat Appl) x0 (lift (S i) O w))) (\lambda (H27: (eq T x x0)).(let H28 
+\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H27) in (eq_ind T x 
+(\lambda (t: T).(sn3 c (THead (Flat Appl) t (lift (S i) O w)))) (sn3_sing c 
+(THead (Flat Appl) x (lift (S i) O w)) H6) x0 H27))) (\lambda (H27: (((eq T x 
+x0) \to (\forall (P: Prop).P)))).(H6 (THead (Flat Appl) x0 (lift (S i) O w)) 
+(\lambda (H28: (eq T (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat 
+Appl) x0 (lift (S i) O w)))).(\lambda (P: Prop).(let H29 \def (f_equal T T 
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t])) 
+(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O 
+w)) H28) in (let H30 \def (eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to 
+(\forall (P0: Prop).P0))) H27 x H29) in (let H31 \def (eq_ind_r T x0 (\lambda 
+(t: T).(pr2 c x t)) H12 x H29) in (H30 (refl_equal T x) P)))))) (pr3_pr2 c 
+(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O 
+w)) (pr2_head_1 c x x0 H12 (Flat Appl) (lift (S i) O w))))) H26)))) x3 
+H22)))) H21))) x1 H18)))))) H16)) H15)) t2 H11))))))) H10)) (\lambda (H10: 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 
+t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind 
+b) u) z1 t3))))))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (H11: (eq T (TLRef i) (THead (Bind Abst) x0 
+x1))).(\lambda (H12: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c 
+x x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) 
+u) x1 x3))))).(let H15 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat 
+Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 (THead (Bind Abbr) x2 
+x3) H12) in (eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) 
+(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 
+x1) H11) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H16)) t2 
+H12)))))))))) H10)) (\lambda (H10: (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind 
+B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq 
+T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) 
+(sn3 c t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 
+Abst))).(\lambda (H12: (eq T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda 
+(H13: (eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) 
+x3)))).(\lambda (_: (pr2 c x x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: 
+(pr2 (CHead c (Bind x0) x5) x2 x3)).(let H17 \def (eq_ind T t2 (\lambda (t: 
+T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 
+(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) H13) in 
+(eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) 
+(\lambda (t: T).(sn3 c t)) (let H18 \def (eq_ind T (TLRef i) (\lambda (ee: 
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
+(THead (Bind x0) x1 x2) H12) in (False_ind (sn3 c (THead (Bind x0) x5 (THead 
+(Flat Appl) (lift (S O) O x4) x3))) H18)) t2 H13)))))))))))))) H10)) 
+H9))))))))))))) y H1)))) H0))))))).
+
 theorem sn3_appl_cast:
  \forall (c: C).(\forall (v: T).(\forall (u: T).((sn3 c (THead (Flat Appl) v 
 u)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) v t)) \to (sn3 c (THead 
@@ -523,6 +1183,451 @@ H18) in (False_ind (sn3 c (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O)
 O x6) x5))) H24)) t2 H19)))))))))))))) H16)) H15))))))))))))))) y0 H5)))) 
 H4))))))))) y H0))))) H)))).
 
+theorem sn3_appl_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: 
+T).((sn3 c u) \to (\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) u) 
+(THead (Flat Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v 
+(THead (Bind b) u t))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda 
+(u: T).(\lambda (H0: (sn3 c u)).(sn3_ind c (\lambda (t: T).(\forall (t0: 
+T).(\forall (v: T).((sn3 (CHead c (Bind b) t) (THead (Flat Appl) (lift (S O) 
+O v) t0)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t t0))))))) 
+(\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall 
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
+(\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) t2) (THead (Flat 
+Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t2 
+t))))))))))).(\lambda (t: T).(\lambda (v: T).(\lambda (H3: (sn3 (CHead c 
+(Bind b) t1) (THead (Flat Appl) (lift (S O) O v) t))).(insert_eq T (THead 
+(Flat Appl) (lift (S O) O v) t) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) 
+t0)) (sn3 c (THead (Flat Appl) v (THead (Bind b) t1 t))) (\lambda (y: 
+T).(\lambda (H4: (sn3 (CHead c (Bind b) t1) y)).(unintro T t (\lambda (t0: 
+T).((eq T y (THead (Flat Appl) (lift (S O) O v) t0)) \to (sn3 c (THead (Flat 
+Appl) v (THead (Bind b) t1 t0))))) (unintro T v (\lambda (t0: T).(\forall (x: 
+T).((eq T y (THead (Flat Appl) (lift (S O) O t0) x)) \to (sn3 c (THead (Flat 
+Appl) t0 (THead (Bind b) t1 x)))))) (sn3_ind (CHead c (Bind b) t1) (\lambda 
+(t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat Appl) (lift 
+(S O) O x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) t1 x0))))))) 
+(\lambda (t2: T).(\lambda (H5: ((\forall (t3: T).((((eq T t2 t3) \to (\forall 
+(P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind 
+b) t1) t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t2 t3) \to (\forall 
+(P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (\forall (x: 
+T).(\forall (x0: T).((eq T t3 (THead (Flat Appl) (lift (S O) O x) x0)) \to 
+(sn3 c (THead (Flat Appl) x (THead (Bind b) t1 x0))))))))))).(\lambda (x: 
+T).(\lambda (x0: T).(\lambda (H7: (eq T t2 (THead (Flat Appl) (lift (S O) O 
+x) x0))).(let H8 \def (eq_ind T t2 (\lambda (t0: T).(\forall (t3: T).((((eq T 
+t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t0 t3) \to 
+(\forall (x1: T).(\forall (x2: T).((eq T t3 (THead (Flat Appl) (lift (S O) O 
+x1) x2)) \to (sn3 c (THead (Flat Appl) x1 (THead (Bind b) t1 x2)))))))))) H6 
+(THead (Flat Appl) (lift (S O) O x) x0) H7) in (let H9 \def (eq_ind T t2 
+(\lambda (t0: T).(\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) 
+\to ((pr3 (CHead c (Bind b) t1) t0 t3) \to (sn3 (CHead c (Bind b) t1) t3))))) 
+H5 (THead (Flat Appl) (lift (S O) O x) x0) H7) in (sn3_pr2_intro c (THead 
+(Flat Appl) x (THead (Bind b) t1 x0)) (\lambda (t3: T).(\lambda (H10: (((eq T 
+(THead (Flat Appl) x (THead (Bind b) t1 x0)) t3) \to (\forall (P: 
+Prop).P)))).(\lambda (H11: (pr2 c (THead (Flat Appl) x (THead (Bind b) t1 
+x0)) t3)).(let H12 \def (pr2_gen_appl c x (THead (Bind b) t1 x0) t3 H11) in 
+(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat 
+Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: 
+T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4)))) (ex4_4 T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
+Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) 
+u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
+b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead 
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
+b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2)))))))) (sn3 c t3) 
+(\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
+(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
+(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) 
+t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
+(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
+(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))) (sn3 c 
+t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H14: (eq T t3 (THead (Flat 
+Appl) x1 x2))).(\lambda (H15: (pr2 c x x1)).(\lambda (H16: (pr2 c (THead 
+(Bind b) t1 x0) x2)).(let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T 
+(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) 
+H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat Appl) x1 x2) 
+(\lambda (t0: T).(sn3 c t0)) (let H_x \def (pr3_gen_bind b H c t1 x0 x2) in 
+(let H18 \def (H_x (pr3_pr2 c (THead (Bind b) t1 x0) x2 H16)) in (or_ind 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 
+t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: 
+T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4)))) (pr3 (CHead c (Bind 
+b) t1) x0 (lift (S O) O x2)) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (H19: 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 
+t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: 
+T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 t4)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 
+(CHead c (Bind b) t1) x0 t4))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda 
+(x3: T).(\lambda (x4: T).(\lambda (H20: (eq T x2 (THead (Bind b) x3 
+x4))).(\lambda (H21: (pr3 c t1 x3)).(\lambda (H22: (pr3 (CHead c (Bind b) t1) 
+x0 x4)).(let H23 \def (eq_ind T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) 
+x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 t0)) \to (\forall (P: 
+Prop).P))) H17 (THead (Bind b) x3 x4) H20) in (eq_ind_r T (THead (Bind b) x3 
+x4) (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 t0))) (let H_x0 \def 
+(term_dec t1 x3) in (let H24 \def H_x0 in (or_ind (eq T t1 x3) ((eq T t1 x3) 
+\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind b) x3 
+x4))) (\lambda (H25: (eq T t1 x3)).(let H26 \def (eq_ind_r T x3 (\lambda (t0: 
+T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 
+(THead (Bind b) t0 x4))) \to (\forall (P: Prop).P))) H23 t1 H25) in (let H27 
+\def (eq_ind_r T x3 (\lambda (t0: T).(pr3 c t1 t0)) H21 t1 H25) in (eq_ind T 
+t1 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 (THead (Bind b) t0 x4)))) 
+(let H_x1 \def (term_dec x0 x4) in (let H28 \def H_x1 in (or_ind (eq T x0 x4) 
+((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead 
+(Bind b) t1 x4))) (\lambda (H29: (eq T x0 x4)).(let H30 \def (eq_ind_r T x4 
+(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead 
+(Flat Appl) x1 (THead (Bind b) t1 t0))) \to (\forall (P: Prop).P))) H26 x0 
+H29) in (let H31 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) 
+t1) x0 t0)) H22 x0 H29) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat 
+Appl) x1 (THead (Bind b) t1 t0)))) (let H_x2 \def (term_dec x x1) in (let H32 
+\def H_x2 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 
+c (THead (Flat Appl) x1 (THead (Bind b) t1 x0))) (\lambda (H33: (eq T x 
+x1)).(let H34 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) 
+x (THead (Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to 
+(\forall (P: Prop).P))) H30 x H33) in (let H35 \def (eq_ind_r T x1 (\lambda 
+(t0: T).(pr2 c x t0)) H15 x H33) in (eq_ind T x (\lambda (t0: T).(sn3 c 
+(THead (Flat Appl) t0 (THead (Bind b) t1 x0)))) (H34 (refl_equal T (THead 
+(Flat Appl) x (THead (Bind b) t1 x0))) (sn3 c (THead (Flat Appl) x (THead 
+(Bind b) t1 x0)))) x1 H33)))) (\lambda (H33: (((eq T x x1) \to (\forall (P: 
+Prop).P)))).(H8 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H34: (eq T 
+(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
+x0))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: T).(match e 
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map 
+(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
+lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow 
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
+t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
+(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
+t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) 
+\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
+(lift (S O) O x1) x0) H34) in (let H36 \def (eq_ind_r T x1 (\lambda (t0: 
+T).((eq T x t0) \to (\forall (P0: Prop).P0))) H33 x (lift_inj x x1 (S O) O 
+H35)) in (let H37 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x 
+(lift_inj x x1 (S O) O H35)) in (H36 (refl_equal T x) P)))))) (pr3_flat 
+(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c 
+(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 
+(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl) x1 x0 
+(refl_equal T (THead (Flat Appl) (lift (S O) O x1) x0)))) H32))) x4 H29)))) 
+(\lambda (H29: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H8 (THead (Flat 
+Appl) (lift (S O) O x1) x4) (\lambda (H30: (eq T (THead (Flat Appl) (lift (S 
+O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P: 
+Prop).(let H31 \def (f_equal T T (\lambda (e: T).(match e in T return 
+(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat 
+\to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
+lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow 
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
+t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
+(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
+t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) 
+\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
+(lift (S O) O x1) x4) H30) in ((let H32 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat 
+Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H30) in 
+(\lambda (H33: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H34 \def 
+(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) 
+H29 x0 H32) in (let H35 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T (THead 
+(Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 (THead (Bind b) 
+t1 t0))) \to (\forall (P0: Prop).P0))) H26 x0 H32) in (let H36 \def (eq_ind_r 
+T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) H22 x0 H32) in (let 
+H37 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead 
+(Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to (\forall 
+(P0: Prop).P0))) H35 x (lift_inj x x1 (S O) O H33)) in (let H38 \def 
+(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O 
+H33)) in (H34 (refl_equal T x0) P)))))))) H31)))) (pr3_flat (CHead c (Bind b) 
+t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S 
+O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) 
+x0 x4 H22 Appl) x1 x4 (refl_equal T (THead (Flat Appl) (lift (S O) O x1) 
+x4)))) H28))) x3 H25)))) (\lambda (H25: (((eq T t1 x3) \to (\forall (P: 
+Prop).P)))).(H2 x3 H25 H21 x4 x1 (sn3_cpr3_trans c t1 x3 H21 (Bind b) (THead 
+(Flat Appl) (lift (S O) O x1) x4) (let H_x1 \def (term_dec x0 x4) in (let H26 
+\def H_x1 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) 
+(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x4)) (\lambda 
+(H27: (eq T x0 x4)).(let H28 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead 
+c (Bind b) t1) x0 t0)) H22 x0 H27) in (eq_ind T x0 (\lambda (t0: T).(sn3 
+(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) t0))) (let H_x2 
+\def (term_dec x x1) in (let H29 \def H_x2 in (or_ind (eq T x x1) ((eq T x 
+x1) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) 
+(lift (S O) O x1) x0)) (\lambda (H30: (eq T x x1)).(let H31 \def (eq_ind_r T 
+x1 (\lambda (t0: T).(pr2 c x t0)) H15 x H30) in (eq_ind T x (\lambda (t0: 
+T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0))) 
+(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9) 
+x1 H30))) (\lambda (H30: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9 
+(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H31: (eq T (THead (Flat 
+Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
+x0))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e 
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map 
+(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
+lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow 
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
+t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
+(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
+t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) 
+\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
+(lift (S O) O x1) x0) H31) in (let H33 \def (eq_ind_r T x1 (\lambda (t0: 
+T).((eq T x t0) \to (\forall (P0: Prop).P0))) H30 x (lift_inj x x1 (S O) O 
+H32)) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x 
+(lift_inj x x1 (S O) O H32)) in (H33 (refl_equal T x) P)))))) (pr3_flat 
+(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c 
+(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 
+(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl))) 
+H29))) x4 H27))) (\lambda (H27: (((eq T x0 x4) \to (\forall (P: 
+Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x1) x4) (\lambda (H28: (eq T 
+(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
+x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e 
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map 
+(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
+lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow 
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
+t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
+(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
+t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) 
+\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
+(lift (S O) O x1) x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat 
+Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H28) in 
+(\lambda (H31: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H32 \def 
+(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) 
+H27 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c 
+(Bind b) t1) x0 t0)) H22 x0 H30) in (let H34 \def (eq_ind_r T x1 (\lambda 
+(t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H31)) in (H32 (refl_equal 
+T x0) P)))))) H29)))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift 
+(S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c 
+c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x4 H22 Appl))) H26)))))) 
+H24))) x2 H20))))))) H19)) (\lambda (H19: (pr3 (CHead c (Bind b) t1) x0 (lift 
+(S O) O x2))).(sn3_gen_lift (CHead c (Bind b) t1) (THead (Flat Appl) x1 x2) 
+(S O) O (eq_ind_r T (THead (Flat Appl) (lift (S O) O x1) (lift (S O) (s (Flat 
+Appl) O) x2)) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) t0)) (sn3_pr3_trans 
+(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x0) (let H_x0 \def 
+(term_dec x x1) in (let H20 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to 
+(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S 
+O) O x1) x0)) (\lambda (H21: (eq T x x1)).(let H22 \def (eq_ind_r T x1 
+(\lambda (t0: T).(pr2 c x t0)) H15 x H21) in (eq_ind T x (\lambda (t0: 
+T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0))) 
+(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9) 
+x1 H21))) (\lambda (H21: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9 
+(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H22: (eq T (THead (Flat 
+Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
+x0))).(\lambda (P: Prop).(let H23 \def (f_equal T T (\lambda (e: T).(match e 
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map 
+(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
+lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (TLRef _) \Rightarrow 
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
+t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
+(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
+t4))]) in lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (THead _ t0 _) 
+\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
+(lift (S O) O x1) x0) H22) in (let H24 \def (eq_ind_r T x1 (\lambda (t0: 
+T).((eq T x t0) \to (\forall (P0: Prop).P0))) H21 x (lift_inj x x1 (S O) O 
+H23)) in (let H25 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x 
+(lift_inj x x1 (S O) O H23)) in (H24 (refl_equal T x) P)))))) (pr3_flat 
+(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c 
+(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 
+(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl))) 
+H20))) (THead (Flat Appl) (lift (S O) O x1) (lift (S O) O x2)) (pr3_thin_dx 
+(CHead c (Bind b) t1) x0 (lift (S O) O x2) H19 (lift (S O) O x1) Appl)) (lift 
+(S O) O (THead (Flat Appl) x1 x2)) (lift_head (Flat Appl) x1 x2 (S O) O)) c 
+(drop_drop (Bind b) O c c (drop_refl c) t1))) H18))) t3 H14))))))) H13)) 
+(\lambda (H13: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: 
+T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: 
+T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))))).(ex4_4_ind T T T T (\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
+b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x 
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: 
+T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))) 
+(sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: 
+T).(\lambda (H14: (eq T (THead (Bind b) t1 x0) (THead (Bind Abst) x1 
+x2))).(\lambda (H15: (eq T t3 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c 
+x x3)).(\lambda (H17: ((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind 
+b0) u0) x2 x4))))).(let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T (THead 
+(Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) H10 
+(THead (Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4) 
+(\lambda (t0: T).(sn3 c t0)) (let H19 \def (f_equal T B (\lambda (e: 
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | 
+(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return 
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
+b])])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in ((let H20 
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
+with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) 
+\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in 
+((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
+T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ 
+t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) 
+in (\lambda (_: (eq T t1 x1)).(\lambda (H23: (eq B b Abst)).(let H24 \def 
+(eq_ind_r T x2 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead 
+c (Bind b0) u0) t0 x4)))) H17 x0 H21) in (let H25 \def (eq_ind B b (\lambda 
+(b0: B).((eq T (THead (Flat Appl) x (THead (Bind b0) t1 x0)) (THead (Bind 
+Abbr) x3 x4)) \to (\forall (P: Prop).P))) H18 Abst H23) in (let H26 \def 
+(eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T (THead (Flat Appl) 
+(lift (S O) O x) x0) t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind 
+b0) t1) (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (sn3 (CHead c (Bind 
+b0) t1) t4))))) H9 Abst H23) in (let H27 \def (eq_ind B b (\lambda (b0: 
+B).(\forall (t4: T).((((eq T (THead (Flat Appl) (lift (S O) O x) x0) t4) \to 
+(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) (THead (Flat Appl) 
+(lift (S O) O x) x0) t4) \to (\forall (x5: T).(\forall (x6: T).((eq T t4 
+(THead (Flat Appl) (lift (S O) O x5) x6)) \to (sn3 c (THead (Flat Appl) x5 
+(THead (Bind b0) t1 x6)))))))))) H8 Abst H23) in (let H28 \def (eq_ind B b 
+(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P))) 
+\to ((pr3 c t1 t4) \to (\forall (t0: T).(\forall (v0: T).((sn3 (CHead c (Bind 
+b0) t4) (THead (Flat Appl) (lift (S O) O v0) t0)) \to (sn3 c (THead (Flat 
+Appl) v0 (THead (Bind b0) t4 t0)))))))))) H2 Abst H23) in (let H29 \def 
+(eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H Abst H23) in (let H30 
+\def (match (H29 (refl_equal B Abst)) in False return (\lambda (_: 
+False).(sn3 c (THead (Bind Abbr) x3 x4))) with []) in H30)))))))))) H20)) 
+H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 B T T T T T (\lambda (b0: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) 
+t1 x0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+t3 (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 
+Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind 
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b0) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b0: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2))))))) (sn3 c t3) (\lambda (x1: 
+B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: 
+T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15: (eq T 
+(THead (Bind b) t1 x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T t3 
+(THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda 
+(H17: (pr2 c x x5)).(\lambda (H18: (pr2 c x2 x6)).(\lambda (H19: (pr2 (CHead 
+c (Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t0: T).((eq T 
+(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) 
+H10 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) H16) in 
+(eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) 
+(\lambda (t0: T).(sn3 c t0)) (let H21 \def (f_equal T B (\lambda (e: 
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | 
+(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return 
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
+b])])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in ((let H22 \def 
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
+[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) 
+\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in 
+((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
+T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ 
+t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in 
+(\lambda (H24: (eq T t1 x2)).(\lambda (H25: (eq B b x1)).(let H26 \def 
+(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H19 x0 
+H23) in (let H27 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H18 t1 
+H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b0: B).(pr2 (CHead c (Bind b0) 
+x6) x0 x4)) H26 b H25) in (eq_ind B b (\lambda (b0: B).(sn3 c (THead (Bind 
+b0) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (sn3_pr3_trans c (THead 
+(Bind b) t1 (THead (Flat Appl) (lift (S O) O x5) x4)) (sn3_bind b c t1 
+(sn3_sing c t1 H1) (THead (Flat Appl) (lift (S O) O x5) x4) (let H_x \def 
+(term_dec x x5) in (let H29 \def H_x in (or_ind (eq T x x5) ((eq T x x5) \to 
+(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S 
+O) O x5) x4)) (\lambda (H30: (eq T x x5)).(let H31 \def (eq_ind_r T x5 
+(\lambda (t0: T).(pr2 c x t0)) H17 x H30) in (eq_ind T x (\lambda (t0: 
+T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x4))) (let 
+H_x0 \def (term_dec x0 x4) in (let H32 \def H_x0 in (or_ind (eq T x0 x4) ((eq 
+T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat 
+Appl) (lift (S O) O x) x4)) (\lambda (H33: (eq T x0 x4)).(let H34 \def 
+(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0 
+H33) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat 
+Appl) (lift (S O) O x) t0))) (sn3_sing (CHead c (Bind b) t1) (THead (Flat 
+Appl) (lift (S O) O x) x0) H9) x4 H33))) (\lambda (H33: (((eq T x0 x4) \to 
+(\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x) x4) (\lambda 
+(H34: (eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift 
+(S O) O x) x4))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat 
+Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x) x4) H34) in 
+(let H36 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: 
+Prop).P0))) H33 x0 H35) in (let H37 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2 
+(CHead c (Bind b) x6) x0 t0)) H28 x0 H35) in (H36 (refl_equal T x0) P)))))) 
+(pr3_pr3_pr3_t c t1 x6 (pr3_pr2 c t1 x6 H27) (THead (Flat Appl) (lift (S O) O 
+x) x0) (THead (Flat Appl) (lift (S O) O x) x4) (Bind b) (pr3_pr2 (CHead c 
+(Bind b) x6) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift 
+(S O) O x) x4) (pr2_thin_dx (CHead c (Bind b) x6) x0 x4 H28 (lift (S O) O x) 
+Appl))))) H32))) x5 H30))) (\lambda (H30: (((eq T x x5) \to (\forall (P: 
+Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x5) x4) (\lambda (H31: (eq T 
+(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) 
+x4))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e 
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map 
+(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
+lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (TLRef _) \Rightarrow 
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
+t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
+(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
+t4))]) in lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (THead _ t0 _) 
+\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
+(lift (S O) O x5) x4) H31) in ((let H33 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat 
+Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) x4) H31) in 
+(\lambda (H34: (eq T (lift (S O) O x) (lift (S O) O x5))).(let H35 \def 
+(eq_ind_r T x5 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) 
+H30 x (lift_inj x x5 (S O) O H34)) in (let H36 \def (eq_ind_r T x5 (\lambda 
+(t0: T).(pr2 c x t0)) H17 x (lift_inj x x5 (S O) O H34)) in (let H37 \def 
+(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0 
+H33) in (H35 (refl_equal T x) P)))))) H32)))) (pr3_pr3_pr3_t c t1 x6 (pr3_pr2 
+c t1 x6 H27) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift 
+(S O) O x5) x4) (Bind b) (pr3_flat (CHead c (Bind b) x6) (lift (S O) O x) 
+(lift (S O) O x5) (pr3_lift (CHead c (Bind b) x6) c (S O) O (drop_drop (Bind 
+b) O c c (drop_refl c) x6) x x5 (pr3_pr2 c x x5 H17)) x0 x4 (pr3_pr2 (CHead c 
+(Bind b) x6) x0 x4 H28) Appl)))) H29)))) (THead (Bind b) x6 (THead (Flat 
+Appl) (lift (S O) O x5) x4)) (pr3_pr2 c (THead (Bind b) t1 (THead (Flat Appl) 
+(lift (S O) O x5) x4)) (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O 
+x5) x4)) (pr2_head_1 c t1 x6 H27 (Bind b) (THead (Flat Appl) (lift (S O) O 
+x5) x4)))) x1 H25))))))) H22)) H21)) t3 H16)))))))))))))) H13)) 
+H12)))))))))))))) y H4))))) H3))))))) u H0))))).
+
 theorem sn3_appl_appl:
  \forall (v1: T).(\forall (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in 
 (\forall (c: C).((sn3 c u1) \to (\forall (v2: T).((sn3 c v2) \to (((\forall 
@@ -991,6 +2096,26 @@ H15) in (False_ind (sn3 c (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O)
 O x5) x4))) H21)) t3 H16)))))))))))))) H13)) H12)))))))))))) v2 H4))))))))) y 
 H0))))) H))))).
 
+theorem sn3_appl_beta:
+ \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((sn3 c 
+(THead (Flat Appl) u (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) 
+\to (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind Abst) w 
+t))))))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H: 
+(sn3 c (THead (Flat Appl) u (THead (Bind Abbr) v t)))).(\lambda (w: 
+T).(\lambda (H0: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THead (Bind 
+Abbr) v t) H) in (let H1 \def H_x in (and_ind (sn3 c u) (sn3 c (THead (Bind 
+Abbr) v t)) (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind 
+Abst) w t)))) (\lambda (H2: (sn3 c u)).(\lambda (H3: (sn3 c (THead (Bind 
+Abbr) v t))).(sn3_appl_appl v (THead (Bind Abst) w t) c (sn3_beta c v t H3 w 
+H0) u H2 (\lambda (u2: T).(\lambda (H4: (pr3 c (THead (Flat Appl) v (THead 
+(Bind Abst) w t)) u2)).(\lambda (H5: (((iso (THead (Flat Appl) v (THead (Bind 
+Abst) w t)) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat 
+Appl) u (THead (Bind Abbr) v t)) H (THead (Flat Appl) u u2) (pr3_thin_dx c 
+(THead (Bind Abbr) v t) u2 (pr3_iso_beta v w t c u2 H4 H5) u Appl)))))))) 
+H1))))))))).
+
 theorem sn3_appl_appls:
  \forall (v1: T).(\forall (t1: T).(\forall (vs: TList).(let u1 \def (THeads 
 (Flat Appl) (TCons v1 vs) t1) in (\forall (c: C).((sn3 c u1) \to (\forall 
@@ -1090,6 +2215,106 @@ Appl) (TCons t1 t2) (THead (Flat Cast) u t3)) u2)).(\lambda (H12: (((iso
 Appl) (TCons t1 t2) t3) u2 (pr3_iso_appls_cast c u t3 (TCons t1 t2) u2 H11 
 H12) t Appl))))))))) H7)))))) H3))))))))))) t0))) vs)).
 
+theorem sn3_appls_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: 
+T).((sn3 c u) \to (\forall (vs: TList).(\forall (t: T).((sn3 (CHead c (Bind 
+b) u) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to (sn3 c (THeads (Flat 
+Appl) vs (THead (Bind b) u t))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda 
+(u: T).(\lambda (H0: (sn3 c u)).(\lambda (vs: TList).(TList_ind (\lambda (t: 
+TList).(\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) (lifts 
+(S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u t0)))))) 
+(\lambda (t: T).(\lambda (H1: (sn3 (CHead c (Bind b) u) t)).(sn3_bind b c u 
+H0 t H1))) (\lambda (v: T).(\lambda (vs0: TList).(TList_ind (\lambda (t: 
+TList).(((\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) 
+(lifts (S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u 
+t0)))))) \to (\forall (t0: T).((sn3 (CHead c (Bind b) u) (THead (Flat Appl) 
+(lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t) t0))) \to (sn3 c 
+(THead (Flat Appl) v (THeads (Flat Appl) t (THead (Bind b) u t0)))))))) 
+(\lambda (_: ((\forall (t: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) 
+(lifts (S O) O TNil) t)) \to (sn3 c (THeads (Flat Appl) TNil (THead (Bind b) 
+u t))))))).(\lambda (t: T).(\lambda (H2: (sn3 (CHead c (Bind b) u) (THead 
+(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O TNil) 
+t)))).(sn3_appl_bind b H c u H0 t v H2)))) (\lambda (t: T).(\lambda (t0: 
+TList).(\lambda (_: ((((\forall (t1: T).((sn3 (CHead c (Bind b) u) (THeads 
+(Flat Appl) (lifts (S O) O t0) t1)) \to (sn3 c (THeads (Flat Appl) t0 (THead 
+(Bind b) u t1)))))) \to (\forall (t1: T).((sn3 (CHead c (Bind b) u) (THead 
+(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t0) t1))) \to 
+(sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (THead (Bind b) u 
+t1))))))))).(\lambda (H2: ((\forall (t1: T).((sn3 (CHead c (Bind b) u) 
+(THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) \to (sn3 c (THeads 
+(Flat Appl) (TCons t t0) (THead (Bind b) u t1))))))).(\lambda (t1: 
+T).(\lambda (H3: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O 
+v) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))).(let H_x \def 
+(sn3_gen_flat Appl (CHead c (Bind b) u) (lift (S O) O v) (THeads (Flat Appl) 
+(lifts (S O) O (TCons t t0)) t1) H3) in (let H4 \def H_x in (and_ind (sn3 
+(CHead c (Bind b) u) (lift (S O) O v)) (sn3 (CHead c (Bind b) u) (THeads 
+(Flat Appl) (lifts (S O) O (TCons t t0)) t1)) (sn3 c (THead (Flat Appl) v 
+(THeads (Flat Appl) (TCons t t0) (THead (Bind b) u t1)))) (\lambda (H5: (sn3 
+(CHead c (Bind b) u) (lift (S O) O v))).(\lambda (H6: (sn3 (CHead c (Bind b) 
+u) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1))).(let H_y \def 
+(sn3_gen_lift (CHead c (Bind b) u) v (S O) O H5 c) in (sn3_appl_appls t 
+(THead (Bind b) u t1) t0 c (H2 t1 H6) v (H_y (drop_drop (Bind b) O c c 
+(drop_refl c) u)) (\lambda (u2: T).(\lambda (H7: (pr3 c (THeads (Flat Appl) 
+(TCons t t0) (THead (Bind b) u t1)) u2)).(\lambda (H8: (((iso (THeads (Flat 
+Appl) (TCons t t0) (THead (Bind b) u t1)) u2) \to (\forall (P: 
+Prop).P)))).(let H9 \def (pr3_iso_appls_bind b H (TCons t t0) u t1 c u2 H7 
+H8) in (sn3_pr3_trans c (THead (Flat Appl) v (THead (Bind b) u (THeads (Flat 
+Appl) (lifts (S O) O (TCons t t0)) t1))) (sn3_appl_bind b H c u H0 (THeads 
+(Flat Appl) (lifts (S O) O (TCons t t0)) t1) v H3) (THead (Flat Appl) v u2) 
+(pr3_flat c v v (pr3_refl c v) (THead (Bind b) u (THeads (Flat Appl) (lifts 
+(S O) O (TCons t t0)) t1)) u2 H9 Appl)))))))))) H4))))))))) vs0))) vs)))))).
+
+theorem sn3_appls_beta:
+ \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (us: TList).((sn3 c 
+(THeads (Flat Appl) us (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c 
+w) \to (sn3 c (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) 
+w t))))))))))
+\def
+ \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (us: 
+TList).(TList_ind (\lambda (t0: TList).((sn3 c (THeads (Flat Appl) t0 (THead 
+(Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat 
+Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) (\lambda (H: 
+(sn3 c (THead (Bind Abbr) v t))).(\lambda (w: T).(\lambda (H0: (sn3 c 
+w)).(sn3_beta c v t H w H0)))) (\lambda (u: T).(\lambda (us0: 
+TList).(TList_ind (\lambda (t0: TList).((((sn3 c (THeads (Flat Appl) t0 
+(THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads 
+(Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 
+c (THead (Flat Appl) u (THeads (Flat Appl) t0 (THead (Bind Abbr) v t)))) \to 
+(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat 
+Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))) (\lambda (_: 
+(((sn3 c (THeads (Flat Appl) TNil (THead (Bind Abbr) v t))) \to (\forall (w: 
+T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) TNil (THead (Flat Appl) v (THead 
+(Bind Abst) w t))))))))).(\lambda (H0: (sn3 c (THead (Flat Appl) u (THeads 
+(Flat Appl) TNil (THead (Bind Abbr) v t))))).(\lambda (w: T).(\lambda (H1: 
+(sn3 c w)).(sn3_appl_beta c u v t H0 w H1))))) (\lambda (t0: T).(\lambda (t1: 
+TList).(\lambda (_: (((((sn3 c (THeads (Flat Appl) t1 (THead (Bind Abbr) v 
+t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) t1 (THead 
+(Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 c (THead (Flat Appl) u 
+(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)))) \to (\forall (w: T).((sn3 c 
+w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) t1 (THead (Flat Appl) 
+v (THead (Bind Abst) w t))))))))))).(\lambda (H0: (((sn3 c (THeads (Flat 
+Appl) (TCons t0 t1) (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) 
+\to (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead 
+(Bind Abst) w t))))))))).(\lambda (H1: (sn3 c (THead (Flat Appl) u (THeads 
+(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t))))).(\lambda (w: 
+T).(\lambda (H2: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THeads 
+(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t)) H1) in (let H3 \def H_x in 
+(and_ind (sn3 c u) (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Bind 
+Abbr) v t))) (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) 
+(THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H4: (sn3 c 
+u)).(\lambda (H5: (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) 
+v t)))).(sn3_appl_appls t0 (THead (Flat Appl) v (THead (Bind Abst) w t)) t1 c 
+(H0 H5 w H2) u H4 (\lambda (u2: T).(\lambda (H6: (pr3 c (THeads (Flat Appl) 
+(TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w t))) u2)).(\lambda 
+(H7: (((iso (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead 
+(Bind Abst) w t))) u2) \to (\forall (P: Prop).P)))).(let H8 \def 
+(pr3_iso_appls_beta (TCons t0 t1) v w t c u2 H6 H7) in (sn3_pr3_trans c 
+(THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v 
+t))) H1 (THead (Flat Appl) u u2) (pr3_thin_dx c (THeads (Flat Appl) (TCons t0 
+t1) (THead (Bind Abbr) v t)) u2 H8 u Appl))))))))) H3)))))))))) us0))) us)))).
+
 theorem sn3_lift:
  \forall (d: C).(\forall (t: T).((sn3 d t) \to (\forall (c: C).(\forall (h: 
 nat).(\forall (i: nat).((drop h i c d) \to (sn3 c (lift h i t))))))))
@@ -1155,6 +2380,44 @@ H10) in (eq_ind T v (\lambda (t: T).(sn3 c (lift (S i) O t))) (let H13 \def
 H11) in (sn3_lift d v H0 c (S i) O (getl_drop Abbr c d v i H13))) x1 H10)))) 
 H9))) t2 H6)))))) H4)) H3))))))))))).
 
+theorem sn3_appls_abbr:
+ \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).((sn3 c (THeads (Flat Appl) 
+vs (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda 
+(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind 
+(\lambda (t: TList).((sn3 c (THeads (Flat Appl) t (lift (S i) O w))) \to (sn3 
+c (THeads (Flat Appl) t (TLRef i))))) (\lambda (H0: (sn3 c (lift (S i) O 
+w))).(let H_y \def (sn3_gen_lift c w (S i) O H0 d (getl_drop Abbr c d w i H)) 
+in (sn3_abbr c d w i H H_y))) (\lambda (v: T).(\lambda (vs0: 
+TList).(TList_ind (\lambda (t: TList).((((sn3 c (THeads (Flat Appl) t (lift 
+(S i) O w))) \to (sn3 c (THeads (Flat Appl) t (TLRef i))))) \to ((sn3 c 
+(THead (Flat Appl) v (THeads (Flat Appl) t (lift (S i) O w)))) \to (sn3 c 
+(THead (Flat Appl) v (THeads (Flat Appl) t (TLRef i))))))) (\lambda (_: 
+(((sn3 c (THeads (Flat Appl) TNil (lift (S i) O w))) \to (sn3 c (THeads (Flat 
+Appl) TNil (TLRef i)))))).(\lambda (H1: (sn3 c (THead (Flat Appl) v (THeads 
+(Flat Appl) TNil (lift (S i) O w))))).(sn3_appl_abbr c d w i H v H1))) 
+(\lambda (t: T).(\lambda (t0: TList).(\lambda (_: (((((sn3 c (THeads (Flat 
+Appl) t0 (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) t0 (TLRef i))))) 
+\to ((sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (lift (S i) O w)))) 
+\to (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (TLRef 
+i)))))))).(\lambda (H1: (((sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i) 
+O w))) \to (sn3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)))))).(\lambda 
+(H2: (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (lift (S i) 
+O w))))).(let H_x \def (sn3_gen_flat Appl c v (THeads (Flat Appl) (TCons t 
+t0) (lift (S i) O w)) H2) in (let H3 \def H_x in (and_ind (sn3 c v) (sn3 c 
+(THeads (Flat Appl) (TCons t t0) (lift (S i) O w))) (sn3 c (THead (Flat Appl) 
+v (THeads (Flat Appl) (TCons t t0) (TLRef i)))) (\lambda (H4: (sn3 c 
+v)).(\lambda (H5: (sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i) O 
+w)))).(sn3_appl_appls t (TLRef i) t0 c (H1 H5) v H4 (\lambda (u2: T).(\lambda 
+(H6: (pr3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2)).(\lambda (H7: 
+(((iso (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2) \to (\forall (P: 
+Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) v (THeads (Flat Appl) (TCons 
+t t0) (lift (S i) O w))) H2 (THead (Flat Appl) v u2) (pr3_thin_dx c (THeads 
+(Flat Appl) (TCons t t0) (lift (S i) O w)) u2 (pr3_iso_appls_abbr c d w i H 
+(TCons t t0) u2 H6 H7) v Appl)))))))) H3)))))))) vs0))) vs)))))).
+
 theorem sns3_lifts:
  \forall (c: C).(\forall (d: C).(\forall (h: nat).(\forall (i: nat).((drop h 
 i c d) \to (\forall (ts: TList).((sns3 d ts) \to (sns3 c (lifts h i ts))))))))
@@ -1169,3 +2432,68 @@ H1 in (and_ind (sn3 d t) (sns3 d t0) (land (sn3 c (lift h i t)) (sns3 c
 (sn3 c (lift h i t)) (sns3 c (lifts h i t0)) (sn3_lift d t H3 c h i H) (H0 
 H4)))) H2)))))) ts)))))).
 
+theorem sn3_gen_def:
+ \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abbr) v)) \to ((sn3 c (TLRef i)) \to (sn3 d v))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda 
+(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 c (TLRef 
+i))).(sn3_gen_lift c v (S i) O (sn3_pr3_trans c (TLRef i) H0 (lift (S i) O v) 
+(pr3_pr2 c (TLRef i) (lift (S i) O v) (pr2_delta c d v i H (TLRef i) (TLRef 
+i) (pr0_refl (TLRef i)) (lift (S i) O v) (subst0_lref v i)))) d (getl_drop 
+Abbr c d v i H))))))).
+
+theorem sn3_cdelta:
+ \forall (v: T).(\forall (t: T).(\forall (i: nat).(((\forall (w: T).(ex T 
+(\lambda (u: T).(subst0 i w t u))))) \to (\forall (c: C).(\forall (d: 
+C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to (sn3 d v))))))))
+\def
+ \lambda (v: T).(\lambda (t: T).(\lambda (i: nat).(\lambda (H: ((\forall (w: 
+T).(ex T (\lambda (u: T).(subst0 i w t u)))))).(let H_x \def (H v) in (let H0 
+\def H_x in (ex_ind T (\lambda (u: T).(subst0 i v t u)) (\forall (c: 
+C).(\forall (d: C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to 
+(sn3 d v))))) (\lambda (x: T).(\lambda (H1: (subst0 i v t x)).(subst0_ind 
+(\lambda (n: nat).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_: T).(\forall 
+(c: C).(\forall (d: C).((getl n c (CHead d (Bind Abbr) t0)) \to ((sn3 c t1) 
+\to (sn3 d t0))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (c: 
+C).(\lambda (d: C).(\lambda (H2: (getl i0 c (CHead d (Bind Abbr) 
+v0))).(\lambda (H3: (sn3 c (TLRef i0))).(sn3_gen_def c d v0 i0 H2 H3))))))) 
+(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: 
+nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c: 
+C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to 
+(sn3 d v0))))))).(\lambda (t0: T).(\lambda (k: K).(\lambda (c: C).(\lambda 
+(d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) v0))).(\lambda (H5: (sn3 
+c (THead k u1 t0))).(let H_y \def (sn3_gen_head k c u1 t0 H5) in (H3 c d H4 
+H_y)))))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: T).(\lambda 
+(t1: T).(\lambda (i0: nat).(\lambda (H2: (subst0 (s k i0) v0 t1 t2)).(\lambda 
+(H3: ((\forall (c: C).(\forall (d: C).((getl (s k i0) c (CHead d (Bind Abbr) 
+v0)) \to ((sn3 c t1) \to (sn3 d v0))))))).(\lambda (u: T).(\lambda (c: 
+C).(\lambda (d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) 
+v0))).(\lambda (H5: (sn3 c (THead k u t1))).(K_ind (\lambda (k0: K).((subst0 
+(s k0 i0) v0 t1 t2) \to (((\forall (c0: C).(\forall (d0: C).((getl (s k0 i0) 
+c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0)))))) \to ((sn3 
+c (THead k0 u t1)) \to (sn3 d v0))))) (\lambda (b: B).(\lambda (_: (subst0 (s 
+(Bind b) i0) v0 t1 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: 
+C).((getl (s (Bind b) i0) c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to 
+(sn3 d0 v0))))))).(\lambda (H8: (sn3 c (THead (Bind b) u t1))).(let H_x0 \def 
+(sn3_gen_bind b c u t1 H8) in (let H9 \def H_x0 in (and_ind (sn3 c u) (sn3 
+(CHead c (Bind b) u) t1) (sn3 d v0) (\lambda (_: (sn3 c u)).(\lambda (H11: 
+(sn3 (CHead c (Bind b) u) t1)).(H7 (CHead c (Bind b) u) d (getl_clear_bind b 
+(CHead c (Bind b) u) c u (clear_bind b c u) (CHead d (Bind Abbr) v0) i0 H4) 
+H11))) H9))))))) (\lambda (f: F).(\lambda (_: (subst0 (s (Flat f) i0) v0 t1 
+t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: C).((getl (s (Flat f) i0) 
+c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0))))))).(\lambda 
+(H8: (sn3 c (THead (Flat f) u t1))).(let H_x0 \def (sn3_gen_flat f c u t1 H8) 
+in (let H9 \def H_x0 in (and_ind (sn3 c u) (sn3 c t1) (sn3 d v0) (\lambda (_: 
+(sn3 c u)).(\lambda (H11: (sn3 c t1)).(H7 c d H4 H11))) H9))))))) k H2 H3 
+H5))))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda 
+(i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c: 
+C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to 
+(sn3 d v0))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: ((\forall (c: C).(\forall (d: 
+C).((getl (s k i0) c (CHead d (Bind Abbr) v0)) \to ((sn3 c t1) \to (sn3 d 
+v0))))))).(\lambda (c: C).(\lambda (d: C).(\lambda (H6: (getl i0 c (CHead d 
+(Bind Abbr) v0))).(\lambda (H7: (sn3 c (THead k u1 t1))).(let H_y \def 
+(sn3_gen_head k c u1 t1 H7) in (H3 c d H6 H_y))))))))))))))))) i v t x H1))) 
+H0)))))).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/fwd.ma
index bb669667e..ccaad15d2 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/fwd.ma
@@ -20,137 +20,17 @@ include "subst0/defs.ma".
 
 include "lift/props.ma".
 
-theorem subst0_gen_sort:
+axiom subst0_gen_sort:
  \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 
 i v (TSort n) x) \to (\forall (P: Prop).P)))))
-\def
- \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda 
-(H: (subst0 i v (TSort n) x)).(\lambda (P: Prop).(let H0 \def (match H in 
-subst0 return (\lambda (n0: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda 
-(t1: T).(\lambda (_: (subst0 n0 t t0 t1)).((eq nat n0 i) \to ((eq T t v) \to 
-((eq T t0 (TSort n)) \to ((eq T t1 x) \to P))))))))) with [(subst0_lref v0 
-i0) \Rightarrow (\lambda (H0: (eq nat i0 i)).(\lambda (H1: (eq T v0 
-v)).(\lambda (H2: (eq T (TLRef i0) (TSort n))).(\lambda (H3: (eq T (lift (S 
-i0) O v0) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (TLRef 
-n0) (TSort n)) \to ((eq T (lift (S n0) O v0) x) \to P)))) (\lambda (H4: (eq T 
-v0 v)).(eq_ind T v (\lambda (t: T).((eq T (TLRef i) (TSort n)) \to ((eq T 
-(lift (S i) O t) x) \to P))) (\lambda (H5: (eq T (TLRef i) (TSort n))).(let 
-H6 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (TSort n) H5) in (False_ind ((eq T (lift 
-(S i) O v) x) \to P) H6))) v0 (sym_eq T v0 v H4))) i0 (sym_eq nat i0 i H0) H1 
-H2 H3))))) | (subst0_fst v0 u2 u1 i0 H0 t k) \Rightarrow (\lambda (H1: (eq 
-nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u1 t) 
-(TSort n))).(\lambda (H4: (eq T (THead k u2 t) x)).(eq_ind nat i (\lambda 
-(n0: nat).((eq T v0 v) \to ((eq T (THead k u1 t) (TSort n)) \to ((eq T (THead 
-k u2 t) x) \to ((subst0 n0 v0 u1 u2) \to P))))) (\lambda (H5: (eq T v0 
-v)).(eq_ind T v (\lambda (t0: T).((eq T (THead k u1 t) (TSort n)) \to ((eq T 
-(THead k u2 t) x) \to ((subst0 i t0 u1 u2) \to P)))) (\lambda (H6: (eq T 
-(THead k u1 t) (TSort n))).(let H7 \def (eq_ind T (THead k u1 t) (\lambda (e: 
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
-(TSort n) H6) in (False_ind ((eq T (THead k u2 t) x) \to ((subst0 i v u1 u2) 
-\to P)) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 
-H0))))) | (subst0_snd k v0 t2 t1 i0 H0 u) \Rightarrow (\lambda (H1: (eq nat 
-i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u t1) (TSort 
-n))).(\lambda (H4: (eq T (THead k u t2) x)).(eq_ind nat i (\lambda (n0: 
-nat).((eq T v0 v) \to ((eq T (THead k u t1) (TSort n)) \to ((eq T (THead k u 
-t2) x) \to ((subst0 (s k n0) v0 t1 t2) \to P))))) (\lambda (H5: (eq T v0 
-v)).(eq_ind T v (\lambda (t: T).((eq T (THead k u t1) (TSort n)) \to ((eq T 
-(THead k u t2) x) \to ((subst0 (s k i) t t1 t2) \to P)))) (\lambda (H6: (eq T 
-(THead k u t1) (TSort n))).(let H7 \def (eq_ind T (THead k u t1) (\lambda (e: 
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
-(TSort n) H6) in (False_ind ((eq T (THead k u t2) x) \to ((subst0 (s k i) v 
-t1 t2) \to P)) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 
-H0))))) | (subst0_both v0 u1 u2 i0 H0 k t1 t2 H1) \Rightarrow (\lambda (H2: 
-(eq nat i0 i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq T (THead k u1 t1) 
-(TSort n))).(\lambda (H5: (eq T (THead k u2 t2) x)).(eq_ind nat i (\lambda 
-(n0: nat).((eq T v0 v) \to ((eq T (THead k u1 t1) (TSort n)) \to ((eq T 
-(THead k u2 t2) x) \to ((subst0 n0 v0 u1 u2) \to ((subst0 (s k n0) v0 t1 t2) 
-\to P)))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T 
-(THead k u1 t1) (TSort n)) \to ((eq T (THead k u2 t2) x) \to ((subst0 i t u1 
-u2) \to ((subst0 (s k i) t t1 t2) \to P))))) (\lambda (H7: (eq T (THead k u1 
-t1) (TSort n))).(let H8 \def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match 
-e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) 
-H7) in (False_ind ((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2) \to 
-((subst0 (s k i) v t1 t2) \to P))) H8))) v0 (sym_eq T v0 v H6))) i0 (sym_eq 
-nat i0 i H2) H3 H4 H5 H0 H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v) 
-(refl_equal T (TSort n)) (refl_equal T x)))))))).
+.
 
-theorem subst0_gen_lref:
+axiom subst0_gen_lref:
  \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 
 i v (TLRef n) x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))
-\def
- \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda 
-(H: (subst0 i v (TLRef n) x)).(let H0 \def (match H in subst0 return (\lambda 
-(n0: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_: 
-(subst0 n0 t t0 t1)).((eq nat n0 i) \to ((eq T t v) \to ((eq T t0 (TLRef n)) 
-\to ((eq T t1 x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))))))) 
-with [(subst0_lref v0 i0) \Rightarrow (\lambda (H0: (eq nat i0 i)).(\lambda 
-(H1: (eq T v0 v)).(\lambda (H2: (eq T (TLRef i0) (TLRef n))).(\lambda (H3: 
-(eq T (lift (S i0) O v0) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) 
-\to ((eq T (TLRef n0) (TLRef n)) \to ((eq T (lift (S n0) O v0) x) \to (land 
-(eq nat n i) (eq T x (lift (S n) O v))))))) (\lambda (H4: (eq T v0 
-v)).(eq_ind T v (\lambda (t: T).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift 
-(S i) O t) x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))) (\lambda 
-(H5: (eq T (TLRef i) (TLRef n))).(let H6 \def (f_equal T nat (\lambda (e: 
-T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | 
-(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef 
-n) H5) in (eq_ind nat n (\lambda (n0: nat).((eq T (lift (S n0) O v) x) \to 
-(land (eq nat n n0) (eq T x (lift (S n) O v))))) (\lambda (H7: (eq T (lift (S 
-n) O v) x)).(eq_ind T (lift (S n) O v) (\lambda (t: T).(land (eq nat n n) (eq 
-T t (lift (S n) O v)))) (conj (eq nat n n) (eq T (lift (S n) O v) (lift (S n) 
-O v)) (refl_equal nat n) (refl_equal T (lift (S n) O v))) x H7)) i (sym_eq 
-nat i n H6)))) v0 (sym_eq T v0 v H4))) i0 (sym_eq nat i0 i H0) H1 H2 H3))))) 
-| (subst0_fst v0 u2 u1 i0 H0 t k) \Rightarrow (\lambda (H1: (eq nat i0 
-i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u1 t) (TLRef 
-n))).(\lambda (H4: (eq T (THead k u2 t) x)).(eq_ind nat i (\lambda (n0: 
-nat).((eq T v0 v) \to ((eq T (THead k u1 t) (TLRef n)) \to ((eq T (THead k u2 
-t) x) \to ((subst0 n0 v0 u1 u2) \to (land (eq nat n i) (eq T x (lift (S n) O 
-v)))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq T 
-(THead k u1 t) (TLRef n)) \to ((eq T (THead k u2 t) x) \to ((subst0 i t0 u1 
-u2) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))) (\lambda (H6: (eq T 
-(THead k u1 t) (TLRef n))).(let H7 \def (eq_ind T (THead k u1 t) (\lambda (e: 
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
-(TLRef n) H6) in (False_ind ((eq T (THead k u2 t) x) \to ((subst0 i v u1 u2) 
-\to (land (eq nat n i) (eq T x (lift (S n) O v))))) H7))) v0 (sym_eq T v0 v 
-H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_snd k v0 t2 t1 i0 H0 
-u) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 
-v)).(\lambda (H3: (eq T (THead k u t1) (TLRef n))).(\lambda (H4: (eq T (THead 
-k u t2) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (THead k 
-u t1) (TLRef n)) \to ((eq T (THead k u t2) x) \to ((subst0 (s k n0) v0 t1 t2) 
-\to (land (eq nat n i) (eq T x (lift (S n) O v)))))))) (\lambda (H5: (eq T v0 
-v)).(eq_ind T v (\lambda (t: T).((eq T (THead k u t1) (TLRef n)) \to ((eq T 
-(THead k u t2) x) \to ((subst0 (s k i) t t1 t2) \to (land (eq nat n i) (eq T 
-x (lift (S n) O v))))))) (\lambda (H6: (eq T (THead k u t1) (TLRef n))).(let 
-H7 \def (eq_ind T (THead k u t1) (\lambda (e: T).(match e in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H6) in 
-(False_ind ((eq T (THead k u t2) x) \to ((subst0 (s k i) v t1 t2) \to (land 
-(eq nat n i) (eq T x (lift (S n) O v))))) H7))) v0 (sym_eq T v0 v H5))) i0 
-(sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_both v0 u1 u2 i0 H0 k t1 t2 
-H1) \Rightarrow (\lambda (H2: (eq nat i0 i)).(\lambda (H3: (eq T v0 
-v)).(\lambda (H4: (eq T (THead k u1 t1) (TLRef n))).(\lambda (H5: (eq T 
-(THead k u2 t2) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T 
-(THead k u1 t1) (TLRef n)) \to ((eq T (THead k u2 t2) x) \to ((subst0 n0 v0 
-u1 u2) \to ((subst0 (s k n0) v0 t1 t2) \to (land (eq nat n i) (eq T x (lift 
-(S n) O v))))))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: 
-T).((eq T (THead k u1 t1) (TLRef n)) \to ((eq T (THead k u2 t2) x) \to 
-((subst0 i t u1 u2) \to ((subst0 (s k i) t t1 t2) \to (land (eq nat n i) (eq 
-T x (lift (S n) O v)))))))) (\lambda (H7: (eq T (THead k u1 t1) (TLRef 
-n))).(let H8 \def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H7) in 
-(False_ind ((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2) \to ((subst0 (s 
-k i) v t1 t2) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))) H8))) v0 
-(sym_eq T v0 v H6))) i0 (sym_eq nat i0 i H2) H3 H4 H5 H0 H1)))))]) in (H0 
-(refl_equal nat i) (refl_equal T v) (refl_equal T (TLRef n)) (refl_equal T 
-x))))))).
+.
 
-theorem subst0_gen_head:
+axiom subst0_gen_head:
  \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall 
 (x: T).(\forall (i: nat).((subst0 i v (THead k u1 t1) x) \to (or3 (ex2 T 
 (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 
@@ -158,646 +38,25 @@ u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2:
 T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
 T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 
 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))))
-\def
- \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda 
-(x: T).(\lambda (i: nat).(\lambda (H: (subst0 i v (THead k u1 t1) x)).(let H0 
-\def (match H in subst0 return (\lambda (n: nat).(\lambda (t: T).(\lambda 
-(t0: T).(\lambda (t2: T).(\lambda (_: (subst0 n t t0 t2)).((eq nat n i) \to 
-((eq T t v) \to ((eq T t0 (THead k u1 t1)) \to ((eq T t2 x) \to (or3 (ex2 T 
-(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 
-u2))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: 
-T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))))))))) 
-with [(subst0_lref v0 i0) \Rightarrow (\lambda (H0: (eq nat i0 i)).(\lambda 
-(H1: (eq T v0 v)).(\lambda (H2: (eq T (TLRef i0) (THead k u1 t1))).(\lambda 
-(H3: (eq T (lift (S i0) O v0) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0 
-v) \to ((eq T (TLRef n) (THead k u1 t1)) \to ((eq T (lift (S n) O v0) x) \to 
-(or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: 
-T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) 
-(\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 
-t2))))))))) (\lambda (H4: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T 
-(TLRef i) (THead k u1 t1)) \to ((eq T (lift (S i) O t) x) \to (or3 (ex2 T 
-(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 
-u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: 
-T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 
-u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))) 
-(\lambda (H5: (eq T (TLRef i) (THead k u1 t1))).(let H6 \def (eq_ind T (TLRef 
-i) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
-False])) I (THead k u1 t1) H5) in (False_ind ((eq T (lift (S i) O v) x) \to 
-(or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: 
-T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) 
-(\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 
-t2)))))) H6))) v0 (sym_eq T v0 v H4))) i0 (sym_eq nat i0 i H0) H1 H2 H3))))) 
-| (subst0_fst v0 u2 u0 i0 H0 t k0) \Rightarrow (\lambda (H1: (eq nat i0 
-i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k0 u0 t) (THead k 
-u1 t1))).(\lambda (H4: (eq T (THead k0 u2 t) x)).(eq_ind nat i (\lambda (n: 
-nat).((eq T v0 v) \to ((eq T (THead k0 u0 t) (THead k u1 t1)) \to ((eq T 
-(THead k0 u2 t) x) \to ((subst0 n v0 u0 u2) \to (or3 (ex2 T (\lambda (u3: 
-T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T 
-(\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v 
-t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3 
-t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))) (\lambda (H5: (eq T v0 
-v)).(eq_ind T v (\lambda (t0: T).((eq T (THead k0 u0 t) (THead k u1 t1)) \to 
-((eq T (THead k0 u2 t) x) \to ((subst0 i t0 u0 u2) \to (or3 (ex2 T (\lambda 
-(u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 
-T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) 
-v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3 
-t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))) (\lambda (H6: (eq T 
-(THead k0 u0 t) (THead k u1 t1))).(let H7 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | 
-(TLRef _) \Rightarrow t | (THead _ _ t0) \Rightarrow t0])) (THead k0 u0 t) 
-(THead k u1 t1) H6) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in 
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) 
-\Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead k0 u0 t) (THead k u1 
-t1) H6) in ((let H9 \def (f_equal T K (\lambda (e: T).(match e in T return 
-(\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 
-| (THead k1 _ _) \Rightarrow k1])) (THead k0 u0 t) (THead k u1 t1) H6) in 
-(eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq T t t1) \to ((eq T (THead 
-k1 u2 t) x) \to ((subst0 i v u0 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x 
-(THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: 
-T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))) (\lambda (H10: (eq T u0 
-u1)).(eq_ind T u1 (\lambda (t0: T).((eq T t t1) \to ((eq T (THead k u2 t) x) 
-\to ((subst0 i v t0 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 
-t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T x 
-(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: 
-T).(subst0 (s k i) v t1 t2))))))))) (\lambda (H11: (eq T t t1)).(eq_ind T t1 
-(\lambda (t0: T).((eq T (THead k u2 t0) x) \to ((subst0 i v u1 u2) \to (or3 
-(ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i 
-v u1 u3))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: 
-T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
-T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 
-u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))) 
-(\lambda (H12: (eq T (THead k u2 t1) x)).(eq_ind T (THead k u2 t1) (\lambda 
-(t0: T).((subst0 i v u1 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T t0 (THead 
-k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq 
-T t0 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T t0 (THead k u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: 
-T).(subst0 (s k i) v t1 t2))))))) (\lambda (H13: (subst0 i v u1 
-u2)).(or3_intro0 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 
-t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T 
-(THead k u2 t1) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t1) (THead k 
-u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))) (ex_intro2 T (\lambda (u3: 
-T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 
-u3)) u2 (refl_equal T (THead k u2 t1)) H13))) x H12)) t (sym_eq T t t1 H11))) 
-u0 (sym_eq T u0 u1 H10))) k0 (sym_eq K k0 k H9))) H8)) H7))) v0 (sym_eq T v0 
-v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_snd k0 v0 t2 t0 i0 
-H0 u) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 
-v)).(\lambda (H3: (eq T (THead k0 u t0) (THead k u1 t1))).(\lambda (H4: (eq T 
-(THead k0 u t2) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0 v) \to ((eq T 
-(THead k0 u t0) (THead k u1 t1)) \to ((eq T (THead k0 u t2) x) \to ((subst0 
-(s k0 n) v0 t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) 
-(\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead 
-k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda 
-(_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) 
-v t1 t3)))))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq 
-T (THead k0 u t0) (THead k u1 t1)) \to ((eq T (THead k0 u t2) x) \to ((subst0 
-(s k0 i) t t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) 
-(\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead 
-k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda 
-(_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) 
-v t1 t3))))))))) (\lambda (H6: (eq T (THead k0 u t0) (THead k u1 t1))).(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 k0 u t0) (THead k u1 t1) H6) in ((let H8 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) 
-\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H6) in ((let H9 \def 
-(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with 
-[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) 
-\Rightarrow k1])) (THead k0 u t0) (THead k u1 t1) H6) in (eq_ind K k (\lambda 
-(k1: K).((eq T u u1) \to ((eq T t0 t1) \to ((eq T (THead k1 u t2) x) \to 
-((subst0 (s k1 i) v t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k 
-u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T 
-x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s k i) v t1 t3)))))))))) (\lambda (H10: (eq T u u1)).(eq_ind T u1 
-(\lambda (t: T).((eq T t0 t1) \to ((eq T (THead k t t2) x) \to ((subst0 (s k 
-i) v t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) 
-(\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead 
-k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda 
-(_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) 
-v t1 t3))))))))) (\lambda (H11: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: 
-T).((eq T (THead k u1 t2) x) \to ((subst0 (s k i) v t t2) \to (or3 (ex2 T 
-(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 
-u2))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: 
-T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))) 
-(\lambda (H12: (eq T (THead k u1 t2) x)).(eq_ind T (THead k u1 t2) (\lambda 
-(t: T).((subst0 (s k i) v t1 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T t 
-(THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: 
-T).(eq T t (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead k u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))))))) (\lambda (H13: (subst0 (s 
-k i) v t1 t2)).(or3_intro1 (ex2 T (\lambda (u2: T).(eq T (THead k u1 t2) 
-(THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: 
-T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v 
-t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead k u1 t2) 
-(THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex_intro2 T 
-(\lambda (t3: T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3: 
-T).(subst0 (s k i) v t1 t3)) t2 (refl_equal T (THead k u1 t2)) H13))) x H12)) 
-t0 (sym_eq T t0 t1 H11))) u (sym_eq T u u1 H10))) k0 (sym_eq K k0 k H9))) 
-H8)) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | 
-(subst0_both v0 u0 u2 i0 H0 k0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq nat i0 
-i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq T (THead k0 u0 t0) (THead k 
-u1 t1))).(\lambda (H5: (eq T (THead k0 u2 t2) x)).(eq_ind nat i (\lambda (n: 
-nat).((eq T v0 v) \to ((eq T (THead k0 u0 t0) (THead k u1 t1)) \to ((eq T 
-(THead k0 u2 t2) x) \to ((subst0 n v0 u0 u2) \to ((subst0 (s k0 n) v0 t0 t2) 
-\to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: 
-T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) 
-(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 
-t3))))))))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T 
-(THead k0 u0 t0) (THead k u1 t1)) \to ((eq T (THead k0 u2 t2) x) \to ((subst0 
-i t u0 u2) \to ((subst0 (s k0 i) t t0 t2) \to (or3 (ex2 T (\lambda (u3: 
-T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T 
-(\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v 
-t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead k u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))))) (\lambda (H7: (eq T 
-(THead k0 u0 t0) (THead k u1 t1))).(let H8 \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 k0 u0 t0) 
-(THead k u1 t1) H7) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e in 
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) 
-\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k0 u0 t0) (THead k u1 
-t1) H7) in ((let H10 \def (f_equal T K (\lambda (e: T).(match e in T return 
-(\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 
-| (THead k1 _ _) \Rightarrow k1])) (THead k0 u0 t0) (THead k u1 t1) H7) in 
-(eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T 
-(THead k1 u2 t2) x) \to ((subst0 i v u0 u2) \to ((subst0 (s k1 i) v t0 t2) 
-\to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: 
-T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) 
-(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 
-t3))))))))))) (\lambda (H11: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq 
-T t0 t1) \to ((eq T (THead k u2 t2) x) \to ((subst0 i v t u2) \to ((subst0 (s 
-k i) v t0 t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) 
-(\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead 
-k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda 
-(u3: T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda 
-(_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) 
-v t1 t3)))))))))) (\lambda (H12: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: 
-T).((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2) \to ((subst0 (s k i) v t 
-t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: 
-T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) 
-(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 
-t3))))))))) (\lambda (H13: (eq T (THead k u2 t2) x)).(eq_ind T (THead k u2 
-t2) (\lambda (t: T).((subst0 i v u1 u2) \to ((subst0 (s k i) v t1 t2) \to 
-(or3 (ex2 T (\lambda (u3: T).(eq T t (THead k u3 t1))) (\lambda (u3: 
-T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T t (THead k u1 t3))) 
-(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T t (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 
-t3)))))))) (\lambda (H14: (subst0 i v u1 u2)).(\lambda (H15: (subst0 (s k i) 
-v t1 t2)).(or3_intro2 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t2) (THead k 
-u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T 
-(THead k u2 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k 
-u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex3_2_intro T T (\lambda 
-(u3: T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda 
-(u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s k i) v t1 t3))) u2 t2 (refl_equal T (THead k u2 t2)) H14 
-H15)))) x H13)) t0 (sym_eq T t0 t1 H12))) u0 (sym_eq T u0 u1 H11))) k0 
-(sym_eq K k0 k H10))) H9)) H8))) v0 (sym_eq T v0 v H6))) i0 (sym_eq nat i0 i 
-H2) H3 H4 H5 H0 H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v) 
-(refl_equal T (THead k u1 t1)) (refl_equal T x))))))))).
+.
 
-theorem subst0_gen_lift_lt:
+axiom subst0_gen_lift_lt:
  \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall 
 (h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t1) 
 x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda 
 (t2: T).(subst0 i u t1 t2)))))))))
-\def
- \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: 
-T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift h d 
-u) (lift h (S (plus i d)) t) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h 
-(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2))))))))) (\lambda (n: 
-nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S (plus i d)) (TSort n)) 
-x)).(let H0 \def (eq_ind T (lift h (S (plus i d)) (TSort n)) (\lambda (t: 
-T).(subst0 i (lift h d u) t x)) H (TSort n) (lift_sort n h (S (plus i d)))) 
-in (subst0_gen_sort (lift h d u) x i n H0 (ex2 T (\lambda (t2: T).(eq T x 
-(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TSort n) 
-t2))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: nat).(\lambda 
-(h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S 
-(plus i d)) (TLRef n)) x)).(lt_le_e n (S (plus i d)) (ex2 T (\lambda (t2: 
-T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef 
-n) t2))) (\lambda (H0: (lt n (S (plus i d)))).(let H1 \def (eq_ind T (lift h 
-(S (plus i d)) (TLRef n)) (\lambda (t: T).(subst0 i (lift h d u) t x)) H 
-(TLRef n) (lift_lref_lt n h (S (plus i d)) H0)) in (and_ind (eq nat n i) (eq 
-T x (lift (S n) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S 
-(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: 
-(eq nat n i)).(\lambda (H3: (eq T x (lift (S n) O (lift h d u)))).(eq_ind_r T 
-(lift (S n) O (lift h d u)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t 
-(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2)))) 
-(eq_ind_r nat i (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S n0) 
-O (lift h d u)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u 
-(TLRef n0) t2)))) (eq_ind T (lift h (plus (S i) d) (lift (S i) O u)) (\lambda 
-(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) (\lambda 
-(t2: T).(subst0 i u (TLRef i) t2)))) (ex_intro2 T (\lambda (t2: T).(eq T 
-(lift h (S (plus i d)) (lift (S i) O u)) (lift h (S (plus i d)) t2))) 
-(\lambda (t2: T).(subst0 i u (TLRef i) t2)) (lift (S i) O u) (refl_equal T 
-(lift h (S (plus i d)) (lift (S i) O u))) (subst0_lref u i)) (lift (S i) O 
-(lift h d u)) (lift_d u h (S i) d O (le_O_n d))) n H2) x H3))) 
-(subst0_gen_lref (lift h d u) x i n H1)))) (\lambda (H0: (le (S (plus i d)) 
-n)).(let H1 \def (eq_ind T (lift h (S (plus i d)) (TLRef n)) (\lambda (t: 
-T).(subst0 i (lift h d u) t x)) H (TLRef (plus n h)) (lift_lref_ge n h (S 
-(plus i d)) H0)) in (and_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n 
-h)) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) 
-t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: (eq nat 
-(plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n h)) O (lift h d 
-u)))).(let H4 \def (eq_ind_r nat i (\lambda (n0: nat).(le (S (plus n0 d)) n)) 
-H0 (plus n h) H2) in (le_false n (plus (plus n h) d) (ex2 T (\lambda (t2: 
-T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef 
-n) t2))) (le_plus_trans n (plus n h) d (le_plus_l n h)) H4)))) 
-(subst0_gen_lref (lift h d u) x i (plus n h) H1))))))))))) (\lambda (k: 
-K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall 
-(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t) 
-x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda 
-(t2: T).(subst0 i u t t2)))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall 
-(x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift 
-h d u) (lift h (S (plus i d)) t0) x) \to (ex2 T (\lambda (t2: T).(eq T x 
-(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t0 
-t2)))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (H1: (subst0 i (lift h d u) (lift h (S (plus i d)) (THead k t 
-t0)) x)).(let H2 \def (eq_ind T (lift h (S (plus i d)) (THead k t t0)) 
-(\lambda (t2: T).(subst0 i (lift h d u) t2 x)) H1 (THead k (lift h (S (plus i 
-d)) t) (lift h (s k (S (plus i d))) t0)) (lift_head k t t0 h (S (plus i d)))) 
-in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k (S (plus 
-i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S (plus i d)) 
-t) u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) 
-t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i 
-d))) t0) t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k 
-u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S 
-(plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h 
-d u) (lift h (s k (S (plus i d))) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x 
-(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) 
-t2))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k 
-(S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S 
-(plus i d)) t) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h 
-(s k (S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h 
-(S (plus i d)) t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) 
-t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0: 
-T).(\lambda (H4: (eq T x (THead k x0 (lift h (s k (S (plus i d))) 
-t0)))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) t) 
-x0)).(eq_ind_r T (THead k x0 (lift h (s k (S (plus i d))) t0)) (\lambda (t2: 
-T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda 
-(t3: T).(subst0 i u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T 
-x0 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T 
-(\lambda (t2: T).(eq T (THead k x0 (lift h (s k (S (plus i d))) t0)) (lift h 
-(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) 
-(\lambda (x1: T).(\lambda (H6: (eq T x0 (lift h (S (plus i d)) x1))).(\lambda 
-(H7: (subst0 i u t x1)).(eq_ind_r T (lift h (S (plus i d)) x1) (\lambda (t2: 
-T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 (lift h (s k (S (plus i d))) 
-t0)) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) 
-t3)))) (eq_ind T (lift h (S (plus i d)) (THead k x1 t0)) (\lambda (t2: 
-T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda 
-(t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T 
-(lift h (S (plus i d)) (THead k x1 t0)) (lift h (S (plus i d)) t2))) (\lambda 
-(t2: T).(subst0 i u (THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h 
-(S (plus i d)) (THead k x1 t0))) (subst0_fst u x1 t i H7 t0 k)) (THead k 
-(lift h (S (plus i d)) x1) (lift h (s k (S (plus i d))) t0)) (lift_head k x1 
-t0 h (S (plus i d)))) x0 H6)))) (H x0 i h d H5)) x H4)))) H3)) (\lambda (H3: 
-(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) t2))) 
-(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) 
-t0) t2)))).(ex2_ind T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i 
-d)) t) t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S 
-(plus i d))) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) 
-t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0: 
-T).(\lambda (H4: (eq T x (THead k (lift h (S (plus i d)) t) x0))).(\lambda 
-(H5: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) 
-x0)).(eq_ind_r T (THead k (lift h (S (plus i d)) t) x0) (\lambda (t2: T).(ex2 
-T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: 
-T).(subst0 i u (THead k t t0) t3)))) (let H6 \def (eq_ind nat (s k (S (plus i 
-d))) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x0)) H5 (S 
-(s k (plus i d))) (s_S k (plus i d))) in (let H7 \def (eq_ind nat (s k (plus 
-i d)) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x0)) 
-H6 (plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x0 
-(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2)) 
-(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) x0) (lift h 
-(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) 
-(\lambda (x1: T).(\lambda (H8: (eq T x0 (lift h (S (plus (s k i) d)) 
-x1))).(\lambda (H9: (subst0 (s k i) u t0 x1)).(eq_ind_r T (lift h (S (plus (s 
-k i) d)) x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h 
-(S (plus i d)) t) t2) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i 
-u (THead k t t0) t3)))) (eq_ind nat (s k (plus i d)) (\lambda (n: nat).(ex2 T 
-(\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h (S n) x1)) 
-(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) 
-t2)))) (eq_ind nat (s k (S (plus i d))) (\lambda (n: nat).(ex2 T (\lambda 
-(t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h n x1)) (lift h (S 
-(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind 
-T (lift h (S (plus i d)) (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda 
-(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u 
-(THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift h (S (plus i 
-d)) (THead k t x1)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u 
-(THead k t t0) t2)) (THead k t x1) (refl_equal T (lift h (S (plus i d)) 
-(THead k t x1))) (subst0_snd k u x1 t0 i H9 t)) (THead k (lift h (S (plus i 
-d)) t) (lift h (s k (S (plus i d))) x1)) (lift_head k t x1 h (S (plus i d)))) 
-(S (s k (plus i d))) (s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x0 
-H8)))) (H0 x0 (s k i) h d H7)))) x H4)))) H3)) (\lambda (H3: (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S (plus i d)) t) u2))) 
-(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S 
-(plus i d))) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq 
-T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d 
-u) (lift h (S (plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 
-(s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) t2))) (ex2 T (\lambda 
-(t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u 
-(THead k t t0) t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T x 
-(THead k x0 x1))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) 
-t) x0)).(\lambda (H6: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i 
-d))) t0) x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda 
-(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u 
-(THead k t t0) t3)))) (let H7 \def (eq_ind nat (s k (S (plus i d))) (\lambda 
-(n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x1)) H6 (S (s k (plus i 
-d))) (s_S k (plus i d))) in (let H8 \def (eq_ind nat (s k (plus i d)) 
-(\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x1)) H7 
-(plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x1 
-(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2)) 
-(ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h (S (plus i d)) t2))) 
-(\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x2: T).(\lambda 
-(H9: (eq T x1 (lift h (S (plus (s k i) d)) x2))).(\lambda (H10: (subst0 (s k 
-i) u t0 x2)).(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h (S (plus i d)) 
-t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T (\lambda (t2: T).(eq T 
-(THead k x0 x1) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u 
-(THead k t t0) t2))) (\lambda (x3: T).(\lambda (H11: (eq T x0 (lift h (S 
-(plus i d)) x3))).(\lambda (H12: (subst0 i u t x3)).(eq_ind_r T (lift h (S 
-(plus i d)) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 
-x1) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) 
-t3)))) (eq_ind_r T (lift h (S (plus (s k i) d)) x2) (\lambda (t2: T).(ex2 T 
-(\lambda (t3: T).(eq T (THead k (lift h (S (plus i d)) x3) t2) (lift h (S 
-(plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (eq_ind 
-nat (s k (plus i d)) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k 
-(lift h (S (plus i d)) x3) (lift h (S n) x2)) (lift h (S (plus i d)) t2))) 
-(\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind nat (s k (S (plus 
-i d))) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S 
-(plus i d)) x3) (lift h n x2)) (lift h (S (plus i d)) t2))) (\lambda (t2: 
-T).(subst0 i u (THead k t t0) t2)))) (eq_ind T (lift h (S (plus i d)) (THead 
-k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus 
-i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T 
-(\lambda (t2: T).(eq T (lift h (S (plus i d)) (THead k x3 x2)) (lift h (S 
-(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)) (THead k 
-x3 x2) (refl_equal T (lift h (S (plus i d)) (THead k x3 x2))) (subst0_both u 
-t x3 i H12 k t0 x2 H10)) (THead k (lift h (S (plus i d)) x3) (lift h (s k (S 
-(plus i d))) x2)) (lift_head k x3 x2 h (S (plus i d)))) (S (s k (plus i d))) 
-(s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x1 H9) x0 H11)))) (H x0 
-i h d H5))))) (H0 x1 (s k i) h d H8)))) x H4)))))) H3)) (subst0_gen_head k 
-(lift h d u) (lift h (S (plus i d)) t) (lift h (s k (S (plus i d))) t0) x i 
-H2))))))))))))) t1)).
+.
 
-theorem subst0_gen_lift_false:
+axiom subst0_gen_lift_false:
  \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall 
 (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst0 i u 
 (lift h d t) x) \to (\forall (P: Prop).P)))))))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (u: T).(\forall (x: 
-T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i 
-(plus d h)) \to ((subst0 i u (lift h d t0) x) \to (\forall (P: 
-Prop).P)))))))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (x: T).(\lambda 
-(h: nat).(\lambda (d: nat).(\lambda (i: nat).(\lambda (_: (le d i)).(\lambda 
-(_: (lt i (plus d h))).(\lambda (H1: (subst0 i u (lift h d (TSort n)) 
-x)).(\lambda (P: Prop).(let H2 \def (eq_ind T (lift h d (TSort n)) (\lambda 
-(t0: T).(subst0 i u t0 x)) H1 (TSort n) (lift_sort n h d)) in 
-(subst0_gen_sort u x i n H2 P)))))))))))) (\lambda (n: nat).(\lambda (u: 
-T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (i: 
-nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d h))).(\lambda (H1: 
-(subst0 i u (lift h d (TLRef n)) x)).(\lambda (P: Prop).(lt_le_e n d P 
-(\lambda (H2: (lt n d)).(let H3 \def (eq_ind T (lift h d (TLRef n)) (\lambda 
-(t0: T).(subst0 i u t0 x)) H1 (TLRef n) (lift_lref_lt n h d H2)) in (and_ind 
-(eq nat n i) (eq T x (lift (S n) O u)) P (\lambda (H4: (eq nat n i)).(\lambda 
-(_: (eq T x (lift (S n) O u))).(let H6 \def (eq_ind nat n (\lambda (n0: 
-nat).(lt n0 d)) H2 i H4) in (le_false d i P H H6)))) (subst0_gen_lref u x i n 
-H3)))) (\lambda (H2: (le d n)).(let H3 \def (eq_ind T (lift h d (TLRef n)) 
-(\lambda (t0: T).(subst0 i u t0 x)) H1 (TLRef (plus n h)) (lift_lref_ge n h d 
-H2)) in (and_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n h)) O u)) P 
-(\lambda (H4: (eq nat (plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n 
-h)) O u))).(let H6 \def (eq_ind_r nat i (\lambda (n0: nat).(lt n0 (plus d 
-h))) H0 (plus n h) H4) in (le_false d n P H2 (lt_le_S n d (simpl_lt_plus_r h 
-n d H6)))))) (subst0_gen_lref u x i (plus n h) H3))))))))))))))) (\lambda (k: 
-K).(\lambda (t0: T).(\lambda (H: ((\forall (u: T).(\forall (x: T).(\forall 
-(h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) 
-\to ((subst0 i u (lift h d t0) x) \to (\forall (P: 
-Prop).P))))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (u: T).(\forall 
-(x: T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to 
-((lt i (plus d h)) \to ((subst0 i u (lift h d t1) x) \to (\forall (P: 
-Prop).P))))))))))).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda 
-(d: nat).(\lambda (i: nat).(\lambda (H1: (le d i)).(\lambda (H2: (lt i (plus 
-d h))).(\lambda (H3: (subst0 i u (lift h d (THead k t0 t1)) x)).(\lambda (P: 
-Prop).(let H4 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: 
-T).(subst0 i u t2 x)) H3 (THead k (lift h d t0) (lift h (s k d) t1)) 
-(lift_head k t0 t1 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k 
-u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2))) 
-(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: 
-T).(subst0 (s k i) u (lift h (s k d) t1) t2))) (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 
-(s k i) u (lift h (s k d) t1) t2)))) P (\lambda (H5: (ex2 T (\lambda (u2: 
-T).(eq T x (THead k u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u 
-(lift h d t0) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h 
-(s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2)) P (\lambda 
-(x0: T).(\lambda (_: (eq T x (THead k x0 (lift h (s k d) t1)))).(\lambda (H7: 
-(subst0 i u (lift h d t0) x0)).(H u x0 h d i H1 H2 H7 P)))) H5)) (\lambda 
-(H5: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda 
-(t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2)))).(ex2_ind T (\lambda (t2: 
-T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: T).(subst0 (s k i) u 
-(lift h (s k d) t1) t2)) P (\lambda (x0: T).(\lambda (_: (eq T x (THead k 
-(lift h d t0) x0))).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t1) 
-x0)).(H0 u x0 h (s k d) (s k i) (s_le k d i H1) (eq_ind nat (s k (plus d h)) 
-(\lambda (n: nat).(lt (s k i) n)) (lt_le_S (s k i) (s k (plus d h)) (s_lt k i 
-(plus d h) H2)) (plus (s k d) h) (s_plus k d h)) H7 P)))) H5)) (\lambda (H5: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))))).(ex3_2_ind 
-T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda 
-(u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))) P (\lambda 
-(x0: T).(\lambda (x1: T).(\lambda (_: (eq T x (THead k x0 x1))).(\lambda (H7: 
-(subst0 i u (lift h d t0) x0)).(\lambda (_: (subst0 (s k i) u (lift h (s k d) 
-t1) x1)).(H u x0 h d i H1 H2 H7 P)))))) H5)) (subst0_gen_head k u (lift h d 
-t0) (lift h (s k d) t1) x i H4))))))))))))))))) t).
+.
 
-theorem subst0_gen_lift_ge:
+axiom subst0_gen_lift_ge:
  \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall 
 (h: nat).(\forall (d: nat).((subst0 i u (lift h d t1) x) \to ((le (plus d h) 
 i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: 
 T).(subst0 (minus i h) u t1 t2))))))))))
-\def
- \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: 
-T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h 
-d t) x) \to ((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d 
-t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)))))))))) (\lambda (n: 
-nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (H: (subst0 i u (lift h d (TSort n)) x)).(\lambda (_: (le (plus 
-d h) i)).(let H1 \def (eq_ind T (lift h d (TSort n)) (\lambda (t: T).(subst0 
-i u t x)) H (TSort n) (lift_sort n h d)) in (subst0_gen_sort u x i n H1 (ex2 
-T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i 
-h) u (TSort n) t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: 
-nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i u (lift h d 
-(TLRef n)) x)).(\lambda (H0: (le (plus d h) i)).(lt_le_e n d (ex2 T (\lambda 
-(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef 
-n) t2))) (\lambda (H1: (lt n d)).(let H2 \def (eq_ind T (lift h d (TLRef n)) 
-(\lambda (t: T).(subst0 i u t x)) H (TLRef n) (lift_lref_lt n h d H1)) in 
-(and_ind (eq nat n i) (eq T x (lift (S n) O u)) (ex2 T (\lambda (t2: T).(eq T 
-x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef n) t2))) 
-(\lambda (H3: (eq nat n i)).(\lambda (_: (eq T x (lift (S n) O u))).(let H5 
-\def (eq_ind nat n (\lambda (n0: nat).(lt n0 d)) H1 i H3) in (le_false (plus 
-d h) i (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: 
-T).(subst0 (minus i h) u (TLRef n) t2))) H0 (le_plus_trans (S i) d h H5))))) 
-(subst0_gen_lref u x i n H2)))) (\lambda (H1: (le d n)).(let H2 \def (eq_ind 
-T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t x)) H (TLRef (plus n h)) 
-(lift_lref_ge n h d H1)) in (and_ind (eq nat (plus n h) i) (eq T x (lift (S 
-(plus n h)) O u)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda 
-(t2: T).(subst0 (minus i h) u (TLRef n) t2))) (\lambda (H3: (eq nat (plus n 
-h) i)).(\lambda (H4: (eq T x (lift (S (plus n h)) O u))).(eq_ind nat (plus n 
-h) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) 
-(\lambda (t2: T).(subst0 (minus n0 h) u (TLRef n) t2)))) (eq_ind_r T (lift (S 
-(plus n h)) O u) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d 
-t2))) (\lambda (t2: T).(subst0 (minus (plus n h) h) u (TLRef n) t2)))) 
-(eq_ind_r nat n (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S 
-(plus n h)) O u) (lift h d t2))) (\lambda (t2: T).(subst0 n0 u (TLRef n) 
-t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift (S (plus n h)) O u) (lift h 
-d t2))) (\lambda (t2: T).(subst0 n u (TLRef n) t2)) (lift (S n) O u) 
-(eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t: T).(eq T (lift (S (plus n 
-h)) O u) t)) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(eq T (lift (S n0) O 
-u) (lift (plus h (S n)) O u))) (eq_ind_r nat (plus h (S n)) (\lambda (n0: 
-nat).(eq T (lift n0 O u) (lift (plus h (S n)) O u))) (refl_equal T (lift 
-(plus h (S n)) O u)) (S (plus h n)) (plus_n_Sm h n)) (plus n h) (plus_comm n 
-h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans d (S n) 
-(plus O (S n)) (le_S d n H1) (le_n (plus O (S n)))) (le_O_n d))) (subst0_lref 
-u n)) (minus (plus n h) h) (minus_plus_r n h)) x H4) i H3))) (subst0_gen_lref 
-u x i (plus n h) H2)))))))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: 
-((\forall (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: 
-nat).((subst0 i u (lift h d t) x) \to ((le (plus d h) i) \to (ex2 T (\lambda 
-(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t 
-t2))))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (x: T).(\forall (i: 
-nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h d t0) x) \to 
-((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) 
-(\lambda (t2: T).(subst0 (minus i h) u t0 t2))))))))))).(\lambda (x: 
-T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: 
-(subst0 i u (lift h d (THead k t t0)) x)).(\lambda (H2: (le (plus d h) 
-i)).(let H3 \def (eq_ind T (lift h d (THead k t t0)) (\lambda (t2: T).(subst0 
-i u t2 x)) H1 (THead k (lift h d t) (lift h (s k d) t0)) (lift_head k t t0 h 
-d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k d) 
-t0)))) (\lambda (u2: T).(subst0 i u (lift h d t) u2))) (ex2 T (\lambda (t2: 
-T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0 (s k i) u 
-(lift h (s k d) t0) t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T 
-x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d 
-t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) 
-t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: 
-T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (H4: (ex2 T (\lambda 
-(u2: T).(eq T x (THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i 
-u (lift h d t) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h 
-(s k d) t0)))) (\lambda (u2: T).(subst0 i u (lift h d t) u2)) (ex2 T (\lambda 
-(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead 
-k t t0) t2))) (\lambda (x0: T).(\lambda (H5: (eq T x (THead k x0 (lift h (s k 
-d) t0)))).(\lambda (H6: (subst0 i u (lift h d t) x0)).(eq_ind_r T (THead k x0 
-(lift h (s k d) t0)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift 
-h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) 
-(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h d t2))) (\lambda (t2: T).(subst0 
-(minus i h) u t t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 (lift h (s k 
-d) t0)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) 
-t2))) (\lambda (x1: T).(\lambda (H7: (eq T x0 (lift h d x1))).(\lambda (H8: 
-(subst0 (minus i h) u t x1)).(eq_ind_r T (lift h d x1) (\lambda (t2: T).(ex2 
-T (\lambda (t3: T).(eq T (THead k t2 (lift h (s k d) t0)) (lift h d t3))) 
-(\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind T (lift 
-h d (THead k x1 t0)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift 
-h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) 
-(ex_intro2 T (\lambda (t2: T).(eq T (lift h d (THead k x1 t0)) (lift h d 
-t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2)) (THead k x1 
-t0) (refl_equal T (lift h d (THead k x1 t0))) (subst0_fst u x1 t (minus i h) 
-H8 t0 k)) (THead k (lift h d x1) (lift h (s k d) t0)) (lift_head k x1 t0 h 
-d)) x0 H7)))) (H x0 i h d H6 H2)) x H5)))) H4)) (\lambda (H4: (ex2 T (\lambda 
-(t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0 (s k i) 
-u (lift h (s k d) t0) t2)))).(ex2_ind T (\lambda (t2: T).(eq T x (THead k 
-(lift h d t) t2))) (\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) 
-t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 
-(minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda (H5: (eq T x 
-(THead k (lift h d t) x0))).(\lambda (H6: (subst0 (s k i) u (lift h (s k d) 
-t0) x0)).(eq_ind_r T (THead k (lift h d t) x0) (\lambda (t2: T).(ex2 T 
-(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i 
-h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T x0 (lift h (s k 
-d) t2))) (\lambda (t2: T).(subst0 (minus (s k i) h) u t0 t2)) (ex2 T (\lambda 
-(t2: T).(eq T (THead k (lift h d t) x0) (lift h d t2))) (\lambda (t2: 
-T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x1: T).(\lambda (H7: 
-(eq T x0 (lift h (s k d) x1))).(\lambda (H8: (subst0 (minus (s k i) h) u t0 
-x1)).(eq_ind_r T (lift h (s k d) x1) (\lambda (t2: T).(ex2 T (\lambda (t3: 
-T).(eq T (THead k (lift h d t) t2) (lift h d t3))) (\lambda (t3: T).(subst0 
-(minus i h) u (THead k t t0) t3)))) (eq_ind T (lift h d (THead k t x1)) 
-(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda 
-(t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (let H9 \def (eq_ind_r 
-nat (minus (s k i) h) (\lambda (n: nat).(subst0 n u t0 x1)) H8 (s k (minus i 
-h)) (s_minus k i h (le_trans_plus_r d h i H2))) in (ex_intro2 T (\lambda (t2: 
-T).(eq T (lift h d (THead k t x1)) (lift h d t2))) (\lambda (t2: T).(subst0 
-(minus i h) u (THead k t t0) t2)) (THead k t x1) (refl_equal T (lift h d 
-(THead k t x1))) (subst0_snd k u x1 t0 (minus i h) H9 t))) (THead k (lift h d 
-t) (lift h (s k d) x1)) (lift_head k t x1 h d)) x0 H7)))) (H0 x0 (s k i) h (s 
-k d) H6 (eq_ind nat (s k (plus d h)) (\lambda (n: nat).(le n (s k i))) (s_le 
-k (plus d h) i H2) (plus (s k d) h) (s_plus k d h)))) x H5)))) H4)) (\lambda 
-(H4: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d t) u2))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))))).(ex3_2_ind 
-T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda 
-(u2: T).(\lambda (_: T).(subst0 i u (lift h d t) u2))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))) (ex2 T 
-(\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) 
-u (THead k t t0) t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T 
-x (THead k x0 x1))).(\lambda (H6: (subst0 i u (lift h d t) x0)).(\lambda (H7: 
-(subst0 (s k i) u (lift h (s k d) t0) x1)).(eq_ind_r T (THead k x0 x1) 
-(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda 
-(t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: 
-T).(eq T x1 (lift h (s k d) t2))) (\lambda (t2: T).(subst0 (minus (s k i) h) 
-u t0 t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h d t2))) 
-(\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x2: 
-T).(\lambda (H8: (eq T x1 (lift h (s k d) x2))).(\lambda (H9: (subst0 (minus 
-(s k i) h) u t0 x2)).(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h d t2))) 
-(\lambda (t2: T).(subst0 (minus i h) u t t2)) (ex2 T (\lambda (t2: T).(eq T 
-(THead k x0 x1) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead 
-k t t0) t2))) (\lambda (x3: T).(\lambda (H10: (eq T x0 (lift h d 
-x3))).(\lambda (H11: (subst0 (minus i h) u t x3)).(eq_ind_r T (lift h d x3) 
-(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 x1) (lift h d 
-t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind_r 
-T (lift h (s k d) x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k 
-(lift h d x3) t2) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u 
-(THead k t t0) t3)))) (eq_ind T (lift h d (THead k x3 x2)) (\lambda (t2: 
-T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 
-(minus i h) u (THead k t t0) t3)))) (let H12 \def (eq_ind_r nat (minus (s k 
-i) h) (\lambda (n: nat).(subst0 n u t0 x2)) H9 (s k (minus i h)) (s_minus k i 
-h (le_trans_plus_r d h i H2))) in (ex_intro2 T (\lambda (t2: T).(eq T (lift h 
-d (THead k x3 x2)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u 
-(THead k t t0) t2)) (THead k x3 x2) (refl_equal T (lift h d (THead k x3 x2))) 
-(subst0_both u t x3 (minus i h) H11 k t0 x2 H12))) (THead k (lift h d x3) 
-(lift h (s k d) x2)) (lift_head k x3 x2 h d)) x1 H8) x0 H10)))) (H x0 i h d 
-H6 H2))))) (H0 x1 (s k i) h (s k d) H7 (eq_ind nat (s k (plus d h)) (\lambda 
-(n: nat).(le n (s k i))) (s_le k (plus d h) i H2) (plus (s k d) h) (s_plus k 
-d h)))) x H5)))))) H4)) (subst0_gen_head k u (lift h d t) (lift h (s k d) t0) 
-x i H3)))))))))))))) t1)).
+.
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst1/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst1/defs.ma
index d806b04c9..49102e3e0 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst1/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst1/defs.ma
@@ -18,7 +18,7 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/subst1/defs".
 
 include "subst0/defs.ma".
 
-inductive subst1 (i:nat) (v:T) (t1:T): T \to Prop \def
+inductive subst1 (i: nat) (v: T) (t1: T): T \to Prop \def
 | subst1_refl: subst1 i v t1 t1
 | subst1_single: \forall (t2: T).((subst0 i v t1 t2) \to (subst1 i v t1 t2)).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tau0/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tau0/defs.ma
index 14373ffcd..84f1a28ca 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tau0/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tau0/defs.ma
@@ -20,7 +20,7 @@ include "G/defs.ma".
 
 include "getl/defs.ma".
 
-inductive tau0 (g:G): C \to (T \to (T \to Prop)) \def
+inductive tau0 (g: G): C \to (T \to (T \to Prop)) \def
 | tau0_sort: \forall (c: C).(\forall (n: nat).(tau0 g c (TSort n) (TSort 
 (next g n))))
 | tau0_abbr: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tau1/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tau1/defs.ma
index 52f32692d..c5eeb0138 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tau1/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tau1/defs.ma
@@ -18,7 +18,7 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/tau1/defs".
 
 include "tau0/defs.ma".
 
-inductive tau1 (g:G) (c:C) (t1:T): T \to Prop \def
+inductive tau1 (g: G) (c: C) (t1: T): T \to Prop \def
 | tau1_tau0: \forall (t2: T).((tau0 g c t1 t2) \to (tau1 g c t1 t2))
 | tau1_sing: \forall (t: T).((tau1 g c t1 t) \to (\forall (t2: T).((tau0 g c 
 t t2) \to (tau1 g c t1 t2)))).
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma
index 2c1ae46cc..b632ec9d4 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma
@@ -26,6 +26,10 @@ include "s/defs.ma".
 
 include "s/props.ma".
 
+include "tlist/defs.ma".
+
+include "tlist/props.ma".
+
 include "tlt/defs.ma".
 
 include "tlt/props.ma".
@@ -310,9 +314,21 @@ include "sc3/defs.ma".
 
 include "sc3/props.ma".
 
-include "ceqc/defs.ma".
+include "csubc/defs.ma".
+
+include "csubc/props.ma".
+
+include "csubc/csuba.ma".
+
+include "csubc/drop.ma".
+
+include "csubc/drop1.ma".
+
+include "csubc/clear.ma".
+
+include "csubc/getl.ma".
 
-include "ceqc/props.ma".
+include "csubc/arity.ma".
 
 include "sc3/arity.ma".
 
@@ -346,21 +362,21 @@ include "ty3/fsubst0.ma".
 
 include "ty3/subst1.ma".
 
-include "csub3/defs.ma".
+include "csubt/defs.ma".
 
-include "csub3/fwd.ma".
+include "csubt/fwd.ma".
 
-include "csub3/props.ma".
+include "csubt/props.ma".
 
-include "csub3/clear.ma".
+include "csubt/clear.ma".
 
-include "csub3/drop.ma".
+include "csubt/drop.ma".
 
-include "csub3/getl.ma".
+include "csubt/getl.ma".
 
-include "csub3/pc3.ma".
+include "csubt/pc3.ma".
 
-include "csub3/ty3.ma".
+include "csubt/ty3.ma".
 
 include "ty3/pr3.ma".
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tlist/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tlist/defs.ma
new file mode 100644
index 000000000..ab7e3258b
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tlist/defs.ma
@@ -0,0 +1,49 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/tlist/defs".
+
+include "T/defs.ma".
+
+inductive TList: Set \def
+| TNil: TList
+| TCons: T \to (TList \to TList).
+
+definition THeads:
+ K \to (TList \to (T \to T))
+\def
+ let rec THeads (k: K) (us: TList) on us: (T \to T) \def (\lambda (t: 
+T).(match us with [TNil \Rightarrow t | (TCons u ul) \Rightarrow (THead k u 
+(THeads k ul t))])) in THeads.
+
+definition TApp:
+ TList \to (T \to TList)
+\def
+ let rec TApp (ts: TList) on ts: (T \to TList) \def (\lambda (v: T).(match ts 
+with [TNil \Rightarrow (TCons v TNil) | (TCons t ts0) \Rightarrow (TCons t 
+(TApp ts0 v))])) in TApp.
+
+definition tslen:
+ TList \to nat
+\def
+ let rec tslen (ts: TList) on ts: nat \def (match ts with [TNil \Rightarrow O 
+| (TCons _ ts0) \Rightarrow (S (tslen ts0))]) in tslen.
+
+definition tslt:
+ TList \to (TList \to Prop)
+\def
+ \lambda (ts1: TList).(\lambda (ts2: TList).(lt (tslen ts1) (tslen ts2))).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tlist/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tlist/props.ma
new file mode 100644
index 000000000..5a91e56b8
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tlist/props.ma
@@ -0,0 +1,120 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/tlist/props".
+
+include "tlist/defs.ma".
+
+theorem tslt_wf__q_ind:
+ \forall (P: ((TList \to Prop))).(((\forall (n: nat).((\lambda (P0: ((TList 
+\to Prop))).(\lambda (n0: nat).(\forall (ts: TList).((eq nat (tslen ts) n0) 
+\to (P0 ts))))) P n))) \to (\forall (ts: TList).(P ts)))
+\def
+ let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts: 
+TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to 
+Prop))).(\lambda (H: ((\forall (n: nat).(\forall (ts: TList).((eq nat (tslen 
+ts) n) \to (P ts)))))).(\lambda (ts: TList).(H (tslen ts) ts (refl_equal nat 
+(tslen ts)))))).
+
+theorem tslt_wf_ind:
+ \forall (P: ((TList \to Prop))).(((\forall (ts2: TList).(((\forall (ts1: 
+TList).((tslt ts1 ts2) \to (P ts1)))) \to (P ts2)))) \to (\forall (ts: 
+TList).(P ts)))
+\def
+ let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts: 
+TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to 
+Prop))).(\lambda (H: ((\forall (ts2: TList).(((\forall (ts1: TList).((lt 
+(tslen ts1) (tslen ts2)) \to (P ts1)))) \to (P ts2))))).(\lambda (ts: 
+TList).(tslt_wf__q_ind (\lambda (t: TList).(P t)) (\lambda (n: 
+nat).(lt_wf_ind n (Q (\lambda (t: TList).(P t))) (\lambda (n0: nat).(\lambda 
+(H0: ((\forall (m: nat).((lt m n0) \to (Q (\lambda (t: TList).(P t)) 
+m))))).(\lambda (ts0: TList).(\lambda (H1: (eq nat (tslen ts0) n0)).(let H2 
+\def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall (m: nat).((lt m n1) \to 
+(\forall (ts1: TList).((eq nat (tslen ts1) m) \to (P ts1)))))) H0 (tslen ts0) 
+H1) in (H ts0 (\lambda (ts1: TList).(\lambda (H3: (lt (tslen ts1) (tslen 
+ts0))).(H2 (tslen ts1) H3 ts1 (refl_equal nat (tslen ts1))))))))))))) ts)))).
+
+theorem theads_tapp:
+ \forall (k: K).(\forall (vs: TList).(\forall (v: T).(\forall (t: T).(eq T 
+(THeads k (TApp vs v) t) (THeads k vs (THead k v t))))))
+\def
+ \lambda (k: K).(\lambda (vs: TList).(TList_ind (\lambda (t: TList).(\forall 
+(v: T).(\forall (t0: T).(eq T (THeads k (TApp t v) t0) (THeads k t (THead k v 
+t0)))))) (\lambda (v: T).(\lambda (t: T).(refl_equal T (THead k v t)))) 
+(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (v: T).(\forall 
+(t1: T).(eq T (THeads k (TApp t0 v) t1) (THeads k t0 (THead k v 
+t1))))))).(\lambda (v: T).(\lambda (t1: T).(eq_ind_r T (THeads k t0 (THead k 
+v t1)) (\lambda (t2: T).(eq T (THead k t t2) (THead k t (THeads k t0 (THead k 
+v t1))))) (refl_equal T (THead k t (THeads k t0 (THead k v t1)))) (THeads k 
+(TApp t0 v) t1) (H v t1))))))) vs)).
+
+theorem tcons_tapp_ex:
+ \forall (ts1: TList).(\forall (t1: T).(ex2_2 TList T (\lambda (ts2: 
+TList).(\lambda (t2: T).(eq TList (TCons t1 ts1) (TApp ts2 t2)))) (\lambda 
+(ts2: TList).(\lambda (_: T).(eq nat (tslen ts1) (tslen ts2))))))
+\def
+ \lambda (ts1: TList).(TList_ind (\lambda (t: TList).(\forall (t1: T).(ex2_2 
+TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t) (TApp 
+ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t) (tslen 
+ts2))))))) (\lambda (t1: T).(ex2_2_intro TList T (\lambda (ts2: 
+TList).(\lambda (t2: T).(eq TList (TCons t1 TNil) (TApp ts2 t2)))) (\lambda 
+(ts2: TList).(\lambda (_: T).(eq nat O (tslen ts2)))) TNil t1 (refl_equal 
+TList (TApp TNil t1)) (refl_equal nat (tslen TNil)))) (\lambda (t: 
+T).(\lambda (t0: TList).(\lambda (H: ((\forall (t1: T).(ex2_2 TList T 
+(\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t0) (TApp ts2 
+t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) (tslen 
+ts2)))))))).(\lambda (t1: T).(let H_x \def (H t) in (let H0 \def H_x in 
+(ex2_2_ind TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t 
+t0) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) 
+(tslen ts2)))) (ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq 
+TList (TCons t1 (TCons t t0)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda 
+(_: T).(eq nat (S (tslen t0)) (tslen ts2))))) (\lambda (x0: TList).(\lambda 
+(x1: T).(\lambda (H1: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H2: (eq 
+nat (tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t2: 
+TList).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t3: T).(eq TList (TCons 
+t1 t2) (TApp ts2 t3)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (S 
+(tslen t0)) (tslen ts2)))))) (eq_ind_r nat (tslen x0) (\lambda (n: 
+nat).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons 
+t1 (TApp x0 x1)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq 
+nat (S n) (tslen ts2)))))) (ex2_2_intro TList T (\lambda (ts2: 
+TList).(\lambda (t2: T).(eq TList (TCons t1 (TApp x0 x1)) (TApp ts2 t2)))) 
+(\lambda (ts2: TList).(\lambda (_: T).(eq nat (S (tslen x0)) (tslen ts2)))) 
+(TCons t1 x0) x1 (refl_equal TList (TApp (TCons t1 x0) x1)) (refl_equal nat 
+(tslen (TCons t1 x0)))) (tslen t0) H2) (TCons t t0) H1))))) H0))))))) ts1).
+
+theorem tlist_ind_rew:
+ \forall (P: ((TList \to Prop))).((P TNil) \to (((\forall (ts: 
+TList).(\forall (t: T).((P ts) \to (P (TApp ts t)))))) \to (\forall (ts: 
+TList).(P ts))))
+\def
+ \lambda (P: ((TList \to Prop))).(\lambda (H: (P TNil)).(\lambda (H0: 
+((\forall (ts: TList).(\forall (t: T).((P ts) \to (P (TApp ts 
+t))))))).(\lambda (ts: TList).(tslt_wf_ind (\lambda (t: TList).(P t)) 
+(\lambda (ts2: TList).(TList_ind (\lambda (t: TList).(((\forall (ts1: 
+TList).((tslt ts1 t) \to (P ts1)))) \to (P t))) (\lambda (_: ((\forall (ts1: 
+TList).((tslt ts1 TNil) \to (P ts1))))).H) (\lambda (t: T).(\lambda (t0: 
+TList).(\lambda (_: ((((\forall (ts1: TList).((tslt ts1 t0) \to (P ts1)))) 
+\to (P t0)))).(\lambda (H2: ((\forall (ts1: TList).((tslt ts1 (TCons t t0)) 
+\to (P ts1))))).(let H_x \def (tcons_tapp_ex t0 t) in (let H3 \def H_x in 
+(ex2_2_ind TList T (\lambda (ts3: TList).(\lambda (t2: T).(eq TList (TCons t 
+t0) (TApp ts3 t2)))) (\lambda (ts3: TList).(\lambda (_: T).(eq nat (tslen t0) 
+(tslen ts3)))) (P (TCons t t0)) (\lambda (x0: TList).(\lambda (x1: 
+T).(\lambda (H4: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H5: (eq nat 
+(tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t1: TList).(P 
+t1)) (H0 x0 x1 (H2 x0 (eq_ind nat (tslen t0) (\lambda (n: nat).(lt n (tslen 
+(TCons t t0)))) (le_n (tslen (TCons t t0))) (tslen x0) H5))) (TCons t t0) 
+H4))))) H3))))))) ts2)) ts)))).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/defs.ma
index 1eecab2ff..457d1d09b 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/defs.ma
@@ -20,7 +20,7 @@ include "G/defs.ma".
 
 include "pc3/defs.ma".
 
-inductive ty3 (g:G): C \to (T \to (T \to Prop)) \def
+inductive ty3 (g: G): C \to (T \to (T \to Prop)) \def
 | ty3_conv: \forall (c: C).(\forall (t2: T).(\forall (t: T).((ty3 g c t2 t) 
 \to (\forall (u: T).(\forall (t1: T).((ty3 g c u t1) \to ((pc3 c t1 t2) \to 
 (ty3 g c u t2))))))))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3.ma
index 0c14d34f3..55a2457a9 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3.ma
@@ -16,7 +16,7 @@
 
 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3".
 
-include "csub3/ty3.ma".
+include "csubt/ty3.ma".
 
 include "ty3/subst1.ma".
 
@@ -427,9 +427,9 @@ t0))))).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead
 (Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w 
 (pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H20 Abst t0 x0 
 H21 x2 H22)) (THead (Bind Abbr) v2 t4) (THead (Bind Abbr) v2 x3) (ty3_bind g 
-c2 v2 u (H1 c2 H4 v2 H14) Abbr t4 x3 (csub3_ty3_ld g c2 v2 u0 (ty3_conv g c2 
+c2 v2 u (H1 c2 H4 v2 H14) Abbr t4 x3 (csubt_ty3_ld g c2 v2 u0 (ty3_conv g c2 
 u0 x4 H24 v2 u (H1 c2 H4 v2 H14) (pc3_s c2 u u0 H27)) t4 x3 H25) x5 
-(csub3_ty3_ld g c2 v2 u0 (ty3_conv g c2 u0 x4 H24 v2 u (H1 c2 H4 v2 H14) 
+(csubt_ty3_ld g c2 v2 u0 (ty3_conv g c2 u0 x4 H24 v2 u (H1 c2 H4 v2 H14) 
 (pc3_s c2 u u0 H27)) x3 x5 H26)) (pc3_t (THead (Bind Abbr) v2 t0) c2 (THead 
 (Bind Abbr) v2 x3) (pc3_head_2 c2 v2 x3 t0 (Bind Abbr) (H28 Abbr v2)) (THead 
 (Flat Appl) w (THead (Bind Abst) u t0)) (pc3_pr2_x c2 (THead (Bind Abbr) v2