From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Wed, 20 Feb 2008 18:53:47 +0000 (+0000)
Subject: LambdaDelta-1 regenerated as a subdevel ov LAMBDA-TYPES
X-Git-Tag: make_still_working~5588
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=e92710b1d9774a6491122668c8463b8658114610;p=helm.git

LambdaDelta-1 regenerated as a subdevel ov LAMBDA-TYPES
some additions and redundant dependences fixed
---

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Base-2/depends b/helm/software/matita/contribs/LAMBDA-TYPES/Base-2/depends
new file mode 100644
index 000000000..7f10402a2
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Base-2/depends
@@ -0,0 +1,9 @@
+Base-2/theory.mma Base-2/preamble.ma Base-1/theory.ma
+Base-2/ext/tactics.mma Base-2/preamble.ma Base-1/ext/tactics.ma
+Base-2/ext/arith.mma Base-2/preamble.ma Base-1/ext/arith.ma
+Base-2/types/defs.mma Base-2/preamble.ma Base-1/types/defs.ma
+Base-2/types/props.mma Base-2/preamble.ma Base-1/types/props.ma
+Base-2/blt/defs.mma Base-2/preamble.ma Base-1/blt/defs.ma
+Base-2/blt/props.mma Base-2/preamble.ma Base-1/blt/props.ma
+Base-2/plist/defs.mma Base-2/preamble.ma Base-1/plist/defs.ma
+Base-2/plist/props.mma Base-2/preamble.ma Base-1/plist/props.ma
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Base-2/depends_mma b/helm/software/matita/contribs/LAMBDA-TYPES/Base-2/depends_mma
deleted file mode 100644
index 7f10402a2..000000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Base-2/depends_mma
+++ /dev/null
@@ -1,9 +0,0 @@
-Base-2/theory.mma Base-2/preamble.ma Base-1/theory.ma
-Base-2/ext/tactics.mma Base-2/preamble.ma Base-1/ext/tactics.ma
-Base-2/ext/arith.mma Base-2/preamble.ma Base-1/ext/arith.ma
-Base-2/types/defs.mma Base-2/preamble.ma Base-1/types/defs.ma
-Base-2/types/props.mma Base-2/preamble.ma Base-1/types/props.ma
-Base-2/blt/defs.mma Base-2/preamble.ma Base-1/blt/defs.ma
-Base-2/blt/props.mma Base-2/preamble.ma Base-1/blt/props.ma
-Base-2/plist/defs.mma Base-2/preamble.ma Base-1/plist/defs.ma
-Base-2/plist/props.mma Base-2/preamble.ma Base-1/plist/props.ma
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/A/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/A/defs.ma
index 1c5def44f..b25d02874 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/A/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/A/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "preamble.ma".
+include "LambdaDelta-1/preamble.ma".
 
 inductive A: Set \def
 | ASort: nat \to (nat \to A)
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/C/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/C/defs.ma
index 042be1801..026539160 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/C/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/C/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "T/defs.ma".
+include "LambdaDelta-1/T/defs.ma".
 
 inductive C: Set \def
 | CSort: nat \to C
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/C/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/C/props.ma
index 06252c222..907c5c373 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/C/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/C/props.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "C/defs.ma".
+include "LambdaDelta-1/C/defs.ma".
 
-include "T/props.ma".
+include "LambdaDelta-1/T/props.ma".
 
 theorem clt_cong:
  \forall (c: C).(\forall (d: C).((clt c d) \to (\forall (k: K).(\forall (t: 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/G/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/G/defs.ma
index dbc7f7ad2..94fe95769 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/G/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/G/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "preamble.ma".
+include "LambdaDelta-1/preamble.ma".
 
 record G : Set \def {
  next: (nat \to nat);
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/Makefile b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/Makefile
deleted file mode 100644
index a57a7281e..000000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/Makefile
+++ /dev/null
@@ -1,15 +0,0 @@
-DIR=$(shell basename $$PWD)
-MATITAOPTIONS=-onepass
-
-$(DIR) all:
-	../../../matitac $(MATITAOPTIONS)
-$(DIR).opt opt all.opt:
-	../../../matitac.opt $(MATITAOPTIONS)
-clean:
-	../../../matitaclean
-clean.opt:
-	../../../matitaclean.opt
-depend:
-	../../../matitadep
-depend.opt:
-	../../../matitadep.opt
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/T/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/T/dec.ma
index 044e04557..adb1b8c1b 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/T/dec.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/T/dec.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "T/defs.ma".
+include "LambdaDelta-1/T/defs.ma".
 
 theorem terms_props__bind_dec:
  \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) ((eq B b1 b2) \to (\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/T/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/T/defs.ma
index 951e1b920..cdc5e05e2 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/T/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/T/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "preamble.ma".
+include "LambdaDelta-1/preamble.ma".
 
 inductive B: Set \def
 | Abbr: B
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/T/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/T/props.ma
index 2fb7361a1..575d918aa 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/T/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/T/props.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "T/defs.ma".
+include "LambdaDelta-1/T/defs.ma".
 
 theorem not_abbr_abst:
  not (eq B Abbr Abst)
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aplus/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aplus/defs.ma
index 98f15dde2..0e08def37 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aplus/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aplus/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "asucc/defs.ma".
+include "LambdaDelta-1/asucc/defs.ma".
 
 definition aplus:
  G \to (A \to (nat \to A))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aplus/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aplus/props.ma
index e0b2bcaca..c7cd372ae 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aplus/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aplus/props.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "aplus/defs.ma".
+include "LambdaDelta-1/aplus/defs.ma".
 
-include "next_plus/props.ma".
+include "LambdaDelta-1/next_plus/props.ma".
 
 theorem aplus_reg_r:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (h1: nat).(\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aprem/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aprem/defs.ma
index bcc17b031..907b30649 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aprem/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aprem/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "A/defs.ma".
+include "LambdaDelta-1/A/defs.ma".
 
 inductive aprem: nat \to (A \to (A \to Prop)) \def
 | aprem_zero: \forall (a1: A).(\forall (a2: A).(aprem O (AHead a1 a2) a1))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aprem/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aprem/props.ma
index 0c38f1e49..0a0a9525e 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aprem/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/aprem/props.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "aprem/defs.ma".
+include "LambdaDelta-1/aprem/defs.ma".
 
-include "leq/defs.ma".
+include "LambdaDelta-1/leq/defs.ma".
 
 theorem aprem_repl:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/aprem.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/aprem.ma
index 9aef2e8a0..6113a2a11 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/aprem.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/aprem.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "arity/props.ma".
+include "LambdaDelta-1/arity/props.ma".
 
-include "arity/cimp.ma".
+include "LambdaDelta-1/arity/cimp.ma".
 
-include "aprem/props.ma".
+include "LambdaDelta-1/aprem/props.ma".
 
 theorem arity_aprem:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/cimp.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/cimp.ma
index dbe488b0e..ab74849bc 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/cimp.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/cimp.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "arity/defs.ma".
+include "LambdaDelta-1/arity/defs.ma".
 
-include "cimp/props.ma".
+include "LambdaDelta-1/cimp/props.ma".
 
 theorem arity_cimp_conf:
  \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/defs.ma
index 5d2979dc4..9843f52a1 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/defs.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "leq/defs.ma".
+include "LambdaDelta-1/leq/defs.ma".
 
-include "getl/defs.ma".
+include "LambdaDelta-1/getl/defs.ma".
 
 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 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/fwd.ma
index fe241d0dc..b0ca1a3b1 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/fwd.ma
@@ -14,13 +14,13 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "arity/defs.ma".
+include "LambdaDelta-1/arity/defs.ma".
 
-include "leq/asucc.ma".
+include "LambdaDelta-1/leq/asucc.ma".
 
-include "leq/fwd.ma".
+include "LambdaDelta-1/leq/fwd.ma".
 
-include "getl/drop.ma".
+include "LambdaDelta-1/getl/drop.ma".
 
 theorem arity_gen_sort:
  \forall (g: G).(\forall (c: C).(\forall (n: nat).(\forall (a: A).((arity g c 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/lift1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/lift1.ma
index c6270d035..c89c57d76 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/lift1.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/lift1.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "arity/props.ma".
+include "LambdaDelta-1/arity/props.ma".
 
-include "drop1/defs.ma".
+include "LambdaDelta-1/drop1/defs.ma".
 
 theorem arity_lift1:
  \forall (g: G).(\forall (a: A).(\forall (c2: C).(\forall (hds: 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/pr3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/pr3.ma
index 15697c87d..b56a8945a 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/pr3.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/pr3.ma
@@ -14,17 +14,17 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csuba/arity.ma".
+include "LambdaDelta-1/csuba/arity.ma".
 
-include "pr3/defs.ma".
+include "LambdaDelta-1/pr3/defs.ma".
 
-include "pr1/defs.ma".
+include "LambdaDelta-1/pr1/defs.ma".
 
-include "wcpr0/getl.ma".
+include "LambdaDelta-1/wcpr0/getl.ma".
 
-include "pr0/fwd.ma".
+include "LambdaDelta-1/pr0/fwd.ma".
 
-include "arity/subst0.ma".
+include "LambdaDelta-1/arity/subst0.ma".
 
 theorem arity_sred_wcpr0_pr0:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/props.ma
index 301c26c6c..2d00abadf 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/props.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "arity/fwd.ma".
+include "LambdaDelta-1/arity/fwd.ma".
 
 theorem node_inh:
  \forall (g: G).(\forall (n: nat).(\forall (k: nat).(ex_2 C T (\lambda (c: 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/subst0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/subst0.ma
index d3f4f1373..006b6c493 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/subst0.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/subst0.ma
@@ -14,19 +14,19 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "arity/props.ma".
+include "LambdaDelta-1/arity/props.ma".
 
-include "fsubst0/fwd.ma".
+include "LambdaDelta-1/fsubst0/fwd.ma".
 
-include "csubst0/getl.ma".
+include "LambdaDelta-1/csubst0/getl.ma".
 
-include "csubst0/props.ma".
+include "LambdaDelta-1/csubst0/props.ma".
 
-include "subst0/dec.ma".
+include "LambdaDelta-1/subst0/dec.ma".
 
-include "subst0/fwd.ma".
+include "LambdaDelta-1/subst0/fwd.ma".
 
-include "getl/getl.ma".
+include "LambdaDelta-1/getl/getl.ma".
 
 theorem arity_gen_cvoid_subst0:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/asucc/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/asucc/defs.ma
index 9dfd88b80..6cf0f1686 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/asucc/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/asucc/defs.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "A/defs.ma".
+include "LambdaDelta-1/A/defs.ma".
 
-include "G/defs.ma".
+include "LambdaDelta-1/G/defs.ma".
 
 definition asucc:
  G \to (A \to A)
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/asucc/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/asucc/fwd.ma
index 0093edc25..f8fae3129 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/asucc/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/asucc/fwd.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "asucc/defs.ma".
+include "LambdaDelta-1/asucc/defs.ma".
 
 theorem asucc_gen_sort:
  \forall (g: G).(\forall (h: nat).(\forall (n: nat).(\forall (a: A).((eq A 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/cimp/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/cimp/defs.ma
index 3182b1231..4cb6fd83f 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/cimp/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/cimp/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "getl/defs.ma".
+include "LambdaDelta-1/getl/defs.ma".
 
 definition cimp:
  C \to (C \to Prop)
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/cimp/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/cimp/props.ma
index f8baf8cae..40e480d6d 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/cimp/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/cimp/props.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "cimp/defs.ma".
+include "LambdaDelta-1/cimp/defs.ma".
 
-include "getl/getl.ma".
+include "LambdaDelta-1/getl/getl.ma".
 
 theorem cimp_flat_sx:
  \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp (CHead c (Flat f) v) 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clear/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clear/defs.ma
index 39deb2fe5..2e2e89790 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clear/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clear/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "C/defs.ma".
+include "LambdaDelta-1/C/defs.ma".
 
 inductive clear: C \to (C \to Prop) \def
 | clear_bind: \forall (b: B).(\forall (e: C).(\forall (u: T).(clear (CHead e 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clear/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clear/drop.ma
index 07fb8137b..5dcff6de4 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clear/drop.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clear/drop.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "clear/fwd.ma".
+include "LambdaDelta-1/clear/fwd.ma".
 
-include "drop/fwd.ma".
+include "LambdaDelta-1/drop/fwd.ma".
 
 theorem drop_clear:
  \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((drop (S i) O c1 c2) \to 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clear/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clear/fwd.ma
index bec178d77..c03161330 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clear/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clear/fwd.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "clear/defs.ma".
+include "LambdaDelta-1/clear/defs.ma".
 
 theorem clear_gen_sort:
  \forall (x: C).(\forall (n: nat).((clear (CSort n) x) \to (\forall (P: 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clear/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clear/props.ma
index 6a99a80d7..134c21448 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clear/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clear/props.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "clear/fwd.ma".
+include "LambdaDelta-1/clear/fwd.ma".
 
 theorem clear_clear:
  \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (clear c2 c2)))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clen/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clen/defs.ma
index a17a17307..f9d73e9da 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clen/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clen/defs.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "C/defs.ma".
+include "LambdaDelta-1/C/defs.ma".
 
-include "s/defs.ma".
+include "LambdaDelta-1/s/defs.ma".
 
 definition clen:
  C \to nat
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clen/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clen/getl.ma
index dcfb3d8bb..6f15e6d31 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clen/getl.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/clen/getl.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "clen/defs.ma".
+include "LambdaDelta-1/clen/defs.ma".
 
-include "getl/props.ma".
+include "LambdaDelta-1/getl/props.ma".
 
 theorem getl_ctail_clen:
  \forall (b: B).(\forall (t: T).(\forall (c: C).(ex nat (\lambda (n: 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/cnt/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/cnt/defs.ma
index a1e32c285..f828cd724 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/cnt/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/cnt/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "T/defs.ma".
+include "LambdaDelta-1/T/defs.ma".
 
 inductive cnt: T \to Prop \def
 | cnt_sort: \forall (n: nat).(cnt (TSort n))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/cnt/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/cnt/props.ma
index 9d4c19f9f..2185c5a8f 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/cnt/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/cnt/props.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "cnt/defs.ma".
+include "LambdaDelta-1/cnt/defs.ma".
 
-include "lift/fwd.ma".
+include "LambdaDelta-1/lift/fwd.ma".
 
 theorem cnt_lift:
  \forall (t: T).((cnt t) \to (\forall (i: nat).(\forall (d: nat).(cnt (lift i 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/arity.ma
index 9b4f9d222..c898adc91 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/arity.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/arity.ma
@@ -14,13 +14,13 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csuba/getl.ma".
+include "LambdaDelta-1/csuba/getl.ma".
 
-include "csuba/props.ma".
+include "LambdaDelta-1/csuba/props.ma".
 
-include "arity/props.ma".
+include "LambdaDelta-1/arity/props.ma".
 
-include "T/props.ma".
+include "LambdaDelta-1/T/props.ma".
 
 theorem csuba_arity:
  \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/clear.ma
index 3a21edd7d..8c424f5d8 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/clear.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/clear.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csuba/defs.ma".
+include "LambdaDelta-1/csuba/defs.ma".
 
-include "clear/fwd.ma".
+include "LambdaDelta-1/clear/fwd.ma".
 
 theorem csuba_clear_conf:
  \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/defs.ma
index 5b1bb92c3..6d09db0c9 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "arity/defs.ma".
+include "LambdaDelta-1/arity/defs.ma".
 
 inductive csuba (g: G): C \to (C \to Prop) \def
 | csuba_sort: \forall (n: nat).(csuba g (CSort n) (CSort n))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/drop.ma
index 4d965603c..599b90c53 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/drop.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/drop.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csuba/fwd.ma".
+include "LambdaDelta-1/csuba/fwd.ma".
 
-include "drop/fwd.ma".
+include "LambdaDelta-1/drop/fwd.ma".
 
 theorem csuba_drop_abbr:
  \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/fwd.ma
index 6a34ae7e7..9635a44a0 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/fwd.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csuba/defs.ma".
+include "LambdaDelta-1/csuba/defs.ma".
 
 theorem csuba_gen_abbr:
  \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g 
@@ -22,54 +22,54 @@ theorem csuba_gen_abbr:
 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))
 \def
  \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: 
-(csuba g (CHead d1 (Bind Abbr) u) c)).(let H0 \def (match H in csuba return 
-(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 
-(CHead d1 (Bind Abbr) u)) \to ((eq C c1 c) \to (ex2 C (\lambda (d2: C).(eq C 
-c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) with 
-[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Bind 
-Abbr) u))).(\lambda (H1: (eq C (CSort n) c)).((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 d1 (Bind Abbr) 
-u) H0) in (False_ind ((eq C (CSort n) c) \to (ex2 C (\lambda (d2: C).(eq C c 
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) H2)) H1))) | 
-(csuba_head c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) 
-(CHead d1 (Bind Abbr) u))).(\lambda (H2: (eq C (CHead c2 k u0) c)).((let H3 
-\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 c1 k 
-u0) (CHead d1 (Bind Abbr) u) 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 u0) (CHead d1 (Bind Abbr) u) 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 c0 _ _) \Rightarrow c0])) 
-(CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) in (eq_ind C d1 (\lambda (c0: 
-C).((eq K k (Bind Abbr)) \to ((eq T u0 u) \to ((eq C (CHead c2 k u0) c) \to 
-((csuba g c0 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) 
-u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (H6: (eq K k (Bind 
-Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead 
-c2 k0 u0) c) \to ((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead 
-d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) (\lambda (H7: (eq 
-T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c2 (Bind Abbr) t) c) \to 
-((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) 
-u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (H8: (eq C (CHead c2 
-(Bind Abbr) u) c)).(eq_ind C (CHead c2 (Bind Abbr) u) (\lambda (c0: 
-C).((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind 
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (H9: (csuba g d1 
-c2)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2 
-(Bind Abbr) u)) H9)) c H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Bind Abbr) 
-H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a 
-H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1 
-(Bind Abbr) u))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) c)).((let H5 
-\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 d1 (Bind Abbr) u) H3) in (False_ind ((eq C 
-(CHead c2 (Bind Abbr) u0) c) \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 c (CHead d2 
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) H5)) H4 H0 H1 H2)))]) 
-in (H0 (refl_equal C (CHead d1 (Bind Abbr) u)) (refl_equal C c))))))).
+(csuba g (CHead d1 (Bind Abbr) u) c)).(insert_eq C (CHead d1 (Bind Abbr) u) 
+(\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq 
+C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (\lambda 
+(y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda 
+(c1: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C 
+c1 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda 
+(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr) u))).(let H2 
+\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: 
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
+False])) I (CHead d1 (Bind Abbr) u) H1) in (False_ind (ex2 C (\lambda (d2: 
+C).(eq C (CSort n) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
+d2))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 
+c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda 
+(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
+d2)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c1 k u0) 
+(CHead d1 (Bind Abbr) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e 
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ 
+_) \Rightarrow c0])) (CHead c1 k u0) (CHead d1 (Bind Abbr) 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 c1 k 
+u0) (CHead d1 (Bind Abbr) u) H3) in ((let H6 \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 c1 k u0) (CHead d1 (Bind Abbr) u) H3) 
+in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r 
+T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k t) (CHead d2 
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (eq_ind_r K (Bind Abbr) 
+(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u) (CHead d2 
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H9 \def (eq_ind C 
+c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda 
+(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
+d2))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 
+c2)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) 
+u) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 
+(refl_equal C (CHead c2 (Bind Abbr) u)) H10))) k H7) u0 H6)))) H5)) 
+H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 
+c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda 
+(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
+d2)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g 
+a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C 
+(CHead c1 (Bind Abst) t) (CHead d1 (Bind Abbr) u))).(let H6 \def (eq_ind C 
+(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee 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) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 
+(Bind Abbr) u0) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) 
+H6)))))))))))) y c H0))) H))))).
 
 theorem csuba_gen_void:
  \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g 
@@ -77,54 +77,54 @@ theorem csuba_gen_void:
 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))))))
 \def
  \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: 
-(csuba g (CHead d1 (Bind Void) u) c)).(let H0 \def (match H in csuba return 
-(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 
-(CHead d1 (Bind Void) u)) \to ((eq C c1 c) \to (ex2 C (\lambda (d2: C).(eq C 
-c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) with 
-[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Bind 
-Void) u))).(\lambda (H1: (eq C (CSort n) c)).((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 d1 (Bind Void) 
-u) H0) in (False_ind ((eq C (CSort n) c) \to (ex2 C (\lambda (d2: C).(eq C c 
-(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))) H2)) H1))) | 
-(csuba_head c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) 
-(CHead d1 (Bind Void) u))).(\lambda (H2: (eq C (CHead c2 k u0) c)).((let H3 
-\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 c1 k 
-u0) (CHead d1 (Bind Void) u) 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 u0) (CHead d1 (Bind Void) u) 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 c0 _ _) \Rightarrow c0])) 
-(CHead c1 k u0) (CHead d1 (Bind Void) u) H1) in (eq_ind C d1 (\lambda (c0: 
-C).((eq K k (Bind Void)) \to ((eq T u0 u) \to ((eq C (CHead c2 k u0) c) \to 
-((csuba g c0 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) 
-u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (H6: (eq K k (Bind 
-Void))).(eq_ind K (Bind Void) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead 
-c2 k0 u0) c) \to ((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead 
-d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))))))) (\lambda (H7: (eq 
-T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c2 (Bind Void) t) c) \to 
-((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) 
-u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (H8: (eq C (CHead c2 
-(Bind Void) u) c)).(eq_ind C (CHead c2 (Bind Void) u) (\lambda (c0: 
-C).((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind 
-Void) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (H9: (csuba g d1 
-c2)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Void) u) (CHead d2 
-(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2 
-(Bind Void) u)) H9)) c H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Bind Void) 
-H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a 
-H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1 
-(Bind Void) u))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) c)).((let H5 
-\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 d1 (Bind Void) u) H3) in (False_ind ((eq C 
-(CHead c2 (Bind Abbr) u0) c) \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 c (CHead d2 
-(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))))))) H5)) H4 H0 H1 H2)))]) 
-in (H0 (refl_equal C (CHead d1 (Bind Void) u)) (refl_equal C c))))))).
+(csuba g (CHead d1 (Bind Void) u) c)).(insert_eq C (CHead d1 (Bind Void) u) 
+(\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq 
+C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))) (\lambda 
+(y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda 
+(c1: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C 
+c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda 
+(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H2 
+\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: 
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
+False])) I (CHead d1 (Bind Void) u) H1) in (False_ind (ex2 C (\lambda (d2: 
+C).(eq C (CSort n) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 
+d2))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 
+c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda 
+(d2: C).(eq C c2 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 
+d2)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c1 k u0) 
+(CHead d1 (Bind Void) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e 
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ 
+_) \Rightarrow c0])) (CHead c1 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 c1 k 
+u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \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 c1 k u0) (CHead d1 (Bind Void) u) H3) 
+in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r 
+T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k t) (CHead d2 
+(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))) (eq_ind_r K (Bind Void) 
+(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u) (CHead d2 
+(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H9 \def (eq_ind C 
+c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda 
+(d2: C).(eq C c2 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 
+d2))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 
+c2)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Void) 
+u) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 
+(refl_equal C (CHead c2 (Bind Void) u)) H10))) k H7) u0 H6)))) H5)) 
+H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 
+c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda 
+(d2: C).(eq C c2 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 
+d2)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g 
+a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C 
+(CHead c1 (Bind Abst) t) (CHead d1 (Bind Void) u))).(let H6 \def (eq_ind C 
+(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee 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 Void) u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 
+(Bind Abbr) u0) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))) 
+H6)))))))))))) y c H0))) H))))).
 
 theorem csuba_gen_abst:
  \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g 
@@ -137,122 +137,117 @@ a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
 a))))))))))
 \def
  \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda 
-(H: (csuba g (CHead d1 (Bind Abst) u1) c)).(let H0 \def (match H in csuba 
-return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C 
-c0 (CHead d1 (Bind Abst) u1)) \to ((eq C c1 c) \to (or (ex2 C (\lambda (d2: 
+(H: (csuba g (CHead d1 (Bind Abst) u1) c)).(insert_eq C (CHead d1 (Bind Abst) 
+u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(or (ex2 C (\lambda (d2: 
 C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
 (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead 
 d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
 A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
 g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 a))))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: 
-(eq C (CSort n) (CHead d1 (Bind Abst) u1))).(\lambda (H1: (eq C (CSort n) 
-c)).((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 d1 (Bind Abst) u1) H0) in (False_ind ((eq C 
-(CSort n) c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) 
+A).(arity g d2 u2 a))))))) (\lambda (y: C).(\lambda (H0: (csuba g y 
+c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind 
+Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) 
 u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) 
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abbr) u2))))) 
 (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
 (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) 
-H2)) H1))) | (csuba_head c1 c2 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead 
-c1 k u) (CHead d1 (Bind Abst) u1))).(\lambda (H2: (eq C (CHead c2 k u) 
-c)).((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 d1 (Bind Abst) u1) 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 d1 
-(Bind Abst) u1) 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 c0 _ _) 
-\Rightarrow c0])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H1) in (eq_ind C 
-d1 (\lambda (c0: C).((eq K k (Bind Abst)) \to ((eq T u u1) \to ((eq C (CHead 
-c2 k u) c) \to ((csuba g c0 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c 
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind 
-Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 
-d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc 
-g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a))))))))))) (\lambda (H6: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) 
-(\lambda (k0: K).((eq T u u1) \to ((eq C (CHead c2 k0 u) c) \to ((csuba g d1 
-c2) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) 
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) 
+(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) 
+u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return 
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) 
+\Rightarrow False])) I (CHead d1 (Bind Abst) u1) H1) in (False_ind (or (ex2 C 
+(\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(eq C (CSort n) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
 C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
 C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))) (\lambda 
-(H7: (eq T u u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 (Bind Abst) 
-t) c) \to ((csuba g d1 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a))))))))) (\lambda (H8: (eq C (CHead c2 (Bind Abst) u1) c)).(eq_ind C (CHead 
-c2 (Bind Abst) u1) (\lambda (c0: C).((csuba g d1 c2) \to (or (ex2 C (\lambda 
-(d2: C).(eq C c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) H2)))) 
+(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda 
+(H2: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: 
+C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 
 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
 A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
 g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 a)))))))) (\lambda (H9: (csuba g d1 c2)).(or_introl (ex2 C 
-(\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) (CHead d2 (Bind Abst) u1))) 
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abst) u1) (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a))))) (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) (CHead 
-d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C 
-(CHead c2 (Bind Abst) u1)) H9))) c H8)) u (sym_eq T u u1 H7))) k (sym_eq K k 
-(Bind Abst) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst 
-c1 c2 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) 
-t) (CHead d1 (Bind Abst) u1))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u) 
-c)).((let H5 \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 d1 (Bind Abst) u1) H3) in ((let H6 \def 
-(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind 
-Abst) t) (CHead d1 (Bind Abst) u1) H3) in (eq_ind C d1 (\lambda (c0: C).((eq 
-T t u1) \to ((eq C (CHead c2 (Bind Abbr) u) c) \to ((csuba g c0 c2) \to 
-((arity g c0 t (asucc g a)) \to ((arity g c2 u a) \to (or (ex2 C (\lambda 
-(d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c 
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity 
-g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: 
-A).(arity g d2 u2 a0)))))))))))) (\lambda (H7: (eq T t u1)).(eq_ind T u1 
-(\lambda (t0: T).((eq C (CHead c2 (Bind Abbr) u) c) \to ((csuba g d1 c2) \to 
-((arity g d1 t0 (asucc g a)) \to ((arity g c2 u a) \to (or (ex2 C (\lambda 
-(d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c 
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity 
-g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: 
-A).(arity g d2 u2 a0))))))))))) (\lambda (H8: (eq C (CHead c2 (Bind Abbr) u) 
-c)).(eq_ind C (CHead c2 (Bind Abbr) u) (\lambda (c0: C).((csuba g d1 c2) \to 
-((arity g d1 u1 (asucc g a)) \to ((arity g c2 u a) \to (or (ex2 C (\lambda 
-(d2: C).(eq C c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity 
-g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: 
-A).(arity g d2 u2 a0)))))))))) (\lambda (H9: (csuba g d1 c2)).(\lambda (H10: 
-(arity g d1 u1 (asucc g a))).(\lambda (H11: (arity g c2 u a)).(or_intror (ex2 
-C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abst) u1))) 
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+A).(arity g d2 u2 a))))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: 
+(eq C (CHead c1 k u) (CHead d1 (Bind Abst) u1))).(let H4 \def (f_equal C C 
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
+\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 k u) (CHead d1 
+(Bind Abst) 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 c1 k u) (CHead d1 (Bind Abst) u1) H3) in ((let H6 
+\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 d1 (Bind Abst) u1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda 
+(H8: (eq C c1 d1)).(eq_ind_r T u1 (\lambda (t: T).(or (ex2 C (\lambda (d2: 
+C).(eq C (CHead c2 k t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g 
+d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C 
+(CHead c2 k t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (eq_ind_r K (Bind Abst) 
+(\lambda (k0: K).(or (ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u1) (CHead d2 
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 k0 u1) (CHead d2 
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
+u2 a))))))) (let H9 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 
+(Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind 
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))) H2 
+d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 
+H8) in (or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abst) 
+u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: 
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 
+(Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) 
+c2 (refl_equal C (CHead c2 (Bind Abst) u1)) H10)))) k H7) u H6)))) H5)) 
+H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 
+c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C 
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba 
+g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq 
+C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: 
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(a: A).(arity g d2 u2 a))))))))).(\lambda (t: T).(\lambda (a: A).(\lambda 
+(H3: (arity g c1 t (asucc g a))).(\lambda (u: T).(\lambda (H4: (arity g c2 u 
+a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) 
+u1))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda 
+(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) 
+(CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1) H5) in ((let H7 \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 d1 (Bind Abst) u1) H5) in (\lambda (H8: (eq C c1 d1)).(let H9 
+\def (eq_ind T t (\lambda (t0: T).(arity g c1 t0 (asucc g a))) H3 u1 H7) in 
+(let H10 \def (eq_ind C c1 (\lambda (c0: C).(arity g c0 u1 (asucc g a))) H9 
+d1 H8) in (let H11 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 
+(Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind 
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))))))) 
+H2 d1 H8) in (let H12 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 
+d1 H8) in (or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) 
+u) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: 
+A).(arity g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(a0: A).(arity g d2 u2 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda 
 (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) 
 u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
 (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g 
 a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 
-a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))) c2 u a 
-(refl_equal C (CHead c2 (Bind Abbr) u)) H9 H10 H11))))) c H8)) t (sym_eq T t 
-u1 H7))) c1 (sym_eq C c1 d1 H6))) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C 
-(CHead d1 (Bind Abst) u1)) (refl_equal C c))))))).
+a0)))) c2 u a (refl_equal C (CHead c2 (Bind Abbr) u)) H12 H10 H4)))))))) 
+H6)))))))))))) y c H0))) H))))).
 
 theorem csuba_gen_flat:
  \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall 
@@ -261,57 +256,57 @@ C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2:
 C).(\lambda (_: T).(csuba g d1 d2)))))))))
 \def
  \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda 
-(f: F).(\lambda (H: (csuba g (CHead d1 (Flat f) u1) c)).(let H0 \def (match H 
-in csuba return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 
-c1)).((eq C c0 (CHead d1 (Flat f) u1)) \to ((eq C c1 c) \to (ex2_2 C T 
+(f: F).(\lambda (H: (csuba g (CHead d1 (Flat f) u1) c)).(insert_eq C (CHead 
+d1 (Flat f) u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(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 d1 d2))))))))) with [(csuba_sort n) 
-\Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Flat f) u1))).(\lambda 
-(H1: (eq C (CSort n) c)).((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 d1 (Flat f) u1) H0) in 
-(False_ind ((eq C (CSort n) c) \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 d1 d2))))) H2)) H1))) | (csuba_head c1 c2 H0 k u) \Rightarrow (\lambda (H1: 
-(eq C (CHead c1 k u) (CHead d1 (Flat f) u1))).(\lambda (H2: (eq C (CHead c2 k 
-u) c)).((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 d1 (Flat f) u1) 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 d1 
-(Flat f) u1) 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 c0 _ _) 
-\Rightarrow c0])) (CHead c1 k u) (CHead d1 (Flat f) u1) H1) in (eq_ind C d1 
-(\lambda (c0: C).((eq K k (Flat f)) \to ((eq T u u1) \to ((eq C (CHead c2 k 
-u) c) \to ((csuba g c0 c2) \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 d1 d2))))))))) (\lambda (H6: (eq K k (Flat f))).(eq_ind K (Flat f) (\lambda 
-(k0: K).((eq T u u1) \to ((eq C (CHead c2 k0 u) c) \to ((csuba g d1 c2) \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 d1 d2)))))))) (\lambda (H7: 
-(eq T u u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 (Flat f) t) c) \to 
-((csuba g d1 c2) \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 d1 
-d2))))))) (\lambda (H8: (eq C (CHead c2 (Flat f) u1) c)).(eq_ind C (CHead c2 
-(Flat f) u1) (\lambda (c0: C).((csuba g d1 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 d1 d2)))))) (\lambda (H9: (csuba g d1 
-c2)).(ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 (Flat 
-f) u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 
-d2))) c2 u1 (refl_equal C (CHead c2 (Flat f) u1)) H9)) c H8)) u (sym_eq T u 
-u1 H7))) k (sym_eq K k (Flat f) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 
-H0))) | (csuba_abst c1 c2 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C 
-(CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u1))).(\lambda (H4: (eq C (CHead 
-c2 (Bind Abbr) u) c)).((let H5 \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 _) \Rightarrow True | (Flat _) \Rightarrow 
-False])])) I (CHead d1 (Flat f) u1) H3) in (False_ind ((eq C (CHead c2 (Bind 
-Abbr) u) c) \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 c (CHead d2 
-(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))) H5)) 
-H4 H0 H1 H2)))]) in (H0 (refl_equal C (CHead d1 (Flat f) u1)) (refl_equal C 
-c)))))))).
+(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (\lambda (y: C).(\lambda (H0: 
+(csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0 
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq 
+C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 
+d2))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f) 
+u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return 
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) 
+\Rightarrow False])) I (CHead d1 (Flat f) u1) H1) 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 d1 d2)))) H2)))) (\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1 
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq 
+C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 
+d2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) 
+(CHead d1 (Flat f) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e 
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ 
+_) \Rightarrow c0])) (CHead c1 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 c1 k 
+u) (CHead d1 (Flat f) u1) H3) in ((let H6 \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 d1 (Flat f) u1) H3) in 
+(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r T u1 
+(\lambda (t: T).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 
+k t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 
+d2))))) (eq_ind_r K (Flat f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(eq C (CHead c2 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (let H9 \def (eq_ind C c1 
+(\lambda (c0: C).((eq C c0 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda 
+(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d1 d2)))))) H2 d1 H8) in (let H10 \def (eq_ind C 
+c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in (ex2_2_intro C T (\lambda 
+(d2: C).(\lambda (u2: T).(eq C (CHead c2 (Flat f) u1) (CHead d2 (Flat f) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))) c2 u1 (refl_equal C 
+(CHead c2 (Flat f) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1 
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq 
+C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 
+d2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g 
+a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C 
+(CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C 
+(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee 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 (ex2_2 C 
+T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 
+(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))) 
+H6)))))))))))) y c H0))) H)))))).
 
 theorem csuba_gen_bind:
  \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall 
@@ -320,92 +315,80 @@ B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2)))))
 (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))))))
 \def
  \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda 
-(v1: T).(\lambda (H: (csuba g (CHead e1 (Bind b1) v1) c2)).(let H0 \def 
-(match H in csuba return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csuba 
-? 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).(csuba g e1 
-e2)))))))))) with [(csuba_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 
+(v1: T).(\lambda (H: (csuba g (CHead e1 (Bind b1) v1) c2)).(insert_eq C 
+(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c c2)) (\lambda (_: 
+C).(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 e1 e2)))))) H2)) H1))) | (csuba_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 ((csuba 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).(csuba 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 ((csuba 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).(csuba 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 ((csuba 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).(csuba g e1 
-e2)))))))) (\lambda (H8: (eq C (CHead c0 (Bind b1) v1) c2)).(eq_ind C (CHead 
-c0 (Bind b1) v1) (\lambda (c: C).((csuba 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).(csuba g e1 e2))))))) 
-(\lambda (H9: (csuba g e1 c0)).(let H10 \def (eq_ind_r C c2 (\lambda (c: 
-C).(csuba 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).(csuba 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))) | (csuba_abst c1 c0 H0 t a 
-H1 u H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead e1 
-(Bind b1) v1))).(\lambda (H4: (eq C (CHead c0 (Bind Abbr) u) c2)).((let H5 
-\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) H3) in ((let H6 \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) H3) in ((let H7 \def 
+T).(csuba g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csuba g y 
+c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind 
+b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
+T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csuba g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq 
+C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) 
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) 
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) 
+v1) H1) 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 e1 e2))))) H2)))) (\lambda (c1: C).(\lambda 
+(c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 
+(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda 
+(v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (k: K).(\lambda (u: 
+T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(let H4 \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) H3) 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 ((csuba 
-g c c0) \to ((arity g c 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 c2 (CHead e2 (Bind 
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 
-e2)))))))))))) (\lambda (H8: (eq B Abst b1)).(eq_ind B Abst (\lambda (_: 
-B).((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csuba g e1 c0) 
-\to ((arity g e1 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 c2 (CHead e2 (Bind 
+[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 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 c1 k u) (CHead e1 (Bind b1) v1) H3) 
+in ((let H6 \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) H3) in (\lambda (H7: (eq K k (Bind 
+b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t) 
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
+T).(csuba g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) 
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
+T).(csuba g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c 
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 H8) in (let 
+H10 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H8) in 
+(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
+(CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda 
+(e2: C).(\lambda (_: T).(csuba g e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3 
+(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
+C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (t: 
+T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc g a))).(\lambda (u: 
+T).(\lambda (_: (arity g c3 u a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) 
+t) (CHead e1 (Bind b1) v1))).(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) H5) in 
+((let H7 \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) H5) in 
+((let H8 \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) H5) in (\lambda (H9: (eq B Abst 
+b1)).(\lambda (H10: (eq C c1 e1)).(let H11 \def (eq_ind T t (\lambda (t0: 
+T).(arity g c1 t0 (asucc g a))) H3 v1 H8) in (let H12 \def (eq_ind C c1 
+(\lambda (c: C).(arity g c v1 (asucc g a))) H11 e1 H10) in (let H13 \def 
+(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C 
+T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind 
 b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 
-e2))))))))))) (\lambda (H9: (eq T t v1)).(eq_ind T v1 (\lambda (t0: T).((eq C 
-(CHead c0 (Bind Abbr) u) c2) \to ((csuba g e1 c0) \to ((arity g e1 t0 (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 c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
-B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))))) (\lambda (H10: 
-(eq C (CHead c0 (Bind Abbr) u) c2)).(eq_ind C (CHead c0 (Bind Abbr) u) 
-(\lambda (c: C).((csuba g e1 c0) \to ((arity g e1 v1 (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 e1 e2))))))))) (\lambda (H11: (csuba g e1 
-c0)).(\lambda (_: (arity g e1 v1 (asucc g a))).(\lambda (_: (arity g c0 u 
-a)).(let H14 \def (eq_ind_r C c2 (\lambda (c: C).(csuba g (CHead e1 (Bind b1) 
-v1) c)) H (CHead c0 (Bind Abbr) u) H10) in (let H15 \def (eq_ind_r B b1 
-(\lambda (b: B).(csuba g (CHead e1 (Bind b) v1) (CHead c0 (Bind Abbr) u))) 
-H14 Abst H8) 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).(csuba g e1 e2)))) Abbr c0 u 
-(refl_equal C (CHead c0 (Bind Abbr) u)) H11)))))) c2 H10)) t (sym_eq T t v1 
-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)))))))).
+e2))))))) H2 e1 H10) in (let H14 \def (eq_ind C c1 (\lambda (c: C).(csuba g c 
+c3)) H1 e1 H10) in (let H15 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 
+(CHead e1 (Bind b) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H13 Abst H9) in 
+(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
+(CHead c3 (Bind Abbr) u) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda 
+(e2: C).(\lambda (_: T).(csuba g e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 
+(Bind Abbr) u)) H14))))))))) H7)) H6)))))))))))) y c2 H0))) H)))))).
 
 theorem csuba_gen_abst_rev:
  \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c 
@@ -413,62 +396,54 @@ theorem csuba_gen_abst_rev:
 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: 
+(csuba g c (CHead d1 (Bind Abst) u))).(insert_eq C (CHead d1 (Bind Abst) u) 
+(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq 
+C c (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (\lambda 
+(y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda 
+(c1: C).((eq C c1 (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 
+(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H2 
+\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: 
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
+False])) I (CHead d1 (Bind Abst) u) H1) in (False_ind (ex2 C (\lambda (d2: 
+C).(eq C (CSort n) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
+d1))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 
+c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda 
+(d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
+d1)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k u0) 
+(CHead d1 (Bind Abst) u))).(let H4 \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 ((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 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)))))))).
+in (\lambda (H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r 
+T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (eq_ind_r K (Bind Abst) 
+(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k0 u) (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H9 \def (eq_ind C 
+c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda 
+(d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
+d1))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 
+c0)) H1 d1 H8) in (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)) H10))) k H7) u0 H6)))) H5)) 
+H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 
+c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda 
+(d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
+d1)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g 
+a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\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 (ee: C).(match ee 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 (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 y H0))) H))))).
 
 theorem csuba_gen_void_rev:
  \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c 
@@ -476,62 +451,54 @@ theorem csuba_gen_void_rev:
 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: 
+(csuba g c (CHead d1 (Bind Void) u))).(insert_eq C (CHead d1 (Bind Void) u) 
+(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq 
+C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (\lambda 
+(y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda 
+(c1: C).((eq C c1 (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 
+(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H2 
+\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: 
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
+False])) I (CHead d1 (Bind Void) u) H1) in (False_ind (ex2 C (\lambda (d2: 
+C).(eq C (CSort n) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 
+d1))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 
+c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda 
+(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 
+d1)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k u0) 
+(CHead d1 (Bind Void) u))).(let H4 \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 ((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 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)))))))).
+in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r 
+T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead d2 
+(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (eq_ind_r K (Bind Void) 
+(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k0 u) (CHead d2 
+(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H9 \def (eq_ind C 
+c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda 
+(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 
+d1))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 
+c0)) H1 d1 H8) in (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)) H10))) k H7) u0 H6)))) H5)) 
+H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 
+c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda 
+(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 
+d1)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g 
+a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\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 (ee: C).(match ee 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 (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 y H0))) H))))).
 
 theorem csuba_gen_abbr_rev:
  \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g c 
@@ -544,132 +511,117 @@ 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: 
+(H: (csuba g c (CHead d1 (Bind Abbr) u1))).(insert_eq C (CHead d1 (Bind Abbr) 
+u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(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 
+(a: A).(arity g d1 u1 a))))))) (\lambda (y: C).(\lambda (H0: (csuba g c 
+y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c1 (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: 
+(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr) 
+u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return 
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) 
+\Rightarrow False])) I (CHead d1 (Bind Abbr) u1) H1) 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)))))) H2)))) (\lambda 
+(c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C 
+c2 (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (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 c1 (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 (u: T).(\lambda (H3: (eq C (CHead c2 k u) 
+(CHead d1 (Bind Abbr) u1))).(let H4 \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 ((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 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 (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r 
+T u1 (\lambda (t: T).(or (ex2 C (\lambda (d2: C).(eq C (CHead c1 k 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 k 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))))))) (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or (ex2 C (\lambda (d2: 
+C).(eq C (CHead c1 k0 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 k0 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))))))))) (\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 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))))) (let H9 \def (eq_ind C c2 
+(\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda 
+(d2: C).(eq C c1 (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 c1 
+(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)))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 
+(\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (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) t) (CHead d2 (Bind Abst) 
+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)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 
+(CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (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 c1 (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 _) 
+a))))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc 
+g a))).(\lambda (u: T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C 
+(CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1))).(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 (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 d1 (Bind Abbr) u1) H5) in ((let H7 \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 (\lambda (H8: (eq C c2 d1)).(let H9 \def (eq_ind T u 
+(\lambda (t0: T).(arity g c2 t0 a)) H4 u1 H7) in (let H10 \def (eq_ind C c2 
+(\lambda (c0: C).(arity g c0 u1 a)) H9 d1 H8) in (let H11 \def (eq_ind C c2 
+(\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda 
+(d2: C).(eq C c1 (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 c1 
 (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) 
+(a0: A).(arity g d1 u1 a0)))))))) H2 d1 H8) in (let H12 \def (eq_ind C c2 
+(\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (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)))) 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)))))))).
+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)) H12 H3 H10)))))))) H6)))))))))))) c y 
+H0))) H))))).
 
 theorem csuba_gen_flat_rev:
  \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall 
@@ -678,65 +630,57 @@ 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 
+(f: F).(\lambda (H: (csuba g c (CHead d1 (Flat f) u1))).(insert_eq C (CHead 
+d1 (Flat f) u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(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 (_: 
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (y: C).(\lambda (H0: 
+(csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c1 
+(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 (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f) 
+u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return 
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) 
+\Rightarrow False])) I (CHead d1 (Flat f) u1) H1) 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)))) H2)))) (\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq 
+C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u) 
+(CHead d1 (Flat f) u1))).(let H4 \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 ((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 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 
+(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r T u1 
+(\lambda (t: T).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 
+k t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))) (eq_ind_r K (Flat f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(eq C (CHead c1 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (let H9 \def (eq_ind C c2 
+(\lambda (c0: C).((eq C c0 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda 
+(d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))))) H2 d1 H8) in (let H10 \def (eq_ind C 
+c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (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)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2 
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq 
+C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g 
+a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\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 (ee: C).(match ee 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))))))))).
+_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind (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 y H0))) H)))))).
 
 theorem csuba_gen_bind_rev:
  \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall 
@@ -745,97 +689,78 @@ 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) 
+(v1: T).(\lambda (H: (csuba g c2 (CHead e1 (Bind b1) v1))).(insert_eq C 
+(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c2 c)) (\lambda (_: 
+C).(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)))))) (\lambda (y: C).(\lambda (H0: (csuba g c2 
+y)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((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 c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csuba g e2 e1)))))))) (\lambda (n: nat).(\lambda (H1: (eq 
+C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) 
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) 
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) 
+v1) H1) 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))))) H2)))) (\lambda (c1: C).(\lambda 
+(c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 (CHead e1 
+(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda 
+(v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (k: K).(\lambda (u: 
+T).(\lambda (H3: (eq C (CHead c3 k u) (CHead e1 (Bind b1) v1))).(let H4 \def 
+(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with 
+[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 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 _ k0 _) \Rightarrow k0])) (CHead c3 k u) (CHead e1 (Bind b1) v1) H3) 
+in ((let H6 \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 c3 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind 
+b1))).(\lambda (H8: (eq C c3 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k t) 
 (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) 
+T).(csuba g e2 e1)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k0 v1) 
 (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: 
+T).(csuba g e2 e1)))))) (let H9 \def (eq_ind C c3 (\lambda (c: C).((eq C c 
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
+C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 H8) in (let 
+H10 \def (eq_ind C c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H8) 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)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
+C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
+C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (t: 
+T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u: 
+T).(\lambda (H4: (arity g c3 u a)).(\lambda (H5: (eq C (CHead c3 (Bind Abbr) 
+u) (CHead e1 (Bind b1) v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match 
+e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c3 | (CHead c _ 
+_) \Rightarrow c])) (CHead c3 (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 c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in 
+((let H8 \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 
+c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in (\lambda (H9: (eq B Abbr 
+b1)).(\lambda (H10: (eq C c3 e1)).(let H11 \def (eq_ind T u (\lambda (t0: 
+T).(arity g c3 t0 a)) H4 v1 H8) in (let H12 \def (eq_ind C c3 (\lambda (c: 
+C).(arity g c v1 a)) H11 e1 H10) in (let H13 \def (eq_ind C c3 (\lambda (c: 
+C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 
+H10) in (let H14 \def (eq_ind C c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H10) 
+in (let H15 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) 
+v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq 
+C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda 
+(_: T).(csuba g e2 e1))))))) H13 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))))))))) (\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))))))))).
+e2 e1)))) Abst c1 t (refl_equal C (CHead c1 (Bind Abst) t)) H14))))))))) H7)) 
+H6)))))))))))) c2 y H0))) H)))))).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/getl.ma
index f75cc859a..6072645a0 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/getl.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/getl.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csuba/drop.ma".
+include "LambdaDelta-1/csuba/drop.ma".
 
-include "csuba/clear.ma".
+include "LambdaDelta-1/csuba/clear.ma".
 
-include "getl/clear.ma".
+include "LambdaDelta-1/getl/clear.ma".
 
 theorem csuba_getl_abbr:
  \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/props.ma
index c92cb5301..9710178ed 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/props.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csuba/defs.ma".
+include "LambdaDelta-1/csuba/defs.ma".
 
 theorem csuba_refl:
  \forall (g: G).(\forall (c: C).(csuba g c c))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/arity.ma
index 003658f0e..7f0630a13 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/arity.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/arity.ma
@@ -14,9 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubc/csuba.ma".
-
-include "arity/defs.ma".
+include "LambdaDelta-1/csubc/csuba.ma".
 
 theorem csubc_arity_conf:
  \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/clear.ma
index a386ad644..f266b8bc8 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/clear.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/clear.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubc/defs.ma".
+include "LambdaDelta-1/csubc/fwd.ma".
 
 theorem csubc_clear_conf:
  \forall (g: G).(\forall (c1: C).(\forall (e1: C).((clear c1 e1) \to (\forall 
@@ -25,123 +25,94 @@ theorem csubc_clear_conf:
 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)))).
+C).(\lambda (H0: (csubc g (CHead e (Bind b) u) c2)).(let H_x \def 
+(csubc_gen_head_l g e c2 u (Bind b) H0) in (let H1 \def H_x in (or_ind (ex2 C 
+(\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g e 
+c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K 
+(Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq 
+C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda 
+(_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 
+g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g 
+a c3 w))))) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g 
+(CHead e (Bind b) u) e2))) (\lambda (H2: (ex2 C (\lambda (c3: C).(eq C c2 
+(CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g e c3)))).(ex2_ind C 
+(\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g e 
+c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead 
+e (Bind b) u) e2))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x (Bind b) 
+u))).(\lambda (H4: (csubc g e x)).(eq_ind_r C (CHead x (Bind b) u) (\lambda 
+(c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead 
+e (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2: C).(clear (CHead x (Bind b) 
+u) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2)) (CHead x (Bind b) 
+u) (clear_bind b x u) (csubc_head g e x H4 (Bind b) u)) c2 H3)))) H2)) 
+(\lambda (H2: (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: 
+A).(eq K (Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda 
+(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda 
+(a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda 
+(w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: 
+C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda 
+(w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda (e2: C).(clear c2 
+e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H3: (eq K (Bind b) (Bind 
+Abst))).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H5: 
+(csubc g e x0)).(\lambda (H6: (sc3 g (asucc g x2) e u)).(\lambda (H7: (sc3 g 
+x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c: C).(ex2 C 
+(\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) 
+e2)))) (let H8 \def (f_equal K B (\lambda (e0: K).(match e0 in K return 
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])) 
+(Bind b) (Bind Abst) H3) in (eq_ind_r B Abst (\lambda (b0: B).(ex2 C (\lambda 
+(e2: C).(clear (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g 
+(CHead e (Bind b0) u) e2)))) (ex_intro2 C (\lambda (e2: C).(clear (CHead x0 
+(Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind Abst) u) e2)) 
+(CHead x0 (Bind Abbr) x1) (clear_bind Abbr x0 x1) (csubc_abst g e x0 H5 u x2 
+H6 x1 H7)) b H8)) c2 H4))))))))) H2)) H1)))))))) (\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 H_x \def (csubc_gen_head_l g e c2 u 
+(Flat f) H2) in (let H3 \def H_x in (or_ind (ex2 C (\lambda (c3: C).(eq C c2 
+(CHead c3 (Flat f) u))) (\lambda (c3: C).(csubc g e c3))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K (Flat f) (Bind 
+Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 
+(Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g 
+e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) e 
+u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) 
+(ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) 
+(\lambda (H4: (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) 
+(\lambda (c3: C).(csubc g e c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 
+(CHead c3 (Flat f) u))) (\lambda (c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: 
+C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x: C).(\lambda 
+(H5: (eq C c2 (CHead x (Flat f) u))).(\lambda (H6: (csubc g e x)).(eq_ind_r C 
+(CHead x (Flat f) u) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) 
+(\lambda (e2: C).(csubc g c e2)))) (let H_x0 \def (H1 x H6) in (let H7 \def 
+H_x0 in (ex2_ind C (\lambda (e2: C).(clear x e2)) (\lambda (e2: C).(csubc g c 
+e2)) (ex2 C (\lambda (e2: C).(clear (CHead x (Flat f) u) e2)) (\lambda (e2: 
+C).(csubc g c e2))) (\lambda (x0: C).(\lambda (H8: (clear x x0)).(\lambda 
+(H9: (csubc g c x0)).(ex_intro2 C (\lambda (e2: C).(clear (CHead x (Flat f) 
+u) e2)) (\lambda (e2: C).(csubc g c e2)) x0 (clear_flat x x0 H8 f u) H9)))) 
+H7))) c2 H5)))) H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda 
+(w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: 
+C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda 
+(w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K (Flat f) (Bind Abst))))) (\lambda 
+(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda 
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda 
+(e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H5: (eq K (Flat f) (Bind 
+Abst))).(\lambda (H6: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (_: 
+(csubc g e x0)).(\lambda (_: (sc3 g (asucc g x2) e u)).(\lambda (_: (sc3 g x2 
+x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C 
+(\lambda (e2: C).(clear c0 e2)) (\lambda (e2: C).(csubc g c e2)))) (let H10 
+\def (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return (\lambda (_: 
+K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) I 
+(Bind Abst) H5) in (False_ind (ex2 C (\lambda (e2: C).(clear (CHead x0 (Bind 
+Abbr) x1) e2)) (\lambda (e2: C).(csubc g c e2))) H10)) c2 H6))))))))) H4)) 
+H3))))))))))) c1 e1 H)))).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/csuba.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/csuba.ma
index 49c062fd1..df5b989c7 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/csuba.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/csuba.ma
@@ -14,11 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubc/defs.ma".
+include "LambdaDelta-1/csubc/defs.ma".
 
-include "sc3/props.ma".
-
-include "csuba/defs.ma".
+include "LambdaDelta-1/sc3/props.ma".
 
 theorem csubc_csuba:
  \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (csuba 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/defs.ma
index bfe52dc36..9f18fca8d 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "sc3/defs.ma".
+include "LambdaDelta-1/sc3/defs.ma".
 
 inductive csubc (g: G): C \to (C \to Prop) \def
 | csubc_sort: \forall (n: nat).(csubc g (CSort n) (CSort n))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/drop.ma
index 4a0803636..887cbc32e 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/drop.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/drop.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubc/defs.ma".
+include "LambdaDelta-1/csubc/fwd.ma".
 
-include "sc3/props.ma".
+include "LambdaDelta-1/sc3/props.ma".
 
 theorem csubc_drop_conf_O:
  \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (h: nat).((drop h 
@@ -51,84 +51,57 @@ c2 e2)) (\lambda (e2: C).(csubc g (CHead c k t) e2)) c2 (drop_refl c2) H1) e1
 (((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)).
+C).(\lambda (H2: (csubc g (CHead c k t) c2)).(let H_x \def (csubc_gen_head_l 
+g c c2 t k H2) in (let H3 \def H_x in (or_ind (ex2 C (\lambda (c3: C).(eq C 
+c2 (CHead c3 k t))) (\lambda (c3: C).(csubc g c c3))) (ex5_3 C T A (\lambda 
+(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda 
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex2 C (\lambda 
+(e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda 
+(H4: (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k t))) (\lambda (c3: 
+C).(csubc g c c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 k t))) 
+(\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (e2: C).(drop (S n) O c2 
+e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: C).(\lambda (H5: (eq C 
+c2 (CHead x k t))).(\lambda (H6: (csubc g c x)).(eq_ind_r C (CHead x k t) 
+(\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2: 
+C).(csubc g e1 e2)))) (let H_x0 \def (H e1 (r k n) (drop_gen_drop k c e1 t n 
+H1) x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (e2: C).(drop (r k n) O 
+x e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: C).(drop (S n) 
+O (CHead x k t) e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: 
+C).(\lambda (H8: (drop (r k n) O x x0)).(\lambda (H9: (csubc g e1 
+x0)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x k t) e2)) (\lambda 
+(e2: C).(csubc g e1 e2)) x0 (drop_drop k n x x0 H8 t) H9)))) H7))) c2 H5)))) 
+H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: 
+A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: 
+A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g c c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(sc3 g (asucc g a) c t)))) (\lambda (c3: C).(\lambda (w: T).(\lambda 
+(a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: 
+T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: 
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda (c3: C).(\lambda 
+(w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda (e2: C).(drop (S n) 
+O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (x2: A).(\lambda (H5: (eq K k (Bind Abst))).(\lambda (H6: (eq C 
+c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (csubc g c x0)).(\lambda (_: 
+(sc3 g (asucc g x2) c t)).(\lambda (_: (sc3 g x2 x0 x1)).(eq_ind_r C (CHead 
+x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop (S n) O c0 
+e2)) (\lambda (e2: C).(csubc g e1 e2)))) (let H10 \def (eq_ind K k (\lambda 
+(k0: K).(drop (r k0 n) O c e1)) (drop_gen_drop k c e1 t n H1) (Bind Abst) H5) 
+in (let H11 \def (eq_ind K k (\lambda (k0: K).((drop n O (CHead c k0 t) e1) 
+\to (\forall (c3: C).((csubc g (CHead c k0 t) c3) \to (ex2 C (\lambda (e2: 
+C).(drop n O c3 e2)) (\lambda (e2: C).(csubc g e1 e2))))))) H0 (Bind Abst) 
+H5) in (let H_x0 \def (H e1 (r (Bind Abst) n) H10 x0 H7) in (let H12 \def 
+H_x0 in (ex2_ind C (\lambda (e2: C).(drop (r (Bind Abst) n) O x0 e2)) 
+(\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: C).(drop (S n) O 
+(CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda 
+(x: C).(\lambda (H13: (drop (r (Bind Abst) n) O x0 x)).(\lambda (H14: (csubc 
+g e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x0 (Bind Abbr) 
+x1) e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind Abbr) n x0 x 
+H13 x1) H14)))) H12))))) c2 H6))))))))) H4)) H3)))))))) h))))))) c1)).
 
 theorem drop_csubc_trans:
  \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall 
@@ -195,95 +168,69 @@ 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 
+c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t0) c1)))) (let H_x \def 
+(csubc_gen_head_l g x0 e1 x1 k H6) in (let H9 \def H_x in (or_ind (ex2 C 
+(\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda (c3: C).(csubc g x0 
+c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k 
+(Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1 
+(CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
+A).(csubc g x0 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g 
+(asucc g a) x0 x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g 
+a c3 w))))) (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))) (\lambda (H10: (ex2 C 
+(\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda (c3: C).(csubc g x0 
+c3)))).(ex2_ind C (\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda (c3: 
+C).(csubc g x0 c3)) (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))) (\lambda (x: 
+C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: (csubc g x0 
+x)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop 
+h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) 
+c1)))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def H_x0 in 
+(ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: C).(csubc g 
+c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda 
+(c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2: 
+C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g c 
+x2)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda 
+(c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)) (CHead x2 k (lift h (r 
+k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g c x2 H15 k (lift h (r k 
+n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind Abbr) w))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0 c3)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0 x1)))) (\lambda 
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
+(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind 
+Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0 
+c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0 
+x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) 
+(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))) (\lambda (x2: C).(\lambda (x3: 
+T).(\lambda (x4: A).(\lambda (H11: (eq K k (Bind Abst))).(\lambda (H12: (eq C 
+e1 (CHead x2 (Bind Abbr) x3))).(\lambda (H13: (csubc g x0 x2)).(\lambda (H14: 
+(sc3 g (asucc g x4) x0 x1)).(\lambda (H15: (sc3 g x4 x2 x3)).(eq_ind_r C 
+(CHead x2 (Bind Abbr) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S 
+n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)))) 
+(let H16 \def (eq_ind 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 (c1: C).(drop h0 n c1 
+e3)) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1)))))))) 
+H8 (Bind Abst) H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r 
+k0 n) c x0)) H5 (Bind Abst) H11) in (eq_ind_r K (Bind Abst) (\lambda (k0: 
+K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3))) 
+(\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1)))) (let H_x0 
+\def (H x0 (r (Bind Abst) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind 
+C (\lambda (c1: C).(drop h (r (Bind Abst) n) c1 x2)) (\lambda (c1: C).(csubc 
+g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3))) 
+(\lambda (c1: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) 
+c1))) (\lambda (x: C).(\lambda (H19: (drop h (r (Bind Abst) n) x 
+x2)).(\lambda (H20: (csubc g c x)).(ex_intro2 C (\lambda (c1: C).(drop h (S 
+n) c1 (CHead x2 (Bind Abbr) x3))) (\lambda (c1: C).(csubc g (CHead c (Bind 
+Abst) (lift h (r (Bind Abst) n) x1)) c1)) (CHead x (Bind Abbr) (lift h n x3)) 
+(drop_skip_bind h n x x2 H19 Abbr x3) (csubc_abst g c x H20 (lift h (r (Bind 
+Abst) n) x1) x4 (sc3_lift g (asucc g x4) x0 x1 H14 c h (r (Bind Abst) n) H17) 
+(lift h n x3) (sc3_lift g x4 x2 x3 H15 x h n H19)))))) H18))) k H11))) e1 
+H12))))))))) H10)) H9))) t H4))))))))) (drop_gen_skip_l c e2 t h n k 
 H1)))))))) d))))))) c2)).
 
 theorem csubc_drop_conf_rev:
@@ -351,98 +298,68 @@ 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)).
+(\lambda (c1: C).(csubc g c1 (CHead c k t0))))) (let H_x \def 
+(csubc_gen_head_r g x0 e1 x1 k H6) in (let H9 \def H_x in (or_ind (ex2 C 
+(\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1 
+x0))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k 
+(Bind Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1 
+(CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: 
+A).(csubc g c1 x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g 
+(asucc g a) c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a 
+x0 x1))))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: 
+C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (H10: (ex2 C 
+(\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1 
+x0)))).(ex2_ind C (\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: 
+C).(csubc g c1 x0)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda 
+(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x: 
+C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: (csubc g x 
+x0)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(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)))))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def H_x0 in 
+(ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: C).(csubc g 
+c1 c)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda 
+(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x2: 
+C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g x2 
+c)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda 
+(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))) (CHead x2 k (lift h (r 
+k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g x2 c H15 k (lift h (r k 
+n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: 
+C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind Abst) v))))) 
+(\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 x0)))) (\lambda 
+(c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))))).(ex5_3_ind C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) 
+(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind 
+Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
+x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))) (ex2 
+C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead 
+c k (lift h (r k n) x1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
+A).(\lambda (H11: (eq K k (Bind Abbr))).(\lambda (H12: (eq C e1 (CHead x2 
+(Bind Abst) x3))).(\lambda (H13: (csubc g x2 x0)).(\lambda (H14: (sc3 g 
+(asucc g x4) x2 x3)).(\lambda (H15: (sc3 g x4 x0 x1)).(eq_ind_r C (CHead x2 
+(Bind Abst) x3) (\lambda (c0: C).(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)))))) (let 
+H16 \def (eq_ind 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 (c1: C).(drop h0 n c1 e3)) (\lambda 
+(c1: C).(csubc g c1 (CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind Abbr) 
+H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5 
+(Bind Abbr) H11) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda 
+(c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc 
+g c1 (CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind 
+Abbr) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind C (\lambda (c1: 
+C).(drop h (r (Bind Abbr) n) c1 x2)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C 
+(\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: 
+C).(csubc g c1 (CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1))))) 
+(\lambda (x: C).(\lambda (H19: (drop h (r (Bind Abbr) n) x x2)).(\lambda 
+(H20: (csubc g x c)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 
+(Bind Abst) x3))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind Abbr) (lift h 
+(r (Bind Abbr) n) x1)))) (CHead x (Bind Abst) (lift h n x3)) (drop_skip_bind 
+h n x x2 H19 Abst x3) (csubc_abst g x c H20 (lift h n x3) x4 (sc3_lift g 
+(asucc g x4) x2 x3 H14 x h n H19) (lift h (r (Bind Abbr) n) x1) (sc3_lift g 
+x4 x0 x1 H15 c h (r (Bind Abbr) n) H17)))))) H18))) k H11))) e1 H12))))))))) 
+H10)) H9))) t H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) 
+c2)).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/drop1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/drop1.ma
index a1f8cbb64..1e4e8fb48 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/drop1.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/drop1.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubc/drop.ma".
+include "LambdaDelta-1/csubc/drop.ma".
 
 theorem drop1_csubc_trans:
  \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/fwd.ma
new file mode 100644
index 000000000..18435fe3e
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/fwd.ma
@@ -0,0 +1,370 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 *********************)
+
+include "LambdaDelta-1/csubc/defs.ma".
+
+theorem csubc_gen_sort_l:
+ \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g (CSort n) x) \to 
+(eq C x (CSort n)))))
+\def
+ \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g 
+(CSort n) x)).(insert_eq C (CSort n) (\lambda (c: C).(csubc g c x)) (\lambda 
+(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g 
+(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) 
+(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def 
+(f_equal C nat (\lambda (e: C).(match e in C return (\lambda (_: C).nat) with 
+[(CSort n1) \Rightarrow n1 | (CHead _ _ _) \Rightarrow n0])) (CSort n0) 
+(CSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq C (CSort n1) (CSort 
+n1))) (refl_equal C (CSort n)) n0 H2)))) (\lambda (c1: C).(\lambda (c2: 
+C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C 
+c2 c1)))).(\lambda (k: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c1 k v) 
+(CSort n))).(let H4 \def (eq_ind C (CHead c1 k v) (\lambda (ee: C).(match ee 
+in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead 
+_ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (eq C (CHead c2 k v) 
+(CHead c1 k v)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: 
+(csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2 
+c1)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: (sc3 g (asucc g a) c1 
+v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 w)).(\lambda (H5: (eq C (CHead 
+c1 (Bind Abst) v) (CSort n))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) v) 
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) 
+\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H5) in 
+(False_ind (eq C (CHead c2 (Bind Abbr) w) (CHead c1 (Bind Abst) v)) 
+H6)))))))))))) y x H0))) H)))).
+
+theorem csubc_gen_head_l:
+ \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (k: 
+K).((csubc g (CHead c1 k v) x) \to (or (ex2 C (\lambda (c2: C).(eq C x (CHead 
+c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c2: 
+C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind Abbr) w))))) 
+(\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda 
+(c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (k: 
+K).(\lambda (H: (csubc g (CHead c1 k v) x)).(insert_eq C (CHead c1 k v) 
+(\lambda (c: C).(csubc g c x)) (\lambda (_: C).(or (ex2 C (\lambda (c2: 
+C).(eq C x (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
+(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind 
+Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
+c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
+v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))))) 
+(\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g (\lambda (c: 
+C).(\lambda (c0: C).((eq C c (CHead c1 k v)) \to (or (ex2 C (\lambda (c2: 
+C).(eq C c0 (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
+(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C c0 (CHead c2 (Bind 
+Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
+c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
+v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))))))) 
+(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead c1 k v))).(let H2 \def 
+(eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: 
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
+False])) I (CHead c1 k v) H1) in (False_ind (or (ex2 C (\lambda (c2: C).(eq C 
+(CSort n) (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
+(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C (CSort n) (CHead c2 
+(Bind Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g 
+c1 c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) 
+c1 v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w)))))) 
+H2)))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (csubc g c0 
+c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or (ex2 C (\lambda (c3: 
+C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
+(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind 
+Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
+c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
+v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 
+w))))))))).(\lambda (k0: K).(\lambda (v0: T).(\lambda (H3: (eq C (CHead c0 k0 
+v0) (CHead c1 k v))).(let H4 \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 k0 v0) (CHead c1 k v) H3) in ((let H5 \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 v0) 
+(CHead c1 k v) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in 
+C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) 
+\Rightarrow t])) (CHead c0 k0 v0) (CHead c1 k v) H3) in (\lambda (H7: (eq K 
+k0 k)).(\lambda (H8: (eq C c0 c1)).(eq_ind_r T v (\lambda (t: T).(or (ex2 C 
+(\lambda (c3: C).(eq C (CHead c2 k0 t) (CHead c3 k v))) (\lambda (c3: 
+C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
+(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: 
+A).(eq C (CHead c2 k0 t) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: 
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))))) (eq_ind_r K k 
+(\lambda (k1: K).(or (ex2 C (\lambda (c3: C).(eq C (CHead c2 k1 v) (CHead c3 
+k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (_: A).(eq C (CHead c2 k1 v) (CHead c3 (Bind 
+Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
+c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
+v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))))) 
+(let H9 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k v)) \to (or 
+(ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g 
+c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k 
+(Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 
+(CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
+A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g 
+(asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g 
+a c3 w)))))))) H2 c1 H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubc 
+g c c2)) H1 c1 H8) in (or_introl (ex2 C (\lambda (c3: C).(eq C (CHead c2 k v) 
+(CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (_: A).(eq C (CHead c2 k v) (CHead c3 (Bind Abbr) 
+w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) 
+(\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) 
+(ex_intro2 C (\lambda (c3: C).(eq C (CHead c2 k v) (CHead c3 k v))) (\lambda 
+(c3: C).(csubc g c1 c3)) c2 (refl_equal C (CHead c2 k v)) H10)))) k0 H7) v0 
+H6)))) H5)) H4))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: 
+(csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or (ex2 C 
+(\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) 
+(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind 
+Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 
+(Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g 
+c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) 
+c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 
+w))))))))).(\lambda (v0: T).(\lambda (a: A).(\lambda (H3: (sc3 g (asucc g a) 
+c0 v0)).(\lambda (w: T).(\lambda (H4: (sc3 g a c2 w)).(\lambda (H5: (eq C 
+(CHead c0 (Bind Abst) v0) (CHead c1 k v))).(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 (Bind Abst) v0) (CHead c1 k v) H5) 
+in ((let H7 \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) v0) (CHead c1 k v) H5) in ((let H8 
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) 
+with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 
+(Bind Abst) v0) (CHead c1 k v) H5) in (\lambda (H9: (eq K (Bind Abst) 
+k)).(\lambda (H10: (eq C c0 c1)).(let H11 \def (eq_ind T v0 (\lambda (t: 
+T).(sc3 g (asucc g a) c0 t)) H3 v H8) in (let H12 \def (eq_ind C c0 (\lambda 
+(c: C).(sc3 g (asucc g a) c v)) H11 c1 H10) in (let H13 \def (eq_ind C c0 
+(\lambda (c: C).((eq C c (CHead c1 k v)) \to (or (ex2 C (\lambda (c3: C).(eq 
+C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
+(\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind 
+Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
+c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) 
+c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 
+w0)))))))) H2 c1 H10) in (let H14 \def (eq_ind C c0 (\lambda (c: C).(csubc g 
+c c2)) H1 c1 H10) in (let H15 \def (eq_ind_r K k (\lambda (k0: K).((eq C c1 
+(CHead c1 k0 v)) \to (or (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k0 v))) 
+(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: 
+T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w0))))) (\lambda (c3: 
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) (\lambda 
+(c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0)))))))) H13 (Bind 
+Abst) H9) in (eq_ind K (Bind Abst) (\lambda (k0: K).(or (ex2 C (\lambda (c3: 
+C).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 k0 v))) (\lambda (c3: C).(csubc g 
+c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K 
+k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C 
+(CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abbr) w0))))) (\lambda (c3: 
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) (\lambda 
+(c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0))))))) (or_intror 
+(ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abst) 
+v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda 
+(_: T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: 
+C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 
+(Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g 
+c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g 
+a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 
+w0))))) (ex5_3_intro C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: 
+A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: 
+T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abbr) 
+w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) 
+(\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0)))) c2 w a 
+(refl_equal K (Bind Abst)) (refl_equal C (CHead c2 (Bind Abbr) w)) H14 H12 
+H4)) k H9))))))))) H7)) H6)))))))))))) y x H0))) H)))))).
+
+theorem csubc_gen_sort_r:
+ \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g x (CSort n)) \to 
+(eq C x (CSort n)))))
+\def
+ \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g x 
+(CSort n))).(insert_eq C (CSort n) (\lambda (c: C).(csubc g x c)) (\lambda 
+(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g 
+(\lambda (c: C).(\lambda (c0: C).((eq C c0 (CSort n)) \to (eq C c c0)))) 
+(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def 
+(f_equal C nat (\lambda (e: C).(match e in C return (\lambda (_: C).nat) with 
+[(CSort n1) \Rightarrow n1 | (CHead _ _ _) \Rightarrow n0])) (CSort n0) 
+(CSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq C (CSort n1) (CSort 
+n1))) (refl_equal C (CSort n)) n0 H2)))) (\lambda (c1: C).(\lambda (c2: 
+C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C 
+c1 c2)))).(\lambda (k: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c2 k v) 
+(CSort n))).(let H4 \def (eq_ind C (CHead c2 k v) (\lambda (ee: C).(match ee 
+in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead 
+_ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (eq C (CHead c1 k v) 
+(CHead c2 k v)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: 
+(csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C c1 
+c2)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: (sc3 g (asucc g a) c1 
+v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 w)).(\lambda (H5: (eq C (CHead 
+c2 (Bind Abbr) w) (CSort n))).(let H6 \def (eq_ind C (CHead c2 (Bind Abbr) w) 
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) 
+\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H5) in 
+(False_ind (eq C (CHead c1 (Bind Abst) v) (CHead c2 (Bind Abbr) w)) 
+H6)))))))))))) x y H0))) H)))).
+
+theorem csubc_gen_head_r:
+ \forall (g: G).(\forall (c2: C).(\forall (x: C).(\forall (w: T).(\forall (k: 
+K).((csubc g x (CHead c2 k w)) \to (or (ex2 C (\lambda (c1: C).(eq C x (CHead 
+c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: 
+C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind Abst) v))))) 
+(\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda 
+(c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w)))))))))))
+\def
+ \lambda (g: G).(\lambda (c2: C).(\lambda (x: C).(\lambda (w: T).(\lambda (k: 
+K).(\lambda (H: (csubc g x (CHead c2 k w))).(insert_eq C (CHead c2 k w) 
+(\lambda (c: C).(csubc g x c)) (\lambda (_: C).(or (ex2 C (\lambda (c1: 
+C).(eq C x (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) 
+(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind 
+Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
+c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))))) 
+(\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g (\lambda (c: 
+C).(\lambda (c0: C).((eq C c0 (CHead c2 k w)) \to (or (ex2 C (\lambda (c1: 
+C).(eq C c (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) 
+(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C c (CHead c1 (Bind 
+Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
+c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))))))) 
+(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead c2 k w))).(let H2 \def 
+(eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: 
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
+False])) I (CHead c2 k w) H1) in (False_ind (or (ex2 C (\lambda (c1: C).(eq C 
+(CSort n) (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) 
+(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C (CSort n) (CHead c1 
+(Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g 
+c1 c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) 
+c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w)))))) 
+H2)))) (\lambda (c1: C).(\lambda (c0: C).(\lambda (H1: (csubc g c1 
+c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w)) \to (or (ex2 C (\lambda (c3: 
+C).(eq C c1 (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) 
+(\lambda (c3: C).(\lambda (v: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind 
+Abst) v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 
+c2)))) (\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 
+w))))))))).(\lambda (k0: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c0 k0 
+v) (CHead c2 k w))).(let H4 \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 k0 v) (CHead c2 k w) H3) in ((let H5 \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 c2 k 
+w) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return 
+(\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow 
+t])) (CHead c0 k0 v) (CHead c2 k w) H3) in (\lambda (H7: (eq K k0 
+k)).(\lambda (H8: (eq C c0 c2)).(eq_ind_r T w (\lambda (t: T).(or (ex2 C 
+(\lambda (c3: C).(eq C (CHead c1 k0 t) (CHead c3 k w))) (\lambda (c3: 
+C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
+(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: 
+A).(eq C (CHead c1 k0 t) (CHead c3 (Bind Abst) v0))))) (\lambda (c3: 
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: 
+C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))))) (eq_ind_r K k 
+(\lambda (k1: K).(or (ex2 C (\lambda (c3: C).(eq C (CHead c1 k1 w) (CHead c3 
+k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: 
+C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k1 w) (CHead c3 (Bind 
+Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 
+c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) 
+c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 
+w))))))) (let H9 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c2 k w)) 
+\to (or (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: 
+C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
+(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: 
+A).(eq C c1 (CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: 
+T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(sc3 g a c2 w)))))))) H2 c2 H8) in (let H10 \def (eq_ind C 
+c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H8) in (or_introl (ex2 C (\lambda 
+(c3: C).(eq C (CHead c1 k w) (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 
+c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k 
+(Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C 
+(CHead c1 k w) (CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: 
+T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(sc3 g a c2 w))))) (ex_intro2 C (\lambda (c3: C).(eq C 
+(CHead c1 k w) (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2)) c1 
+(refl_equal C (CHead c1 k w)) H10)))) k0 H7) v H6)))) H5)) H4))))))))) 
+(\lambda (c1: C).(\lambda (c0: C).(\lambda (H1: (csubc g c1 c0)).(\lambda 
+(H2: (((eq C c0 (CHead c2 k w)) \to (or (ex2 C (\lambda (c3: C).(eq C c1 
+(CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: 
+C).(\lambda (v: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda 
+(c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))))))).(\lambda (v: 
+T).(\lambda (a: A).(\lambda (H3: (sc3 g (asucc g a) c1 v)).(\lambda (w0: 
+T).(\lambda (H4: (sc3 g a c0 w0)).(\lambda (H5: (eq C (CHead c0 (Bind Abbr) 
+w0) (CHead c2 k w))).(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 (Bind Abbr) w0) (CHead c2 k w) H5) in ((let H7 
+\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) w0) (CHead c2 k w) H5) in ((let H8 \def (f_equal C T 
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
+\Rightarrow w0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind Abbr) w0) 
+(CHead c2 k w) H5) in (\lambda (H9: (eq K (Bind Abbr) k)).(\lambda (H10: (eq 
+C c0 c2)).(let H11 \def (eq_ind T w0 (\lambda (t: T).(sc3 g a c0 t)) H4 w H8) 
+in (let H12 \def (eq_ind C c0 (\lambda (c: C).(sc3 g a c w)) H11 c2 H10) in 
+(let H13 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c2 k w)) \to (or 
+(ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: C).(csubc g 
+c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k 
+(Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 
+(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
+A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 
+g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: 
+A).(sc3 g a0 c2 w)))))))) H2 c2 H10) in (let H14 \def (eq_ind C c0 (\lambda 
+(c: C).(csubc g c1 c)) H1 c2 H10) in (let H15 \def (eq_ind_r K k (\lambda 
+(k0: K).((eq C c2 (CHead c2 k0 w)) \to (or (ex2 C (\lambda (c3: C).(eq C c1 
+(CHead c3 k0 w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda 
+(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: 
+C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v0))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda 
+(c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 v0)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w)))))))) H13 
+(Bind Abbr) H9) in (eq_ind K (Bind Abbr) (\lambda (k0: K).(or (ex2 C (\lambda 
+(c3: C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 k0 w))) (\lambda (c3: 
+C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
+(_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda 
+(_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) v0))))) (\lambda 
+(c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: 
+C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 v0)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w))))))) (or_intror (ex2 
+C (\lambda (c3: C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abbr) w))) 
+(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda (c3: 
+C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 
+(Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g 
+c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g 
+a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 
+w))))) (ex5_3_intro C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq 
+K (Bind Abbr) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: 
+A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) v0))))) (\lambda (c3: 
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: 
+C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 v0)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w)))) c1 v a (refl_equal 
+K (Bind Abbr)) (refl_equal C (CHead c1 (Bind Abst) v)) H14 H3 H12)) k 
+H9))))))))) H7)) H6)))))))))))) x y H0))) H)))))).
+
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/getl.ma
index 4ab71a145..54a801f09 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/getl.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/getl.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubc/drop.ma".
+include "LambdaDelta-1/csubc/drop.ma".
 
-include "csubc/clear.ma".
+include "LambdaDelta-1/csubc/clear.ma".
 
 theorem csubc_getl_conf:
  \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (i: nat).((getl i 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/props.ma
index 17e95182b..be0b76e29 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/props.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubc/defs.ma".
+include "LambdaDelta-1/csubc/defs.ma".
 
-include "sc3/props.ma".
+include "LambdaDelta-1/sc3/props.ma".
 
 theorem csubc_refl:
  \forall (g: G).(\forall (c: C).(csubc g c c))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/clear.ma
index 0ecdcc1cf..7a1d5e009 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/clear.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/clear.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubst0/fwd.ma".
+include "LambdaDelta-1/csubst0/fwd.ma".
 
-include "clear/fwd.ma".
+include "LambdaDelta-1/clear/fwd.ma".
 
 theorem csubst0_clear_O:
  \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/defs.ma
index 3b26bb371..3f5e59e67 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/defs.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "subst0/defs.ma".
+include "LambdaDelta-1/subst0/defs.ma".
 
-include "C/defs.ma".
+include "LambdaDelta-1/C/defs.ma".
 
 inductive csubst0: nat \to (T \to (C \to (C \to Prop))) \def
 | csubst0_snd: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/drop.ma
index c3e86929e..d7e31cf82 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/drop.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/drop.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubst0/fwd.ma".
+include "LambdaDelta-1/csubst0/fwd.ma".
 
-include "drop/fwd.ma".
+include "LambdaDelta-1/drop/fwd.ma".
 
-include "s/props.ma".
+include "LambdaDelta-1/s/props.ma".
 
 theorem csubst0_drop_gt:
  \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/fwd.ma
index 31854f925..a5159d6a9 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/fwd.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubst0/defs.ma".
+include "LambdaDelta-1/csubst0/defs.ma".
 
 theorem csubst0_gen_sort:
  \forall (x: C).(\forall (v: T).(\forall (i: nat).(\forall (n: nat).((csubst0 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/getl.ma
index e724aa55d..218664c0f 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/getl.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/getl.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubst0/clear.ma".
+include "LambdaDelta-1/csubst0/clear.ma".
 
-include "csubst0/drop.ma".
+include "LambdaDelta-1/csubst0/drop.ma".
 
-include "getl/fwd.ma".
+include "LambdaDelta-1/getl/fwd.ma".
 
 theorem csubst0_getl_ge:
  \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/props.ma
index 527cc5bb8..2f85df2da 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst0/props.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubst0/defs.ma".
+include "LambdaDelta-1/csubst0/defs.ma".
 
 theorem csubst0_snd_bind:
  \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst1/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst1/defs.ma
index 0684e35b6..3922386b0 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst1/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst1/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubst0/defs.ma".
+include "LambdaDelta-1/csubst0/defs.ma".
 
 inductive csubst1 (i: nat) (v: T) (c1: C): C \to Prop \def
 | csubst1_refl: csubst1 i v c1 c1
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst1/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst1/fwd.ma
index 60f422c5c..290ebdc57 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst1/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst1/fwd.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubst1/defs.ma".
+include "LambdaDelta-1/csubst1/defs.ma".
 
-include "csubst0/fwd.ma".
+include "LambdaDelta-1/csubst0/fwd.ma".
 
-include "subst1/props.ma".
+include "LambdaDelta-1/subst1/props.ma".
 
 theorem csubst1_gen_head:
  \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst1/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst1/getl.ma
index 960f16d78..e04a34a34 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst1/getl.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst1/getl.ma
@@ -14,15 +14,15 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubst1/props.ma".
+include "LambdaDelta-1/csubst1/props.ma".
 
-include "csubst0/getl.ma".
+include "LambdaDelta-1/csubst0/getl.ma".
 
-include "csubst0/props.ma".
+include "LambdaDelta-1/csubst0/props.ma".
 
-include "subst1/props.ma".
+include "LambdaDelta-1/subst1/props.ma".
 
-include "drop/props.ma".
+include "LambdaDelta-1/drop/props.ma".
 
 theorem csubst1_getl_ge:
  \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst1/props.ma
index 0a73f8d22..9a8b4a2bb 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst1/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubst1/props.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubst1/defs.ma".
+include "LambdaDelta-1/csubst1/defs.ma".
 
-include "subst1/defs.ma".
+include "LambdaDelta-1/subst1/defs.ma".
 
 theorem csubst1_head:
  \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/clear.ma
index 6a6c2ec62..2541790a9 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/clear.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/clear.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubt/defs.ma".
+include "LambdaDelta-1/csubt/defs.ma".
 
-include "clear/fwd.ma".
+include "LambdaDelta-1/clear/fwd.ma".
 
 theorem csubt_clear_conf:
  \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/defs.ma
index 6627d5eff..520e3bd70 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "ty3/defs.ma".
+include "LambdaDelta-1/ty3/defs.ma".
 
 inductive csubt (g: G): C \to (C \to Prop) \def
 | csubt_sort: \forall (n: nat).(csubt g (CSort n) (CSort n))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/drop.ma
index 8c834e611..8ceb4bd9d 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/drop.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/drop.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubt/defs.ma".
+include "LambdaDelta-1/csubt/defs.ma".
 
-include "drop/fwd.ma".
+include "LambdaDelta-1/drop/fwd.ma".
 
 theorem csubt_drop_flat:
  \forall (g: G).(\forall (f: F).(\forall (n: nat).(\forall (c1: C).(\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/fwd.ma
index 1aac67dff..9cb070ded 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/fwd.ma
@@ -14,47 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubt/defs.ma".
-
-theorem csubt_inv_coq:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).(\forall (P: ((G \to (C \to 
-(C \to Prop))))).((((csubt g c1 c2) \to (\forall (n: nat).((eq C (CSort n) 
-c1) \to ((eq C (CSort n) c2) \to (P g c1 c2)))))) \to ((((csubt g c1 c2) \to 
-(\forall (c0: C).(\forall (c3: C).(\forall (k: K).(\forall (u: T).((eq C 
-(CHead c0 k u) c1) \to ((eq C (CHead c3 k u) c2) \to ((csubt g c0 c3) \to (P 
-g c1 c2)))))))))) \to ((((csubt g c1 c2) \to (\forall (c0: C).(\forall (c3: 
-C).(\forall (b: B).(\forall (u1: T).(\forall (u2: T).((eq C (CHead c0 (Bind 
-Void) u1) c1) \to ((eq C (CHead c3 (Bind b) u2) c2) \to ((csubt g c0 c3) \to 
-((not (eq B b Void)) \to (P g c1 c2)))))))))))) \to ((((csubt g c1 c2) \to 
-(\forall (c0: C).(\forall (c3: C).(\forall (u: T).(\forall (t: T).((eq C 
-(CHead c0 (Bind Abst) t) c1) \to ((eq C (CHead c3 (Bind Abbr) u) c2) \to 
-((csubt g c0 c3) \to ((ty3 g c3 u t) \to (P g c1 c2))))))))))) \to ((csubt g 
-c1 c2) \to (P g c1 c2)))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (P: ((G \to (C \to 
-(C \to Prop))))).(\lambda (H: (((csubt g c1 c2) \to (\forall (n: nat).((eq C 
-(CSort n) c1) \to ((eq C (CSort n) c2) \to (P g c1 c2))))))).(\lambda (H0: 
-(((csubt g c1 c2) \to (\forall (c0: C).(\forall (c3: C).(\forall (k: 
-K).(\forall (u: T).((eq C (CHead c0 k u) c1) \to ((eq C (CHead c3 k u) c2) 
-\to ((csubt g c0 c3) \to (P g c1 c2))))))))))).(\lambda (H1: (((csubt g c1 
-c2) \to (\forall (c0: C).(\forall (c3: C).(\forall (b: B).(\forall (u1: 
-T).(\forall (u2: T).((eq C (CHead c0 (Bind Void) u1) c1) \to ((eq C (CHead c3 
-(Bind b) u2) c2) \to ((csubt g c0 c3) \to ((not (eq B b Void)) \to (P g c1 
-c2))))))))))))).(\lambda (H2: (((csubt g c1 c2) \to (\forall (c0: C).(\forall 
-(c3: C).(\forall (u: T).(\forall (t: T).((eq C (CHead c0 (Bind Abst) t) c1) 
-\to ((eq C (CHead c3 (Bind Abbr) u) c2) \to ((csubt g c0 c3) \to ((ty3 g c3 u 
-t) \to (P g c1 c2)))))))))))).(\lambda (H3: (csubt g c1 c2)).(let H4 \def 
-(match H3 in csubt return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: 
-(csubt ? c c0)).((eq C c c1) \to ((eq C c0 c2) \to (P g c1 c2)))))) with 
-[(csubt_sort n) \Rightarrow (\lambda (H4: (eq C (CSort n) c1)).(\lambda (H5: 
-(eq C (CSort n) c2)).(H H3 n H4 H5))) | (csubt_head c0 c3 H4 k u) \Rightarrow 
-(\lambda (H5: (eq C (CHead c0 k u) c1)).(\lambda (H6: (eq C (CHead c3 k u) 
-c2)).(H0 H3 c0 c3 k u H5 H6 H4))) | (csubt_void c0 c3 H4 b H5 u1 u2) 
-\Rightarrow (\lambda (H6: (eq C (CHead c0 (Bind Void) u1) c1)).(\lambda (H7: 
-(eq C (CHead c3 (Bind b) u2) c2)).(H1 H3 c0 c3 b u1 u2 H6 H7 H4 H5))) | 
-(csubt_abst c0 c3 H4 u t H5) \Rightarrow (\lambda (H6: (eq C (CHead c0 (Bind 
-Abst) t) c1)).(\lambda (H7: (eq C (CHead c3 (Bind Abbr) u) c2)).(H2 H3 c0 c3 
-u t H6 H7 H4 H5)))]) in (H4 (refl_equal C c1) (refl_equal C c2))))))))))).
+include "LambdaDelta-1/csubt/defs.ma".
 
 theorem csubt_gen_abbr:
  \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csubt g 
@@ -62,84 +22,66 @@ theorem csubt_gen_abbr:
 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))))
 \def
  \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda 
-(H: (csubt g (CHead e1 (Bind Abbr) v) c2)).(csubt_inv_coq g (CHead e1 (Bind 
-Abbr) v) c2 (\lambda (g0: G).(\lambda (_: C).(\lambda (c0: C).(ex2 C (\lambda 
-(e2: C).(eq C c0 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g0 e1 
-e2)))))) (\lambda (H0: (csubt g (CHead e1 (Bind Abbr) v) c2)).(\lambda (n: 
-nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind Abbr) v))).(\lambda (H2: 
-(eq C (CSort n) c2)).(let H3 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g 
-(CHead e1 (Bind Abbr) v) c)) H0 (CSort n) H2) in (let H4 \def (eq_ind_r C c2 
-(\lambda (c: C).(csubt g (CHead e1 (Bind Abbr) v) c)) H (CSort n) H2) in 
-(eq_ind C (CSort n) (\lambda (c: C).(ex2 C (\lambda (e2: C).(eq C c (CHead e2 
-(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (let H5 \def (eq_ind C 
-(CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with 
-[(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 
-(Bind Abbr) v) H1) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CSort n) 
-(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H5)) c2 
-H2))))))) (\lambda (H0: (csubt g (CHead e1 (Bind Abbr) v) c2)).(\lambda (c0: 
-C).(\lambda (c3: C).(\lambda (k: K).(\lambda (u: T).(\lambda (H1: (eq C 
-(CHead c0 k u) (CHead e1 (Bind Abbr) v))).(\lambda (H2: (eq C (CHead c3 k u) 
-c2)).(\lambda (H3: (csubt g c0 c3)).(let H4 \def (eq_ind_r C c2 (\lambda (c: 
-C).(csubt g (CHead e1 (Bind Abbr) v) c)) H0 (CHead c3 k u) H2) in (let H5 
-\def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind Abbr) v) c)) H 
-(CHead c3 k u) H2) in (eq_ind C (CHead c3 k u) (\lambda (c: C).(ex2 C 
-(\lambda (e2: C).(eq C c (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g 
-e1 e2)))) (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 Abbr) v) H1) in ((let H7 \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 Abbr) v) H1) in ((let H8 \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 Abbr) v) H1) in (\lambda (H9: 
-(eq K k (Bind Abbr))).(\lambda (H10: (eq C c0 e1)).(let H11 \def (eq_ind T u 
-(\lambda (t: T).(csubt g (CHead e1 (Bind Abbr) v) (CHead c3 k t))) H5 v H8) 
-in (let H12 \def (eq_ind T u (\lambda (t: T).(csubt g (CHead e1 (Bind Abbr) 
-v) (CHead c3 k t))) H4 v H8) in (eq_ind_r T v (\lambda (t: T).(ex2 C (\lambda 
-(e2: C).(eq C (CHead c3 k t) (CHead e2 (Bind Abbr) v))) (\lambda (e2: 
-C).(csubt g e1 e2)))) (let H13 \def (eq_ind K k (\lambda (k0: K).(csubt g 
-(CHead e1 (Bind Abbr) v) (CHead c3 k0 v))) H11 (Bind Abbr) H9) in (let H14 
-\def (eq_ind K k (\lambda (k0: K).(csubt g (CHead e1 (Bind Abbr) v) (CHead c3 
-k0 v))) H12 (Bind Abbr) H9) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2 
-C (\lambda (e2: C).(eq C (CHead c3 k0 v) (CHead e2 (Bind Abbr) v))) (\lambda 
-(e2: C).(csubt g e1 e2)))) (let H15 \def (eq_ind C c0 (\lambda (c: C).(csubt 
-g c c3)) H3 e1 H10) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind 
-Abbr) v) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)) c3 
-(refl_equal C (CHead c3 (Bind Abbr) v)) H15)) k H9))) u H8)))))) H7)) H6)) c2 
-H2))))))))))) (\lambda (H0: (csubt g (CHead e1 (Bind Abbr) v) c2)).(\lambda 
-(c0: C).(\lambda (c3: C).(\lambda (b: B).(\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (H2: (eq C (CHead c0 (Bind Void) u1) (CHead e1 (Bind Abbr) 
-v))).(\lambda (H3: (eq C (CHead c3 (Bind b) u2) c2)).(\lambda (_: (csubt g c0 
-c3)).(\lambda (_: (not (eq B b Void))).(let H5 \def (eq_ind_r C c2 (\lambda 
-(c: C).(csubt g (CHead e1 (Bind Abbr) v) c)) H0 (CHead c3 (Bind b) u2) H3) in 
-(let H6 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind Abbr) v) 
-c)) H (CHead c3 (Bind b) u2) H3) in (eq_ind C (CHead c3 (Bind b) u2) (\lambda 
-(c: C).(ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abbr) v))) (\lambda 
-(e2: C).(csubt g e1 e2)))) (let H7 \def (eq_ind C (CHead c0 (Bind Void) u1) 
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) 
+(H: (csubt g (CHead e1 (Bind Abbr) v) c2)).(insert_eq C (CHead e1 (Bind Abbr) 
+v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C (\lambda (e2: 
+C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) 
+(\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda (c: 
+C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda 
+(e2: C).(eq C c0 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 
+e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind 
+Abbr) v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C 
+return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) 
+\Rightarrow False])) I (CHead e1 (Bind Abbr) v) H1) in (False_ind (ex2 C 
+(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Bind Abbr) v))) (\lambda (e2: 
+C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: 
+(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 
+C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: 
+C).(csubt g e1 e2)))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C 
+(CHead c1 k u) (CHead e1 (Bind Abbr) v))).(let H4 \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) 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 c1 k u) (CHead e1 (Bind Abbr) v) H3) in ((let H6 \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) H3) in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1 
+e1)).(eq_ind_r T v (\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k 
+t) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K 
+(Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v) 
+(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def 
+(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C 
+(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt 
+g e1 e2))))) H2 e1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g 
+c c3)) H1 e1 H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) 
+v) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)) c3 
+(refl_equal C (CHead c3 (Bind Abbr) v)) H10))) k H7) u H6)))) H5)) 
+H4))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 
+c3)).(\lambda (_: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda 
+(e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 
+e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 
+(Bind Abbr) v))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda 
+(ee: C).(match ee 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 (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2) 
-(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H7)) c2 
-H3))))))))))))) (\lambda (H0: (csubt g (CHead e1 (Bind Abbr) v) c2)).(\lambda 
-(c0: C).(\lambda (c3: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (eq C 
-(CHead c0 (Bind Abst) t) (CHead e1 (Bind Abbr) v))).(\lambda (H3: (eq C 
-(CHead c3 (Bind Abbr) u) c2)).(\lambda (_: (csubt g c0 c3)).(\lambda (_: (ty3 
-g c3 u t)).(let H5 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 
-(Bind Abbr) v) c)) H0 (CHead c3 (Bind Abbr) u) H3) in (let H6 \def (eq_ind_r 
-C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind Abbr) v) c)) H (CHead c3 (Bind 
-Abbr) u) H3) in (eq_ind C (CHead c3 (Bind Abbr) u) (\lambda (c: C).(ex2 C 
-(\lambda (e2: C).(eq C c (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g 
-e1 e2)))) (let H7 \def (eq_ind C (CHead c0 (Bind Abst) t) (\lambda (ee: 
-C).(match ee 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 
-(ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) 
-v))) (\lambda (e2: C).(csubt g e1 e2))) H7)) c2 H3)))))))))))) H))))).
+v) H4) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2) 
+(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H5))))))))))) 
+(\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: 
+(((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 
+(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (u: 
+T).(\lambda (t: T).(\lambda (_: (ty3 g c3 u t)).(\lambda (H4: (eq C (CHead c1 
+(Bind Abst) t) (CHead e1 (Bind Abbr) v))).(let H5 \def (eq_ind C (CHead c1 
+(Bind Abst) t) (\lambda (ee: C).(match ee 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) H4) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind 
+Abbr) u) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) 
+H5)))))))))) y c2 H0))) H))))).
 
 theorem csubt_gen_abst:
  \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csubt g 
@@ -150,122 +92,111 @@ C).(\lambda (_: T).(csubt 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: (csubt g (CHead e1 (Bind Abst) v1) c2)).(csubt_inv_coq g (CHead e1 (Bind 
-Abst) v1) c2 (\lambda (g0: G).(\lambda (_: C).(\lambda (c0: C).(or (ex2 C 
-(\lambda (e2: C).(eq C c0 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt 
-g0 e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c0 (CHead e2 
-(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g0 e1 e2))) 
-(\lambda (e2: C).(\lambda (v2: T).(ty3 g0 e2 v2 v1)))))))) (\lambda (H0: 
-(csubt g (CHead e1 (Bind Abst) v1) c2)).(\lambda (n: nat).(\lambda (H1: (eq C 
-(CSort n) (CHead e1 (Bind Abst) v1))).(\lambda (H2: (eq C (CSort n) c2)).(let 
-H3 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind Abst) v1) c)) 
-H0 (CSort n) H2) in (let H4 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g 
-(CHead e1 (Bind Abst) v1) c)) H (CSort n) H2) in (eq_ind C (CSort n) (\lambda 
-(c: C).(or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abst) v1))) 
-(\lambda (e2: C).(csubt 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).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) 
-(let H5 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return 
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) 
-\Rightarrow False])) I (CHead e1 (Bind Abst) v1) H1) in (False_ind (or (ex2 C 
-(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Bind Abst) v1))) (\lambda (e2: 
-C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C 
+(H: (csubt g (CHead e1 (Bind Abst) v1) c2)).(insert_eq C (CHead e1 (Bind 
+Abst) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(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)))))) (\lambda (y: C).(\lambda (H0: 
+(csubt g y c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead 
+e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c0 (CHead e2 (Bind 
+Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: 
+C).(\lambda (v2: T).(eq C c0 (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)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 
+(Bind Abst) v1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee 
+in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ 
+_ _) \Rightarrow False])) I (CHead e1 (Bind Abst) v1) H1) in (False_ind (or 
+(ex2 C (\lambda (e2: C).(eq C (CSort n) (CHead e2 (Bind Abst) v1))) (\lambda 
+(e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C 
 (CSort n) (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))))) 
-H5)) c2 H2))))))) (\lambda (H0: (csubt g (CHead e1 (Bind Abst) v1) 
-c2)).(\lambda (c0: C).(\lambda (c3: C).(\lambda (k: K).(\lambda (u: 
-T).(\lambda (H1: (eq C (CHead c0 k u) (CHead e1 (Bind Abst) v1))).(\lambda 
-(H2: (eq C (CHead c3 k u) c2)).(\lambda (H3: (csubt g c0 c3)).(let H4 \def 
-(eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind Abst) v1) c)) H0 
-(CHead c3 k u) H2) in (let H5 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g 
-(CHead e1 (Bind Abst) v1) c)) H (CHead c3 k u) H2) in (eq_ind C (CHead c3 k 
-u) (\lambda (c: C).(or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abst) 
-v1))) (\lambda (e2: C).(csubt 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).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) 
-(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 Abst) v1) H1) in ((let H7 \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 Abst) v1) 
-H1) in ((let H8 \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 Abst) v1) H1) in (\lambda (H9: (eq K k 
-(Bind Abst))).(\lambda (H10: (eq C c0 e1)).(let H11 \def (eq_ind T u (\lambda 
-(t: T).(csubt g (CHead e1 (Bind Abst) v1) (CHead c3 k t))) H5 v1 H8) in (let 
-H12 \def (eq_ind T u (\lambda (t: T).(csubt g (CHead e1 (Bind Abst) v1) 
-(CHead c3 k t))) H4 v1 H8) in (eq_ind_r T v1 (\lambda (t: T).(or (ex2 C 
-(\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2 (Bind Abst) v1))) (\lambda 
-(e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C 
-(CHead c3 k t) (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)))))) 
-(let H13 \def (eq_ind K k (\lambda (k0: K).(csubt g (CHead e1 (Bind Abst) v1) 
-(CHead c3 k0 v1))) H11 (Bind Abst) H9) in (let H14 \def (eq_ind K k (\lambda 
-(k0: K).(csubt g (CHead e1 (Bind Abst) v1) (CHead c3 k0 v1))) H12 (Bind Abst) 
-H9) in (eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex2 C (\lambda (e2: 
-C).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt 
-g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 
-v1) (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)))))) (let H15 \def 
-(eq_ind C c0 (\lambda (c: C).(csubt g c c3)) H3 e1 H10) in (or_introl (ex2 C 
-(\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst) v1))) 
+H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 
+c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C 
+(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt 
+g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (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)))))))).(\lambda (k: 
+K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind Abst) 
+v1))).(let H4 \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) 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 c1 k u) (CHead e1 
+(Bind Abst) v1) H3) in ((let H6 \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) H3) in (\lambda 
+(H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 
+(\lambda (t: T).(or (ex2 C (\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2 
+(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: 
+C).(\lambda (v2: T).(eq C (CHead c3 k t) (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)))))) (eq_ind_r K (Bind Abst) (\lambda (k0: K).(or 
+(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abst) v1))) 
 (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: 
-T).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
+T).(eq C (CHead c3 k0 v1) (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)))) (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) 
-(CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal 
-C (CHead c3 (Bind Abst) v1)) H15))) k H9))) u H8)))))) H7)) H6)) c2 
-H2))))))))))) (\lambda (H0: (csubt g (CHead e1 (Bind Abst) v1) c2)).(\lambda 
-(c0: C).(\lambda (c3: C).(\lambda (b: B).(\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (H2: (eq C (CHead c0 (Bind Void) u1) (CHead e1 (Bind Abst) 
-v1))).(\lambda (H3: (eq C (CHead c3 (Bind b) u2) c2)).(\lambda (_: (csubt g 
-c0 c3)).(\lambda (_: (not (eq B b Void))).(let H5 \def (eq_ind_r C c2 
-(\lambda (c: C).(csubt g (CHead e1 (Bind Abst) v1) c)) H0 (CHead c3 (Bind b) 
-u2) H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 
-(Bind Abst) v1) c)) H (CHead c3 (Bind b) u2) H3) in (eq_ind C (CHead c3 (Bind 
-b) u2) (\lambda (c: C).(or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind 
+e2 v2 v1)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 
+(Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind 
 Abst) v1))) (\lambda (e2: C).(csubt 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 (v2: T).(eq C c3 (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)))))) (let H7 \def (eq_ind C (CHead c0 (Bind Void) u1) (\lambda (ee: 
-C).(match ee 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 
-(or (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Bind 
-Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: 
-C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) u2) (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))))) H7)) c2 H3))))))))))))) (\lambda (H0: (csubt g 
-(CHead e1 (Bind Abst) v1) c2)).(\lambda (c0: C).(\lambda (c3: C).(\lambda (u: 
-T).(\lambda (t: T).(\lambda (H2: (eq C (CHead c0 (Bind Abst) t) (CHead e1 
-(Bind Abst) v1))).(\lambda (H3: (eq C (CHead c3 (Bind Abbr) u) c2)).(\lambda 
-(H1: (csubt g c0 c3)).(\lambda (H4: (ty3 g c3 u t)).(let H5 \def (eq_ind_r C 
-c2 (\lambda (c: C).(csubt g (CHead e1 (Bind Abst) v1) c)) H0 (CHead c3 (Bind 
-Abbr) u) H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead 
-e1 (Bind Abst) v1) c)) H (CHead c3 (Bind Abbr) u) H3) in (eq_ind C (CHead c3 
-(Bind Abbr) u) (\lambda (c: C).(or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 
+e2 v2 v1))))))) H2 e1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c: 
+C).(csubt g c c3)) H1 e1 H8) in (or_introl (ex2 C (\lambda (e2: C).(eq C 
+(CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt 
+g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind 
+Abst) v1) (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)))) 
+(ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind 
+Abst) v1))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 
+(Bind Abst) v1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
+C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 
+(CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 
 (Bind Abst) v1))) (\lambda (e2: C).(csubt 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 (v2: T).(eq C c3 (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)))))) (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) t) (CHead e1 (Bind Abst) v1) H2) in ((let H8 \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 c0 (Bind 
-Abst) t) (CHead e1 (Bind Abst) v1) H2) in (\lambda (H9: (eq C c0 e1)).(let 
-H10 \def (eq_ind T t (\lambda (t0: T).(ty3 g c3 u t0)) H4 v1 H8) in (let H11 
-\def (eq_ind C c0 (\lambda (c: C).(csubt g c c3)) H1 e1 H9) in (or_intror 
-(ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abst) 
-v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda 
-(v2: T).(eq C (CHead c3 (Bind Abbr) u) (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)))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (v2: T).(eq 
-C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
+e2 v2 v1)))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda 
+(u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead 
+e1 (Bind Abst) v1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) 
+(\lambda (ee: C).(match ee 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) H4) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) 
+u2) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T 
+(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) u2) (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))))) H5))))))))))) (\lambda 
+(c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C 
+c1 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead 
+e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda 
+(e2: C).(\lambda (v2: T).(eq C c3 (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))) c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H11 H10)))))) H7)) 
-c2 H3)))))))))))) H))))).
+e2 v2 v1)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c3 u 
+t)).(\lambda (H4: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) 
+v1))).(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) H4) in ((let H6 \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) H4) in (\lambda (H7: (eq C c1 e1)).(let H8 
+\def (eq_ind T t (\lambda (t0: T).(ty3 g c3 u t0)) H3 v1 H6) in (let H9 \def 
+(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or (ex2 
+C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: 
+C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 
+(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))))))) H2 e1 H7) in 
+(let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H7) in 
+(or_intror (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 
+(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: 
+C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (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)))) (ex3_2_intro C T (\lambda (e2: 
+C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (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))) c3 u (refl_equal C (CHead c3 (Bind 
+Abbr) u)) H10 H8))))))) H5)))))))))) y c2 H0))) H))))).
 
 theorem csubt_gen_bind:
  \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall 
@@ -274,114 +205,105 @@ 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))))))))))
 \def
  \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda 
-(v1: T).(\lambda (H: (csubt g (CHead e1 (Bind b1) v1) c2)).(csubt_inv_coq g 
-(CHead e1 (Bind b1) v1) c2 (\lambda (g0: G).(\lambda (_: C).(\lambda (c0: 
-C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c0 
+(v1: T).(\lambda (H: (csubt g (CHead e1 (Bind b1) v1) c2)).(insert_eq C 
+(CHead e1 (Bind b1) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: 
+C).(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 g0 e1 e2)))))))) (\lambda (H0: (csubt g (CHead e1 (Bind b1) v1) 
-c2)).(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind b1) 
-v1))).(\lambda (H2: (eq C (CSort n) c2)).(let H3 \def (eq_ind_r C c2 (\lambda 
-(c: C).(csubt g (CHead e1 (Bind b1) v1) c)) H0 (CSort n) H2) in (let H4 \def 
-(eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind b1) v1) c)) H (CSort 
-n) H2) in (eq_ind C (CSort n) (\lambda (c: C).(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).(csubt g e1 e2)))))) (let H5 
-\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: 
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
-False])) I (CHead e1 (Bind b1) v1) H1) 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).(csubt g e1 e2))))) 
-H5)) c2 H2))))))) (\lambda (H0: (csubt g (CHead e1 (Bind b1) v1) 
-c2)).(\lambda (c0: C).(\lambda (c3: C).(\lambda (k: K).(\lambda (u: 
-T).(\lambda (H1: (eq C (CHead c0 k u) (CHead e1 (Bind b1) v1))).(\lambda (H2: 
-(eq C (CHead c3 k u) c2)).(\lambda (H3: (csubt g c0 c3)).(let H4 \def 
-(eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind b1) v1) c)) H0 (CHead 
-c3 k u) H2) in (let H5 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 
-(Bind b1) v1) c)) H (CHead c3 k u) H2) in (eq_ind C (CHead c3 k u) (\lambda 
-(c: C).(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).(csubt g e1 e2)))))) (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) H1) in ((let H7 \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) H1) in ((let H8 \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) H1) in 
-(\lambda (H9: (eq K k (Bind b1))).(\lambda (H10: (eq C c0 e1)).(let H11 \def 
-(eq_ind T u (\lambda (t: T).(csubt g (CHead e1 (Bind b1) v1) (CHead c3 k t))) 
-H5 v1 H8) in (let H12 \def (eq_ind T u (\lambda (t: T).(csubt g (CHead e1 
-(Bind b1) v1) (CHead c3 k t))) H4 v1 H8) in (eq_ind_r T v1 (\lambda (t: 
-T).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
-(CHead c3 k t) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csubt g e1 e2)))))) (let H13 \def (eq_ind K k (\lambda 
-(k0: K).(csubt g (CHead e1 (Bind b1) v1) (CHead c3 k0 v1))) H11 (Bind b1) H9) 
-in (let H14 \def (eq_ind K k (\lambda (k0: K).(csubt g (CHead e1 (Bind b1) 
-v1) (CHead c3 k0 v1))) H12 (Bind b1) H9) in (eq_ind_r K (Bind b1) (\lambda 
-(k0: K).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
-(CHead c3 k0 v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csubt g e1 e2)))))) (let H15 \def (eq_ind C c0 (\lambda 
-(c: C).(csubt g c c3)) H3 e1 H10) in (ex2_3_intro B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b1) v1) (CHead e2 
-(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g 
-e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3 (Bind b1) v1)) H15)) k H9))) u 
-H8)))))) H7)) H6)) c2 H2))))))))))) (\lambda (H0: (csubt g (CHead e1 (Bind 
-b1) v1) c2)).(\lambda (c0: C).(\lambda (c3: C).(\lambda (b: B).(\lambda (u1: 
-T).(\lambda (u2: T).(\lambda (H2: (eq C (CHead c0 (Bind Void) u1) (CHead e1 
-(Bind b1) v1))).(\lambda (H3: (eq C (CHead c3 (Bind b) u2) c2)).(\lambda (H1: 
-(csubt g c0 c3)).(\lambda (_: (not (eq B b Void))).(let H5 \def (eq_ind_r C 
-c2 (\lambda (c: C).(csubt g (CHead e1 (Bind b1) v1) c)) H0 (CHead c3 (Bind b) 
-u2) H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 
-(Bind b1) v1) c)) H (CHead c3 (Bind b) u2) H3) in (eq_ind C (CHead c3 (Bind 
-b) u2) (\lambda (c: C).(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).(csubt g e1 e2)))))) (let H7 \def 
+T).(csubt g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csubt g y 
+c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind 
+b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
+T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csubt g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq 
+C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) 
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) 
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) 
+v1) H1) 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).(csubt g e1 e2))))) H2)))) (\lambda (c1: C).(\lambda 
+(c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 
+(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda 
+(v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (k: K).(\lambda (u: 
+T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(let H4 \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 
-Void) u1) (CHead e1 (Bind b1) v1) H2) in ((let H8 \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 c0 (Bind 
-Void) u1) (CHead e1 (Bind b1) v1) H2) in ((let H9 \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 c0 (Bind Void) u1) (CHead e1 (Bind 
-b1) v1) H2) in (\lambda (H10: (eq B Void b1)).(\lambda (H11: (eq C c0 
-e1)).(let H12 \def (eq_ind C c0 (\lambda (c: C).(csubt g c c3)) H1 e1 H11) in 
-(let H13 \def (eq_ind_r B b1 (\lambda (b0: B).(csubt g (CHead e1 (Bind b0) 
-v1) (CHead c3 (Bind b) u2))) H6 Void H10) in (let H14 \def (eq_ind_r B b1 
-(\lambda (b0: B).(csubt g (CHead e1 (Bind b0) v1) (CHead c3 (Bind b) u2))) H5 
-Void H10) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda 
-(v2: T).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: 
-B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))) b c3 u2 (refl_equal C 
-(CHead c3 (Bind b) u2)) H12))))))) H8)) H7)) c2 H3))))))))))))) (\lambda (H0: 
-(csubt g (CHead e1 (Bind b1) v1) c2)).(\lambda (c0: C).(\lambda (c3: 
-C).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (eq C (CHead c0 (Bind Abst) 
-t) (CHead e1 (Bind b1) v1))).(\lambda (H3: (eq C (CHead c3 (Bind Abbr) u) 
-c2)).(\lambda (H1: (csubt g c0 c3)).(\lambda (H4: (ty3 g c3 u t)).(let H5 
-\def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind b1) v1) c)) H0 
-(CHead c3 (Bind Abbr) u) H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c: 
-C).(csubt g (CHead e1 (Bind b1) v1) c)) H (CHead c3 (Bind Abbr) u) H3) in 
-(eq_ind C (CHead c3 (Bind Abbr) u) (\lambda (c: C).(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).(csubt g e1 e2)))))) (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) t) (CHead e1 (Bind b1) v1) H2) in ((let H8 \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 c0 (Bind Abst) t) (CHead e1 (Bind b1) v1) H2) in ((let H9 \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 c0 (Bind 
-Abst) t) (CHead e1 (Bind b1) v1) H2) in (\lambda (H10: (eq B Abst 
-b1)).(\lambda (H11: (eq C c0 e1)).(let H12 \def (eq_ind T t (\lambda (t0: 
-T).(ty3 g c3 u t0)) H4 v1 H9) in (let H13 \def (eq_ind C c0 (\lambda (c: 
-C).(csubt g c c3)) H1 e1 H11) in (let H14 \def (eq_ind_r B b1 (\lambda (b: 
-B).(csubt g (CHead e1 (Bind b) v1) (CHead c3 (Bind Abbr) u))) H6 Abst H10) in 
-(let H15 \def (eq_ind_r B b1 (\lambda (b: B).(csubt g (CHead e1 (Bind b) v1) 
-(CHead c3 (Bind Abbr) u))) H5 Abst H10) in (ex2_3_intro B C T (\lambda (b2: 
+[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 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 c1 k u) (CHead e1 (Bind b1) v1) H3) 
+in ((let H6 \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) H3) in (\lambda (H7: (eq K k (Bind 
+b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t) 
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
+T).(csubt g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) 
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
+T).(csubt g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c 
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 H8) in (let 
+H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in 
+(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
+(CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda 
+(e2: C).(\lambda (_: T).(csubt g e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3 
+(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
+C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (b: 
+B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) 
+v1))).(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 Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H6 \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) H4) in ((let H7 
+\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) H4) in (\lambda (H8: (eq B Void 
+b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind C c1 (\lambda (c: 
+C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 
+H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H9) 
+in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 (CHead e1 (Bind 
+b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
+T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csubt g e1 e2))))))) H10 Void H8) in (ex2_3_intro B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) 
+u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
+T).(csubt g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 (Bind b) u2)) 
+H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: 
+(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3 
+B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 
+(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g 
+e1 e2)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c3 u 
+t)).(\lambda (H4: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) 
+v1))).(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 b1) v1) H4) in ((let H6 \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) H4) in ((let H7 
+\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) H4) in (\lambda (H8: (eq B Abst 
+b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind T t (\lambda (t0: 
+T).(ty3 g c3 u t0)) H3 v1 H7) in (let H11 \def (eq_ind C c1 (\lambda (c: 
+C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 
+H9) in (let H12 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H9) 
+in (let H13 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) 
+v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq 
+C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda 
+(_: T).(csubt g e1 e2))))))) H11 Abst H8) in (ex2_3_intro B C T (\lambda (b2: 
 B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 
 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g 
-e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H13)))))))) H8)) 
-H7)) c2 H3)))))))))))) H)))))).
+e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H12)))))))) H6)) 
+H5)))))))))) y c2 H0))) H)))))).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/getl.ma
index 1f39ef182..9c5c018af 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/getl.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/getl.ma
@@ -14,13 +14,13 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubt/fwd.ma".
+include "LambdaDelta-1/csubt/fwd.ma".
 
-include "csubt/clear.ma".
+include "LambdaDelta-1/csubt/clear.ma".
 
-include "csubt/drop.ma".
+include "LambdaDelta-1/csubt/drop.ma".
 
-include "getl/clear.ma".
+include "LambdaDelta-1/getl/clear.ma".
 
 theorem csubt_getl_abbr:
  \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/pc3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/pc3.ma
index 342443cf6..a372c47a1 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/pc3.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/pc3.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubt/getl.ma".
+include "LambdaDelta-1/csubt/getl.ma".
 
-include "pc3/left.ma".
+include "LambdaDelta-1/pc3/left.ma".
 
 theorem csubt_pr2:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/props.ma
index 766ba4589..1942b98f9 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/props.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubt/defs.ma".
+include "LambdaDelta-1/csubt/defs.ma".
 
 theorem csubt_refl:
  \forall (g: G).(\forall (c: C).(csubt g c c))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/ty3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/ty3.ma
index cc8839f26..6334713f6 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/ty3.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/ty3.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubt/pc3.ma".
+include "LambdaDelta-1/csubt/pc3.ma".
 
-include "csubt/props.ma".
+include "LambdaDelta-1/csubt/props.ma".
 
 theorem csubt_ty3:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/definitions.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/definitions.ma
index 7b6e09292..1b9906dbb 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/definitions.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/definitions.ma
@@ -14,47 +14,47 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "tlt/defs.ma".
+include "LambdaDelta-1/tlt/defs.ma".
 
-include "iso/defs.ma".
+include "LambdaDelta-1/iso/defs.ma".
 
-include "clen/defs.ma".
+include "LambdaDelta-1/clen/defs.ma".
 
-include "flt/defs.ma".
+include "LambdaDelta-1/flt/defs.ma".
 
-include "cnt/defs.ma".
+include "LambdaDelta-1/cnt/defs.ma".
 
-include "cimp/defs.ma".
+include "LambdaDelta-1/cimp/defs.ma".
 
-include "subst1/defs.ma".
+include "LambdaDelta-1/subst1/defs.ma".
 
-include "csubst1/defs.ma".
+include "LambdaDelta-1/csubst1/defs.ma".
 
-include "fsubst0/defs.ma".
+include "LambdaDelta-1/fsubst0/defs.ma".
 
-include "next_plus/defs.ma".
+include "LambdaDelta-1/next_plus/defs.ma".
 
-include "tau1/defs.ma".
+include "LambdaDelta-1/tau1/defs.ma".
 
-include "llt/defs.ma".
+include "LambdaDelta-1/llt/defs.ma".
 
-include "aprem/defs.ma".
+include "LambdaDelta-1/aprem/defs.ma".
 
-include "ex0/defs.ma".
+include "LambdaDelta-1/ex0/defs.ma".
 
-include "wcpr0/defs.ma".
+include "LambdaDelta-1/wcpr0/defs.ma".
 
-include "csuba/defs.ma".
+include "LambdaDelta-1/csuba/defs.ma".
 
-include "nf2/defs.ma".
+include "LambdaDelta-1/nf2/defs.ma".
 
-include "ex2/defs.ma".
+include "LambdaDelta-1/ex2/defs.ma".
 
-include "csubc/defs.ma".
+include "LambdaDelta-1/csubc/defs.ma".
 
-include "pc1/defs.ma".
+include "LambdaDelta-1/pc1/defs.ma".
 
-include "ex1/defs.ma".
+include "LambdaDelta-1/ex1/defs.ma".
 
-include "csubt/defs.ma".
+include "LambdaDelta-1/csubt/defs.ma".
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/depends b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/depends
deleted file mode 100644
index 32b690aa9..000000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/depends
+++ /dev/null
@@ -1,200 +0,0 @@
-theory.ma ex0/props.ma ex1/props.ma ex2/props.ma pr3/wcpr0.ma subst0/tlt.ma tau1/cnt.ma ty3/dec.ma ty3/nf2.ma ty3/tau0.ma wcpr0/fwd.ma
-spare.ma theory.ma
-preamble.ma ../Base-1/theory.ma
-definitions.ma aprem/defs.ma cimp/defs.ma clen/defs.ma cnt/defs.ma csuba/defs.ma csubc/defs.ma csubst1/defs.ma csubt/defs.ma ex0/defs.ma ex1/defs.ma ex2/defs.ma flt/defs.ma fsubst0/defs.ma iso/defs.ma llt/defs.ma next_plus/defs.ma nf2/defs.ma pc1/defs.ma subst1/defs.ma tau1/defs.ma tlt/defs.ma wcpr0/defs.ma
-ex0/defs.ma A/defs.ma G/defs.ma
-ex0/props.ma aplus/props.ma ex0/defs.ma leq/defs.ma
-ex2/defs.ma C/defs.ma
-ex2/props.ma arity/fwd.ma ex2/defs.ma nf2/defs.ma pr2/fwd.ma
-tlist/defs.ma T/defs.ma
-tlist/props.ma tlist/defs.ma
-csubt/defs.ma ty3/defs.ma
-csubt/fwd.ma csubt/defs.ma
-csubt/props.ma csubt/defs.ma
-csubt/clear.ma clear/fwd.ma csubt/defs.ma
-csubt/drop.ma csubt/defs.ma drop/fwd.ma
-csubt/getl.ma csubt/clear.ma csubt/drop.ma csubt/fwd.ma getl/clear.ma
-csubt/pc3.ma csubt/getl.ma pc3/left.ma
-csubt/ty3.ma csubt/pc3.ma csubt/props.ma
-csubc/defs.ma sc3/defs.ma
-csubc/props.ma csubc/defs.ma sc3/props.ma
-csubc/csuba.ma csuba/defs.ma csubc/defs.ma sc3/props.ma
-csubc/drop.ma csubc/defs.ma sc3/props.ma
-csubc/drop1.ma csubc/drop.ma
-csubc/clear.ma csubc/defs.ma
-csubc/getl.ma csubc/clear.ma csubc/drop.ma
-csubc/arity.ma arity/defs.ma csubc/csuba.ma
-ty3/fsubst0.ma csubst0/props.ma getl/getl.ma pc3/fsubst0.ma ty3/props.ma
-ty3/defs.ma G/defs.ma pc3/defs.ma
-ty3/fwd.ma pc3/props.ma ty3/defs.ma
-ty3/props.ma pc3/fwd.ma ty3/fwd.ma
-ty3/subst1.ma csubst1/fwd.ma csubst1/getl.ma getl/getl.ma pc3/fwd.ma pc3/subst1.ma ty3/props.ma
-ty3/pr3_props.ma ty3/pr3.ma
-ty3/pr3.ma csubt/ty3.ma pc1/props.ma pc3/pc1.ma pc3/wcpr0.ma ty3/fsubst0.ma ty3/subst1.ma
-ty3/tau0.ma tau0/defs.ma ty3/pr3_props.ma
-ty3/arity.ma arity/pr3.ma asucc/fwd.ma ty3/pr3_props.ma
-ty3/dec.ma getl/dec.ma getl/flt.ma pc3/dec.ma ty3/pr3_props.ma
-ty3/arity_props.ma sc3/arity.ma ty3/arity.ma ty3/fwd.ma
-ty3/nf2.ma nf2/arity.ma pc3/nf2.ma ty3/arity.ma
-pc3/defs.ma pr3/defs.ma
-pc3/props.ma pc3/defs.ma pr3/pr3.ma
-pc3/fsubst0.ma csubst0/getl.ma fsubst0/defs.ma pc3/left.ma
-pc3/pc1.ma pc1/defs.ma pc3/defs.ma pr3/pr1.ma
-pc3/wcpr0.ma pc3/props.ma wcpr0/getl.ma
-pc3/left.ma pc3/props.ma
-pc3/fwd.ma pc3/props.ma pr3/fwd.ma
-pc3/nf2.ma nf2/pr3.ma pc3/defs.ma
-pc3/subst1.ma pc3/props.ma pr3/subst1.ma
-pc3/dec.ma nf2/fwd.ma ty3/arity_props.ma ty3/pr3.ma
-pc1/defs.ma pr1/defs.ma
-pc1/props.ma pc1/defs.ma pr1/pr1.ma
-ex1/defs.ma C/defs.ma
-ex1/props.ma arity/defs.ma ex1/defs.ma leq/props.ma nf2/pr3.ma nf2/props.ma pc3/fwd.ma ty3/fwd.ma
-sc3/defs.ma arity/defs.ma drop1/defs.ma sn3/defs.ma
-sc3/props.ma arity/aprem.ma arity/lift1.ma csuba/arity.ma drop1/getl.ma drop1/props.ma lift1/props.ma llt/props.ma nf2/lift1.ma sc3/defs.ma sn3/lift1.ma
-sc3/arity.ma csubc/arity.ma csubc/drop1.ma csubc/getl.ma csubc/props.ma
-sn3/defs.ma pr3/defs.ma
-sn3/nf2.ma nf2/dec.ma nf2/pr3.ma sn3/defs.ma
-sn3/props.ma iso/props.ma nf2/iso.ma pr3/iso.ma sn3/fwd.ma sn3/nf2.ma
-sn3/lift1.ma drop1/defs.ma lift1/fwd.ma sn3/props.ma
-sn3/fwd.ma pr3/props.ma sn3/defs.ma
-nf2/defs.ma pr2/defs.ma
-nf2/fwd.ma T/props.ma nf2/defs.ma pr2/clen.ma subst0/dec.ma
-nf2/props.ma nf2/defs.ma pr2/fwd.ma
-nf2/arity.ma arity/subst0.ma nf2/fwd.ma
-nf2/pr3.ma nf2/defs.ma pr3/pr3.ma
-nf2/lift1.ma drop1/defs.ma nf2/props.ma
-nf2/iso.ma iso/props.ma nf2/pr3.ma pr3/fwd.ma
-nf2/dec.ma C/props.ma nf2/defs.ma pr0/dec.ma pr2/clen.ma pr2/fwd.ma
-csuba/defs.ma arity/defs.ma
-csuba/fwd.ma csuba/defs.ma
-csuba/props.ma csuba/defs.ma
-csuba/arity.ma T/props.ma arity/props.ma csuba/getl.ma csuba/props.ma
-csuba/clear.ma clear/fwd.ma csuba/defs.ma
-csuba/drop.ma csuba/fwd.ma drop/fwd.ma
-csuba/getl.ma csuba/clear.ma csuba/drop.ma getl/clear.ma
-wcpr0/getl.ma getl/props.ma wcpr0/defs.ma
-wcpr0/fwd.ma wcpr0/defs.ma
-wcpr0/defs.ma C/defs.ma pr0/defs.ma
-pr3/defs.ma pr2/defs.ma
-pr3/subst1.ma pr2/subst1.ma pr3/defs.ma
-pr3/props.ma pr1/props.ma pr2/props.ma pr3/pr1.ma
-pr3/pr1.ma pr1/defs.ma pr3/defs.ma
-pr3/wcpr0.ma pr3/props.ma wcpr0/getl.ma
-pr3/fwd.ma pr2/fwd.ma pr3/props.ma
-pr3/pr3.ma pr2/pr2.ma pr3/props.ma
-pr3/iso.ma iso/props.ma pr3/fwd.ma tlist/props.ma
-pr2/defs.ma getl/defs.ma pr0/defs.ma
-pr2/props.ma getl/clear.ma getl/drop.ma pr0/props.ma pr2/defs.ma
-pr2/fwd.ma getl/clear.ma getl/drop.ma pr0/fwd.ma pr2/defs.ma
-pr2/subst1.ma csubst1/fwd.ma csubst1/getl.ma getl/drop.ma pr0/fwd.ma pr0/subst1.ma pr2/defs.ma subst1/subst1.ma
-pr2/clen.ma clen/getl.ma pr2/props.ma
-pr2/pr2.ma getl/props.ma pr0/pr0.ma pr2/defs.ma
-pr1/defs.ma pr0/defs.ma
-pr1/props.ma T/props.ma pr0/subst1.ma pr1/defs.ma subst1/props.ma
-pr1/pr1.ma pr0/pr0.ma pr1/props.ma
-pr0/subst1.ma pr0/props.ma subst1/defs.ma
-pr0/defs.ma subst0/defs.ma
-pr0/fwd.ma pr0/props.ma
-pr0/props.ma pr0/defs.ma subst0/subst0.ma
-pr0/pr0.ma lift/tlt.ma pr0/fwd.ma
-pr0/dec.ma T/dec.ma T/props.ma pr0/fwd.ma subst0/dec.ma
-tau1/defs.ma tau0/defs.ma
-tau1/props.ma tau0/props.ma tau1/defs.ma
-tau1/cnt.ma cnt/props.ma tau1/props.ma
-arity/defs.ma getl/defs.ma leq/defs.ma
-arity/fwd.ma arity/defs.ma getl/drop.ma leq/asucc.ma leq/fwd.ma
-arity/subst0.ma arity/props.ma csubst0/getl.ma csubst0/props.ma fsubst0/fwd.ma getl/getl.ma subst0/dec.ma subst0/fwd.ma
-arity/props.ma arity/fwd.ma
-arity/pr3.ma arity/subst0.ma csuba/arity.ma pr0/fwd.ma pr1/defs.ma pr3/defs.ma wcpr0/getl.ma
-arity/lift1.ma arity/props.ma drop1/defs.ma
-arity/cimp.ma arity/defs.ma cimp/props.ma
-arity/aprem.ma aprem/props.ma arity/cimp.ma arity/props.ma
-aprem/defs.ma A/defs.ma
-aprem/props.ma aprem/defs.ma leq/defs.ma
-llt/defs.ma A/defs.ma
-llt/props.ma leq/defs.ma llt/defs.ma
-leq/defs.ma aplus/defs.ma
-leq/asucc.ma aplus/props.ma leq/props.ma
-leq/props.ma aplus/props.ma leq/defs.ma
-leq/fwd.ma leq/defs.ma
-aplus/defs.ma asucc/defs.ma
-aplus/props.ma aplus/defs.ma next_plus/props.ma
-asucc/defs.ma A/defs.ma G/defs.ma
-asucc/fwd.ma asucc/defs.ma
-A/defs.ma preamble.ma
-fsubst0/defs.ma csubst0/defs.ma subst0/defs.ma
-fsubst0/fwd.ma fsubst0/defs.ma
-csubst1/defs.ma csubst0/defs.ma
-csubst1/props.ma csubst1/defs.ma subst1/defs.ma
-csubst1/fwd.ma csubst0/fwd.ma csubst1/defs.ma subst1/props.ma
-csubst1/getl.ma csubst0/getl.ma csubst0/props.ma csubst1/props.ma drop/props.ma subst1/props.ma
-csubst0/defs.ma C/defs.ma subst0/defs.ma
-csubst0/fwd.ma csubst0/defs.ma
-csubst0/props.ma csubst0/defs.ma
-csubst0/drop.ma csubst0/fwd.ma drop/fwd.ma s/props.ma
-csubst0/clear.ma clear/fwd.ma csubst0/fwd.ma
-csubst0/getl.ma csubst0/clear.ma csubst0/drop.ma getl/fwd.ma
-subst0/defs.ma lift/defs.ma
-subst0/fwd.ma lift/props.ma subst0/defs.ma
-subst0/subst0.ma subst0/props.ma
-subst0/props.ma lift/props.ma subst0/fwd.ma
-subst0/tlt.ma lift/props.ma lift/tlt.ma subst0/defs.ma
-subst0/dec.ma lift/props.ma subst0/defs.ma
-tau0/defs.ma G/defs.ma getl/defs.ma
-tau0/props.ma getl/drop.ma tau0/defs.ma
-subst1/subst1.ma subst0/subst0.ma subst1/fwd.ma
-subst1/defs.ma subst0/defs.ma
-subst1/fwd.ma subst0/props.ma subst1/defs.ma
-subst1/props.ma subst0/props.ma subst1/defs.ma
-cimp/defs.ma getl/defs.ma
-cimp/props.ma cimp/defs.ma getl/getl.ma
-getl/defs.ma clear/defs.ma drop/defs.ma
-getl/props.ma clear/props.ma drop/props.ma getl/fwd.ma
-getl/fwd.ma clear/fwd.ma drop/fwd.ma getl/defs.ma
-getl/drop.ma clear/drop.ma getl/props.ma r/props.ma
-getl/clear.ma clear/drop.ma getl/props.ma
-getl/getl.ma getl/clear.ma getl/drop.ma
-getl/dec.ma getl/props.ma
-getl/flt.ma clear/props.ma flt/props.ma getl/fwd.ma
-clear/defs.ma C/defs.ma
-clear/fwd.ma clear/defs.ma
-clear/props.ma clear/fwd.ma
-clear/drop.ma clear/fwd.ma drop/fwd.ma
-drop1/defs.ma drop/defs.ma lift1/defs.ma
-drop1/props.ma drop/props.ma drop1/defs.ma getl/defs.ma
-drop1/getl.ma drop1/defs.ma getl/drop.ma
-drop/defs.ma C/defs.ma lift/defs.ma r/defs.ma
-drop/fwd.ma drop/defs.ma
-drop/props.ma drop/fwd.ma lift/props.ma r/props.ma
-lift1/defs.ma lift/defs.ma
-lift1/fwd.ma lift/fwd.ma lift1/defs.ma
-lift1/props.ma drop1/defs.ma lift/props.ma lift1/defs.ma
-cnt/defs.ma T/defs.ma
-cnt/props.ma cnt/defs.ma lift/fwd.ma
-lift/defs.ma T/defs.ma s/defs.ma tlist/defs.ma
-lift/fwd.ma lift/defs.ma
-lift/props.ma lift/fwd.ma s/props.ma tlist/defs.ma
-lift/tlt.ma lift/fwd.ma tlt/props.ma
-iso/props.ma iso/fwd.ma
-iso/fwd.ma iso/defs.ma tlist/defs.ma
-iso/defs.ma T/defs.ma
-G/defs.ma preamble.ma
-flt/props.ma C/props.ma flt/defs.ma
-flt/defs.ma C/defs.ma
-tlt/props.ma tlt/defs.ma
-tlt/defs.ma T/defs.ma
-T/props.ma T/defs.ma
-T/dec.ma T/defs.ma
-T/defs.ma preamble.ma
-s/props.ma s/defs.ma
-s/defs.ma T/defs.ma
-C/props.ma C/defs.ma T/props.ma
-C/defs.ma T/defs.ma
-r/props.ma r/defs.ma s/defs.ma
-r/defs.ma T/defs.ma
-next_plus/props.ma next_plus/defs.ma
-next_plus/defs.ma G/defs.ma
-clen/getl.ma clen/defs.ma getl/props.ma
-clen/defs.ma C/defs.ma s/defs.ma
-../Base-1/theory.ma 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop/defs.ma
index 8967b8645..6c534b080 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop/defs.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "C/defs.ma".
+include "LambdaDelta-1/C/defs.ma".
 
-include "lift/defs.ma".
+include "LambdaDelta-1/lift/defs.ma".
 
-include "r/defs.ma".
+include "LambdaDelta-1/r/defs.ma".
 
 inductive drop: nat \to (nat \to (C \to (C \to Prop))) \def
 | drop_refl: \forall (c: C).(drop O O c c)
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop/fwd.ma
index b4653f80f..aff6250a2 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop/fwd.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "drop/defs.ma".
+include "LambdaDelta-1/drop/defs.ma".
 
 theorem drop_gen_sort:
  \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(\forall (x: C).((drop 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop/props.ma
index 39343a178..95761db5c 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop/props.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "drop/fwd.ma".
+include "LambdaDelta-1/drop/fwd.ma".
 
-include "lift/props.ma".
+include "LambdaDelta-1/lift/props.ma".
 
-include "r/props.ma".
+include "LambdaDelta-1/r/props.ma".
 
 theorem drop_skip_bind:
  \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop1/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop1/defs.ma
index 78e13d219..2e4482ec9 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop1/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop1/defs.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "drop/defs.ma".
+include "LambdaDelta-1/drop/defs.ma".
 
-include "lift1/defs.ma".
+include "LambdaDelta-1/lift1/defs.ma".
 
 inductive drop1: PList \to (C \to (C \to Prop)) \def
 | drop1_nil: \forall (c: C).(drop1 PNil c c)
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop1/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop1/getl.ma
index 711735acd..6704bf6f8 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop1/getl.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop1/getl.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "drop1/defs.ma".
+include "LambdaDelta-1/drop1/defs.ma".
 
-include "getl/drop.ma".
+include "LambdaDelta-1/getl/drop.ma".
 
 theorem drop1_getl_trans:
  \forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1) 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop1/props.ma
index 1f6599d96..cb83a4c5b 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop1/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop1/props.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "drop1/defs.ma".
+include "LambdaDelta-1/drop1/defs.ma".
 
-include "drop/props.ma".
+include "LambdaDelta-1/drop/props.ma".
 
-include "getl/defs.ma".
+include "LambdaDelta-1/getl/defs.ma".
 
 theorem drop1_skip_bind:
  \forall (b: B).(\forall (e: C).(\forall (hds: PList).(\forall (c: 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex0/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex0/defs.ma
index 755b481e2..58d08227c 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex0/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex0/defs.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "A/defs.ma".
+include "LambdaDelta-1/A/defs.ma".
 
-include "G/defs.ma".
+include "LambdaDelta-1/G/defs.ma".
 
 definition gz:
  G
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex0/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex0/props.ma
index c035c099e..56620b18f 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex0/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex0/props.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "ex0/defs.ma".
+include "LambdaDelta-1/ex0/defs.ma".
 
-include "leq/defs.ma".
+include "LambdaDelta-1/leq/defs.ma".
 
-include "aplus/props.ma".
+include "LambdaDelta-1/aplus/props.ma".
 
 theorem aplus_gz_le:
  \forall (k: nat).(\forall (h: nat).(\forall (n: nat).((le h k) \to (eq A 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex1/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex1/defs.ma
index e4e1e229d..941ec4adf 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex1/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex1/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "C/defs.ma".
+include "LambdaDelta-1/C/defs.ma".
 
 definition ex1_c:
  C
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex1/props.ma
index e809b11bf..65636c34f 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex1/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex1/props.ma
@@ -14,19 +14,19 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "ex1/defs.ma".
+include "LambdaDelta-1/ex1/defs.ma".
 
-include "ty3/fwd.ma".
+include "LambdaDelta-1/ty3/fwd.ma".
 
-include "pc3/fwd.ma".
+include "LambdaDelta-1/pc3/fwd.ma".
 
-include "nf2/pr3.ma".
+include "LambdaDelta-1/nf2/pr3.ma".
 
-include "nf2/props.ma".
+include "LambdaDelta-1/nf2/props.ma".
 
-include "arity/defs.ma".
+include "LambdaDelta-1/arity/defs.ma".
 
-include "leq/props.ma".
+include "LambdaDelta-1/leq/props.ma".
 
 theorem ex1__leq_sort_SS:
  \forall (g: G).(\forall (k: nat).(\forall (n: nat).(leq g (ASort k n) (asucc 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex2/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex2/defs.ma
index 713a5aac9..1f9cb2dcd 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex2/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex2/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "C/defs.ma".
+include "LambdaDelta-1/C/defs.ma".
 
 definition ex2_c:
  C
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex2/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex2/props.ma
index afcdfbfe5..4372ace2d 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex2/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex2/props.ma
@@ -14,13 +14,13 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "ex2/defs.ma".
+include "LambdaDelta-1/ex2/defs.ma".
 
-include "nf2/defs.ma".
+include "LambdaDelta-1/nf2/defs.ma".
 
-include "pr2/fwd.ma".
+include "LambdaDelta-1/pr2/fwd.ma".
 
-include "arity/fwd.ma".
+include "LambdaDelta-1/arity/fwd.ma".
 
 theorem ex2_nf2:
  nf2 ex2_c ex2_t
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/flt/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/flt/defs.ma
index 020a79788..46515c161 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/flt/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/flt/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "C/defs.ma".
+include "LambdaDelta-1/C/defs.ma".
 
 definition fweight:
  C \to (T \to nat)
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/flt/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/flt/props.ma
index bdb5ccaa0..c526d5b60 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/flt/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/flt/props.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "flt/defs.ma".
+include "LambdaDelta-1/flt/defs.ma".
 
-include "C/props.ma".
+include "LambdaDelta-1/C/props.ma".
 
 theorem flt_thead_sx:
  \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c u c 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/fsubst0/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/fsubst0/defs.ma
index 9da21ae76..bad6dc447 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/fsubst0/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/fsubst0/defs.ma
@@ -14,9 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubst0/defs.ma".
-
-include "subst0/defs.ma".
+include "LambdaDelta-1/csubst0/defs.ma".
 
 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 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/fsubst0/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/fsubst0/fwd.ma
index 85696f0b3..7df3bf8bb 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/fsubst0/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/fsubst0/fwd.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "fsubst0/defs.ma".
+include "LambdaDelta-1/fsubst0/defs.ma".
 
 theorem fsubst0_gen_base:
  \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).(\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/clear.ma
index 22bbb7ec2..a08c5da27 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/clear.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/clear.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "getl/props.ma".
+include "LambdaDelta-1/getl/props.ma".
 
-include "clear/drop.ma".
+include "LambdaDelta-1/clear/drop.ma".
 
 theorem clear_getl_trans:
  \forall (i: nat).(\forall (c2: C).(\forall (c3: C).((getl i c2 c3) \to 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/dec.ma
index 01ce28e0d..93cea0f74 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/dec.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/dec.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "getl/props.ma".
+include "LambdaDelta-1/getl/props.ma".
 
 theorem getl_dec:
  \forall (c: C).(\forall (i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/defs.ma
index 545b48953..fc179d0a8 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/defs.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "drop/defs.ma".
+include "LambdaDelta-1/drop/defs.ma".
 
-include "clear/defs.ma".
+include "LambdaDelta-1/clear/defs.ma".
 
 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 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/drop.ma
index 47d5168a4..7f0ade54d 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/drop.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/drop.ma
@@ -14,11 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "getl/props.ma".
+include "LambdaDelta-1/getl/props.ma".
 
-include "clear/drop.ma".
-
-include "r/props.ma".
+include "LambdaDelta-1/clear/drop.ma".
 
 theorem getl_drop:
  \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h: 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/flt.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/flt.ma
index f4d4ffa8d..9c3f4ffaa 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/flt.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/flt.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "getl/fwd.ma".
+include "LambdaDelta-1/getl/fwd.ma".
 
-include "clear/props.ma".
+include "LambdaDelta-1/clear/props.ma".
 
-include "flt/props.ma".
+include "LambdaDelta-1/flt/props.ma".
 
 theorem getl_flt:
  \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/fwd.ma
index 0f3244947..13f96107e 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/fwd.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "getl/defs.ma".
+include "LambdaDelta-1/getl/defs.ma".
 
-include "drop/fwd.ma".
+include "LambdaDelta-1/drop/fwd.ma".
 
-include "clear/fwd.ma".
+include "LambdaDelta-1/clear/fwd.ma".
 
 theorem getl_gen_all:
  \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex2 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/getl.ma
index 5534efff6..19ab5606a 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/getl.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/getl.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "getl/drop.ma".
+include "LambdaDelta-1/getl/drop.ma".
 
-include "getl/clear.ma".
+include "LambdaDelta-1/getl/clear.ma".
 
 theorem getl_conf_le:
  \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/props.ma
index 3c884173d..4bcf2521c 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/props.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "getl/fwd.ma".
+include "LambdaDelta-1/getl/fwd.ma".
 
-include "drop/props.ma".
+include "LambdaDelta-1/drop/props.ma".
 
-include "clear/props.ma".
+include "LambdaDelta-1/clear/props.ma".
 
 theorem getl_refl:
  \forall (b: B).(\forall (c: C).(\forall (u: T).(getl O (CHead c (Bind b) u) 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/iso/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/iso/defs.ma
index 58bf0424c..78025c0b5 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/iso/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/iso/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "T/defs.ma".
+include "LambdaDelta-1/T/defs.ma".
 
 inductive iso: T \to (T \to Prop) \def
 | iso_sort: \forall (n1: nat).(\forall (n2: nat).(iso (TSort n1) (TSort n2)))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/iso/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/iso/fwd.ma
index 37ddce3ed..069532c05 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/iso/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/iso/fwd.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "iso/defs.ma".
+include "LambdaDelta-1/iso/defs.ma".
 
-include "tlist/defs.ma".
+include "LambdaDelta-1/tlist/defs.ma".
 
 theorem iso_flats_lref_bind_false:
  \forall (f: F).(\forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/iso/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/iso/props.ma
index 215f79c28..961bf3b2a 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/iso/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/iso/props.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "iso/fwd.ma".
+include "LambdaDelta-1/iso/fwd.ma".
 
 theorem iso_refl:
  \forall (t: T).(iso t t)
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/asucc.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/asucc.ma
index ef8360a02..f3fe08b00 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/asucc.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/asucc.ma
@@ -14,9 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "leq/props.ma".
-
-include "aplus/props.ma".
+include "LambdaDelta-1/leq/props.ma".
 
 theorem asucc_repl:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/defs.ma
index dad66914d..f5d466402 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "aplus/defs.ma".
+include "LambdaDelta-1/aplus/defs.ma".
 
 inductive leq (g: G): A \to (A \to Prop) \def
 | leq_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/fwd.ma
index 6b953558d..03c6edd3a 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/fwd.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "leq/defs.ma".
+include "LambdaDelta-1/leq/defs.ma".
 
 theorem leq_gen_sort:
  \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/props.ma
index f74c7e47a..3e5717ef7 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/props.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "leq/defs.ma".
+include "LambdaDelta-1/leq/defs.ma".
 
-include "aplus/props.ma".
+include "LambdaDelta-1/aplus/props.ma".
 
 theorem ahead_inj_snd:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/defs.ma
index 49d9a670d..2f6131325 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/defs.ma
@@ -14,11 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "T/defs.ma".
+include "LambdaDelta-1/tlist/defs.ma".
 
-include "tlist/defs.ma".
-
-include "s/defs.ma".
+include "LambdaDelta-1/s/defs.ma".
 
 definition lref_map:
  ((nat \to nat)) \to (nat \to (T \to T))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/fwd.ma
index 07f801655..af8277b07 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/fwd.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "lift/defs.ma".
+include "LambdaDelta-1/lift/defs.ma".
 
 theorem lift_sort:
  \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (TSort 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/props.ma
index 428146f30..168d3e09c 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/props.ma
@@ -14,11 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "tlist/defs.ma".
+include "LambdaDelta-1/lift/fwd.ma".
 
-include "lift/fwd.ma".
-
-include "s/props.ma".
+include "LambdaDelta-1/s/props.ma".
 
 theorem thead_x_lift_y_y:
  \forall (k: K).(\forall (t: T).(\forall (v: T).(\forall (h: nat).(\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/tlt.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/tlt.ma
index 4c728e102..b80753879 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/tlt.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/tlt.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "lift/fwd.ma".
+include "LambdaDelta-1/lift/fwd.ma".
 
-include "tlt/props.ma".
+include "LambdaDelta-1/tlt/props.ma".
 
 theorem lift_weight_map:
  \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift1/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift1/defs.ma
index 691829e8e..fc8405a3c 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift1/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift1/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "lift/defs.ma".
+include "LambdaDelta-1/lift/defs.ma".
 
 definition trans:
  PList \to (nat \to nat)
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift1/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift1/fwd.ma
index de9bcba64..8a164af04 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift1/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift1/fwd.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "lift1/defs.ma".
+include "LambdaDelta-1/lift1/defs.ma".
 
-include "lift/fwd.ma".
+include "LambdaDelta-1/lift/fwd.ma".
 
 theorem lift1_sort:
  \forall (n: nat).(\forall (is: PList).(eq T (lift1 is (TSort n)) (TSort n)))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift1/props.ma
index 154fd00c8..7e7f6a80a 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift1/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift1/props.ma
@@ -14,11 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "lift1/defs.ma".
+include "LambdaDelta-1/lift/props.ma".
 
-include "lift/props.ma".
-
-include "drop1/defs.ma".
+include "LambdaDelta-1/drop1/defs.ma".
 
 theorem lift1_lift1:
  \forall (is1: PList).(\forall (is2: PList).(\forall (t: T).(eq T (lift1 is1 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/llt/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/llt/defs.ma
index b4d60f377..4822f8cff 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/llt/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/llt/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "A/defs.ma".
+include "LambdaDelta-1/A/defs.ma".
 
 definition lweight:
  A \to nat
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/llt/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/llt/props.ma
index 6cfc3ca11..41a498780 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/llt/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/llt/props.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "llt/defs.ma".
+include "LambdaDelta-1/llt/defs.ma".
 
-include "leq/defs.ma".
+include "LambdaDelta-1/leq/defs.ma".
 
 theorem lweight_repl:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (eq nat 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/next_plus/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/next_plus/defs.ma
index f7650f16d..89e426452 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/next_plus/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/next_plus/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "G/defs.ma".
+include "LambdaDelta-1/G/defs.ma".
 
 definition next_plus:
  G \to (nat \to (nat \to nat))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/next_plus/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/next_plus/props.ma
index b8d9a45ac..7e26f27a7 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/next_plus/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/next_plus/props.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "next_plus/defs.ma".
+include "LambdaDelta-1/next_plus/defs.ma".
 
 theorem next_plus_assoc:
  \forall (g: G).(\forall (n: nat).(\forall (h1: nat).(\forall (h2: nat).(eq 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/arity.ma
index a57a60930..867f573e8 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/arity.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/arity.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "nf2/fwd.ma".
+include "LambdaDelta-1/nf2/fwd.ma".
 
-include "arity/subst0.ma".
+include "LambdaDelta-1/arity/subst0.ma".
 
 theorem arity_nf2_inv_all:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/dec.ma
index 85113c01d..37cc2b116 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/dec.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/dec.ma
@@ -14,15 +14,15 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "nf2/defs.ma".
+include "LambdaDelta-1/nf2/defs.ma".
 
-include "pr2/clen.ma".
+include "LambdaDelta-1/pr2/clen.ma".
 
-include "pr2/fwd.ma".
+include "LambdaDelta-1/pr2/fwd.ma".
 
-include "pr0/dec.ma".
+include "LambdaDelta-1/pr0/dec.ma".
 
-include "C/props.ma".
+include "LambdaDelta-1/C/props.ma".
 
 theorem nf2_dec:
  \forall (c: C).(\forall (t1: T).(or (nf2 c t1) (ex2 T (\lambda (t2: T).((eq 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/defs.ma
index df65c39b9..1983d0926 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr2/defs.ma".
+include "LambdaDelta-1/pr2/defs.ma".
 
 definition nf2:
  C \to (T \to Prop)
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/fwd.ma
index 8f9151fb9..aa5cd7c9e 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/fwd.ma
@@ -14,13 +14,13 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "nf2/defs.ma".
+include "LambdaDelta-1/nf2/defs.ma".
 
-include "pr2/clen.ma".
+include "LambdaDelta-1/pr2/clen.ma".
 
-include "subst0/dec.ma".
+include "LambdaDelta-1/subst0/dec.ma".
 
-include "T/props.ma".
+include "LambdaDelta-1/T/props.ma".
 
 theorem nf2_gen_lref:
  \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/iso.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/iso.ma
index d57d48f52..2b145ce28 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/iso.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/iso.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "nf2/pr3.ma".
+include "LambdaDelta-1/nf2/pr3.ma".
 
-include "pr3/fwd.ma".
+include "LambdaDelta-1/pr3/fwd.ma".
 
-include "iso/props.ma".
+include "LambdaDelta-1/iso/props.ma".
 
 theorem nf2_iso_appls_lref:
  \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/lift1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/lift1.ma
index 48c9ddaa2..5a9920471 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/lift1.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/lift1.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "nf2/props.ma".
+include "LambdaDelta-1/nf2/props.ma".
 
-include "drop1/defs.ma".
+include "LambdaDelta-1/drop1/defs.ma".
 
 theorem nf2_lift1:
  \forall (e: C).(\forall (hds: PList).(\forall (c: C).(\forall (t: T).((drop1 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/pr3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/pr3.ma
index 9f52c0d19..b3eaae1e4 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/pr3.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/pr3.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "nf2/defs.ma".
+include "LambdaDelta-1/nf2/defs.ma".
 
-include "pr3/pr3.ma".
+include "LambdaDelta-1/pr3/pr3.ma".
 
 theorem nf2_pr3_unfold:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to ((nf2 c 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/props.ma
index 05f2b77f1..0bafedff2 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/props.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "nf2/defs.ma".
+include "LambdaDelta-1/nf2/defs.ma".
 
-include "pr2/fwd.ma".
+include "LambdaDelta-1/pr2/fwd.ma".
 
 theorem nf2_sort:
  \forall (c: C).(\forall (n: nat).(nf2 c (TSort n)))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc1/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc1/defs.ma
index efe04085a..b91dd4a75 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc1/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc1/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr1/defs.ma".
+include "LambdaDelta-1/pr1/defs.ma".
 
 definition pc1:
  T \to (T \to Prop)
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc1/props.ma
index caa38f0fc..a156a65b4 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc1/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc1/props.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pc1/defs.ma".
+include "LambdaDelta-1/pc1/defs.ma".
 
-include "pr1/pr1.ma".
+include "LambdaDelta-1/pr1/pr1.ma".
 
 theorem pc1_pr0_r:
  \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pc1 t1 t2)))
@@ -32,6 +32,12 @@ theorem pc1_pr0_x:
 (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t1 (pr1_refl t1) 
 (pr1_pr0 t2 t1 H)))).
 
+theorem pc1_refl:
+ \forall (t: T).(pc1 t t)
+\def
+ \lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr1 t t0)) (\lambda (t0: 
+T).(pr1 t t0)) t (pr1_refl t) (pr1_refl t)).
+
 theorem pc1_pr0_u:
  \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: T).((pc1 t2 
 t3) \to (pc1 t1 t3)))))
@@ -43,12 +49,6 @@ T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(ex_intro2 T (\lambda
 (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x (pr1_sing t2 t1 H x H2) 
 H3)))) H1)))))).
 
-theorem pc1_refl:
- \forall (t: T).(pc1 t t)
-\def
- \lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr1 t t0)) (\lambda (t0: 
-T).(pr1 t t0)) t (pr1_refl t) (pr1_refl t)).
-
 theorem pc1_s:
  \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (pc1 t2 t1)))
 \def
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/dec.ma
index 70a73062f..35ebcfd34 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/dec.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/dec.ma
@@ -14,11 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "ty3/arity_props.ma".
+include "LambdaDelta-1/ty3/arity_props.ma".
 
-include "ty3/pr3.ma".
-
-include "nf2/fwd.ma".
+include "LambdaDelta-1/nf2/fwd.ma".
 
 theorem pc3_dec:
  \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/defs.ma
index 0a2712197..32f545555 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr3/defs.ma".
+include "LambdaDelta-1/pr3/defs.ma".
 
 definition pc3:
  C \to (T \to (T \to Prop))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/fsubst0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/fsubst0.ma
index 610fa416e..6a2cda2f6 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/fsubst0.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/fsubst0.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pc3/left.ma".
+include "LambdaDelta-1/pc3/left.ma".
 
-include "fsubst0/defs.ma".
+include "LambdaDelta-1/fsubst0/defs.ma".
 
-include "csubst0/getl.ma".
+include "LambdaDelta-1/csubst0/getl.ma".
 
 theorem pc3_pr2_fsubst0:
  \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pr2 c1 t1 t) \to (\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/fwd.ma
index cf4fad72f..83db5a284 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/fwd.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pc3/props.ma".
+include "LambdaDelta-1/pc3/props.ma".
 
-include "pr3/fwd.ma".
+include "LambdaDelta-1/pr3/fwd.ma".
 
 theorem pc3_gen_sort:
  \forall (c: C).(\forall (m: nat).(\forall (n: nat).((pc3 c (TSort m) (TSort 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/left.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/left.ma
index 08d060d4d..00e38a85a 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/left.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/left.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pc3/props.ma".
+include "LambdaDelta-1/pc3/props.ma".
 
 theorem pc3_ind_left__pc3_left_pr3:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/nf2.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/nf2.ma
index 73ec672fb..22b552be5 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/nf2.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/nf2.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pc3/defs.ma".
+include "LambdaDelta-1/pc3/defs.ma".
 
-include "nf2/pr3.ma".
+include "LambdaDelta-1/nf2/pr3.ma".
 
 theorem pc3_nf2:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/pc1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/pc1.ma
index a85f348cb..1e2e02799 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/pc1.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/pc1.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pc3/defs.ma".
+include "LambdaDelta-1/pc3/defs.ma".
 
-include "pc1/defs.ma".
+include "LambdaDelta-1/pc1/defs.ma".
 
-include "pr3/pr1.ma".
+include "LambdaDelta-1/pr3/pr1.ma".
 
 theorem pc3_pc1:
  \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (c: C).(pc3 c t1 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/props.ma
index 2685fe355..867542f5a 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/props.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pc3/defs.ma".
+include "LambdaDelta-1/pc3/defs.ma".
 
-include "pr3/pr3.ma".
+include "LambdaDelta-1/pr3/pr3.ma".
 
 theorem clear_pc3_trans:
  \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pc3 c2 t1 t2) \to 
@@ -70,17 +70,6 @@ theorem pc3_pr3_t:
 t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t2 t0)).(ex_intro2 T (\lambda (t: 
 T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) t0 H H0)))))).
 
-theorem pc3_pr2_u:
- \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall 
-(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
-\def
- \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr2 c t1 
-t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in 
-(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c 
-t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3 
-x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t3 t)) 
-x (pr3_sing c t2 t1 H x H2) H3)))) H1))))))).
-
 theorem pc3_refl:
  \forall (c: C).(\forall (t: T).(pc3 c t t))
 \def
@@ -138,6 +127,17 @@ T (\lambda (t: T).(pr3 c (THead k u t1) t)) (\lambda (t: T).(pr3 c (THead k u
 t2) t)) (THead k u x) (pr3_head_12 c u u (pr3_refl c u) k t1 x H1) 
 (pr3_head_12 c u u (pr3_refl c u) k t2 x H2))))) H0))))))).
 
+theorem pc3_pr2_u:
+ \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall 
+(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
+\def
+ \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr2 c t1 
+t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in 
+(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c 
+t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3 
+x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t3 t)) 
+x (pr3_sing c t2 t1 H x H2) H3)))) H1))))))).
+
 theorem pc3_t:
  \forall (t2: T).(\forall (c: C).(\forall (t1: T).((pc3 c t1 t2) \to (\forall 
 (t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
@@ -189,82 +189,69 @@ theorem pc3_pr0_pr2_t:
 \def
  \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u2 u1)).(\lambda (c: 
 C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2 
-(CHead c k u2) t1 t2)).(let H1 \def (match H0 in pr2 return (\lambda (c0: 
-C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 
-(CHead c k u2)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pc3 (CHead c k u1) t1 
-t2)))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0 
-(CHead c k u2))).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 
-t2)).(eq_ind C (CHead c k u2) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) 
-\to ((pr0 t0 t3) \to (pc3 (CHead c k u1) t1 t2))))) (\lambda (H5: (eq T t0 
-t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pc3 
-(CHead c k u1) t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda 
-(t: T).((pr0 t1 t) \to (pc3 (CHead c k u1) t1 t2))) (\lambda (H7: (pr0 t1 
-t2)).(pc3_pr2_r (CHead c k u1) t1 t2 (pr2_free (CHead c k u1) t1 t2 H7))) t3 
-(sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 (CHead c k u2) 
-H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t0 t3 H2 t H3) \Rightarrow (\lambda 
-(H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq 
-T t t2)).(eq_ind C (CHead c k u2) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t 
-t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i 
-u t3 t) \to (pc3 (CHead c k u1) t1 t2))))))) (\lambda (H7: (eq T t0 
-t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c k u2) 
-(CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pc3 
-(CHead c k u1) t1 t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda 
-(t4: T).((getl i (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to 
-((subst0 i u t3 t4) \to (pc3 (CHead c k u1) t1 t2))))) (\lambda (H9: (getl i 
-(CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda 
-(H11: (subst0 i u t3 t2)).(nat_ind (\lambda (n: nat).((getl n (CHead c k u2) 
-(CHead d (Bind Abbr) u)) \to ((subst0 n u t3 t2) \to (pc3 (CHead c k u1) t1 
-t2)))) (\lambda (H12: (getl O (CHead c k u2) (CHead d (Bind Abbr) 
-u))).(\lambda (H13: (subst0 O u t3 t2)).(K_ind (\lambda (k0: K).((clear 
-(CHead c k0 u2) (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t1 t2))) 
-(\lambda (b: B).(\lambda (H14: (clear (CHead c (Bind b) u2) (CHead d (Bind 
-Abbr) u))).(let H15 \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) u2) (clear_gen_bind b c 
-(CHead d (Bind Abbr) u) u2 H14)) in ((let H16 \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 d (Bind 
-Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 
-H14)) in ((let H17 \def (f_equal C T (\lambda (e: C).(match e in C return 
-(\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow 
-t4])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c 
-(CHead d (Bind Abbr) u) u2 H14)) in (\lambda (H18: (eq B Abbr b)).(\lambda 
-(_: (eq C d c)).(let H20 \def (eq_ind T u (\lambda (t4: T).(subst0 O t4 t3 
-t2)) H13 u2 H17) in (eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c (Bind b0) 
-u1) t1 t2)) (ex2_ind T (\lambda (t4: T).(subst0 O u1 t3 t4)) (\lambda (t4: 
-T).(pr0 t2 t4)) (pc3 (CHead c (Bind Abbr) u1) t1 t2) (\lambda (x: T).(\lambda 
-(H21: (subst0 O u1 t3 x)).(\lambda (H22: (pr0 t2 x)).(pc3_pr3_t (CHead c 
-(Bind Abbr) u1) t1 x (pr3_pr2 (CHead c (Bind Abbr) u1) t1 x (pr2_delta (CHead 
-c (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 t3 H10 x H21)) t2 (pr3_pr2 
-(CHead c (Bind Abbr) u1) t2 x (pr2_free (CHead c (Bind Abbr) u1) t2 x 
-H22)))))) (pr0_subst0_fwd u2 t3 t2 O H20 u1 H)) b H18))))) H16)) H15)))) 
-(\lambda (f: F).(\lambda (H14: (clear (CHead c (Flat f) u2) (CHead d (Bind 
-Abbr) u))).(clear_pc3_trans (CHead d (Bind Abbr) u) t1 t2 (pc3_pr2_r (CHead d 
-(Bind Abbr) u) t1 t2 (pr2_delta (CHead d (Bind Abbr) u) d u O (getl_refl Abbr 
-d u) t1 t3 H10 t2 H13)) (CHead c (Flat f) u1) (clear_flat c (CHead d (Bind 
-Abbr) u) (clear_gen_flat f c (CHead d (Bind Abbr) u) u2 H14) f u1)))) k 
-(getl_gen_O (CHead c k u2) (CHead d (Bind Abbr) u) H12)))) (\lambda (i0: 
-nat).(\lambda (IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to 
-((subst0 i0 u t3 t2) \to (pc3 (CHead c k u1) t1 t2))))).(\lambda (H12: (getl 
-(S i0) (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H13: (subst0 (S i0) 
-u t3 t2)).(K_ind (\lambda (k0: K).((((getl i0 (CHead c k0 u2) (CHead d (Bind 
-Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pc3 (CHead c k0 u1) t1 t2)))) \to 
-((getl (r k0 i0) c (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t1 
-t2)))) (\lambda (b: B).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d 
-(Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pc3 (CHead c (Bind b) u1) t1 
-t2))))).(\lambda (H14: (getl (r (Bind b) i0) c (CHead d (Bind Abbr) 
-u))).(pc3_pr2_r (CHead c (Bind b) u1) t1 t2 (pr2_delta (CHead c (Bind b) u1) 
-d u (S i0) (getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) H14 u1) t1 t3 H10 
-t2 H13))))) (\lambda (f: F).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) 
-(CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pc3 (CHead c (Flat f) 
-u1) t1 t2))))).(\lambda (H14: (getl (r (Flat f) i0) c (CHead d (Bind Abbr) 
-u))).(pc3_pr2_r (CHead c (Flat f) u1) t1 t2 (pr2_cflat c t1 t2 (pr2_delta c d 
-u (r (Flat f) i0) H14 t1 t3 H10 t2 H13) f u1))))) k IHi (getl_gen_S k c 
-(CHead d (Bind Abbr) u) u2 i0 H12)))))) i H9 H11)))) t (sym_eq T t t2 H8))) 
-t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0 (CHead c k u2) H4) H5 H6 H1 H2 
-H3))))]) in (H1 (refl_equal C (CHead c k u2)) (refl_equal T t1) (refl_equal T 
-t2)))))))))).
+(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0 
+t1 t2)) (\lambda (_: C).(pc3 (CHead c k u1) 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 k u2)) \to (pc3 (CHead c k u1) t t0))))) (\lambda (c0: 
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3: 
+(eq C c0 (CHead c k u2))).(let H4 \def (f_equal C C (\lambda (e: C).e) c0 
+(CHead c k u2) H3) in (pc3_pr2_r (CHead c k u1) t3 t4 (pr2_free (CHead c k 
+u1) 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 k 
+u2))).(let H6 \def (f_equal C C (\lambda (e: C).e) c0 (CHead c k u2) H5) in 
+(let H7 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) 
+u))) H2 (CHead c k u2) H6) in (nat_ind (\lambda (n: nat).((getl n (CHead c k 
+u2) (CHead d (Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pc3 (CHead c k u1) 
+t3 t)))) (\lambda (H8: (getl O (CHead c k u2) (CHead d (Bind Abbr) 
+u))).(\lambda (H9: (subst0 O u t4 t)).(K_ind (\lambda (k0: K).((clear (CHead 
+c k0 u2) (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t))) (\lambda 
+(b: B).(\lambda (H10: (clear (CHead c (Bind b) u2) (CHead d (Bind Abbr) 
+u))).(let H11 \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) u2) (clear_gen_bind b c (CHead d 
+(Bind Abbr) u) u2 H10)) in ((let H12 \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 d (Bind Abbr) u) 
+(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H10)) in 
+((let H13 \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) u2) (clear_gen_bind b c (CHead d (Bind 
+Abbr) u) u2 H10)) in (\lambda (H14: (eq B Abbr b)).(\lambda (_: (eq C d 
+c)).(let H16 \def (eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H9 u2 H13) 
+in (eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t3 t)) 
+(ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t t0)) 
+(pc3 (CHead c (Bind Abbr) u1) t3 t) (\lambda (x: T).(\lambda (H17: (subst0 O 
+u1 t4 x)).(\lambda (H18: (pr0 t x)).(pc3_pr3_t (CHead c (Bind Abbr) u1) t3 x 
+(pr3_pr2 (CHead c (Bind Abbr) u1) t3 x (pr2_delta (CHead c (Bind Abbr) u1) c 
+u1 O (getl_refl Abbr c u1) t3 t4 H3 x H17)) t (pr3_pr2 (CHead c (Bind Abbr) 
+u1) t x (pr2_free (CHead c (Bind Abbr) u1) t x H18)))))) (pr0_subst0_fwd u2 
+t4 t O H16 u1 H)) b H14))))) H12)) H11)))) (\lambda (f: F).(\lambda (H10: 
+(clear (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(clear_pc3_trans 
+(CHead d (Bind Abbr) u) t3 t (pc3_pr2_r (CHead d (Bind Abbr) u) t3 t 
+(pr2_delta (CHead d (Bind Abbr) u) d u O (getl_refl Abbr d u) t3 t4 H3 t H9)) 
+(CHead c (Flat f) u1) (clear_flat c (CHead d (Bind Abbr) u) (clear_gen_flat f 
+c (CHead d (Bind Abbr) u) u2 H10) f u1)))) k (getl_gen_O (CHead c k u2) 
+(CHead d (Bind Abbr) u) H8)))) (\lambda (i0: nat).(\lambda (IHi: (((getl i0 
+(CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 t) \to (pc3 
+(CHead c k u1) t3 t))))).(\lambda (H8: (getl (S i0) (CHead c k u2) (CHead d 
+(Bind Abbr) u))).(\lambda (H9: (subst0 (S i0) u t4 t)).(K_ind (\lambda (k0: 
+K).((((getl i0 (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 
+t) \to (pc3 (CHead c k0 u1) t3 t)))) \to ((getl (r k0 i0) c (CHead d (Bind 
+Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t)))) (\lambda (b: B).(\lambda (_: 
+(((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u 
+t4 t) \to (pc3 (CHead c (Bind b) u1) t3 t))))).(\lambda (H10: (getl (r (Bind 
+b) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Bind b) u1) t3 t 
+(pr2_delta (CHead c (Bind b) u1) d u (S i0) (getl_head (Bind b) i0 c (CHead d 
+(Bind Abbr) u) H10 u1) t3 t4 H3 t H9))))) (\lambda (f: F).(\lambda (_: 
+(((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u 
+t4 t) \to (pc3 (CHead c (Flat f) u1) t3 t))))).(\lambda (H10: (getl (r (Flat 
+f) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Flat f) u1) t3 t 
+(pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0) H10 t3 t4 H3 t H9) f 
+u1))))) k IHi (getl_gen_S k c (CHead d (Bind Abbr) u) u2 i0 H8)))))) i H7 
+H4)))))))))))))) y t1 t2 H1))) H0)))))))).
 
 theorem pc3_pr2_pr2_t:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u2 u1) \to (\forall 
@@ -272,119 +259,80 @@ theorem pc3_pr2_pr2_t:
 (CHead c k u1) t1 t2))))))))
 \def
  \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u2 
-u1)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t u2) 
-\to ((eq T t0 u1) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 
-(CHead c k u2) t1 t2) \to (pc3 (CHead c k u1) t1 t2)))))))))))) with 
-[(pr2_free c0 t1 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: 
-(eq T t1 u2)).(\lambda (H3: (eq T t2 u1)).(eq_ind C c (\lambda (_: C).((eq T 
-t1 u2) \to ((eq T t2 u1) \to ((pr0 t1 t2) \to (\forall (t3: T).(\forall (t4: 
-T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 
-t4))))))))) (\lambda (H4: (eq T t1 u2)).(eq_ind T u2 (\lambda (t: T).((eq T 
-t2 u1) \to ((pr0 t t2) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: 
-K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4)))))))) (\lambda 
-(H5: (eq T t2 u1)).(eq_ind T u1 (\lambda (t: T).((pr0 u2 t) \to (\forall (t3: 
-T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 
-(CHead c k u1) t3 t4))))))) (\lambda (H6: (pr0 u2 u1)).(\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H7: (pr2 (CHead c k u2) t0 
-t3)).(pc3_pr0_pr2_t u1 u2 H6 c t0 t3 k H7)))))) t2 (sym_eq T t2 u1 H5))) t1 
-(sym_eq T t1 u2 H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u 
-i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq 
-T t1 u2)).(\lambda (H5: (eq T t u1)).(eq_ind C c (\lambda (c1: C).((eq T t1 
-u2) \to ((eq T t u1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1 
-t2) \to ((subst0 i u t2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: 
-K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4))))))))))) 
-(\lambda (H6: (eq T t1 u2)).(eq_ind T u2 (\lambda (t0: T).((eq T t u1) \to 
-((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) 
-\to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 
-t4) \to (pc3 (CHead c k u1) t3 t4)))))))))) (\lambda (H7: (eq T t 
-u1)).(eq_ind T u1 (\lambda (t0: T).((getl i c (CHead d (Bind Abbr) u)) \to 
-((pr0 u2 t2) \to ((subst0 i u t2 t0) \to (\forall (t3: T).(\forall (t4: 
-T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 
-t4))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: 
-(pr0 u2 t2)).(\lambda (H10: (subst0 i u t2 u1)).(\lambda (t0: T).(\lambda 
-(t3: T).(\lambda (k: K).(\lambda (H11: (pr2 (CHead c k u2) t0 t3)).(let H12 
-\def (match H11 in pr2 return (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: 
-T).(\lambda (_: (pr2 c1 t4 t5)).((eq C c1 (CHead c k u2)) \to ((eq T t4 t0) 
-\to ((eq T t5 t3) \to (pc3 (CHead c k u1) t0 t3)))))))) with [(pr2_free c1 t4 
-t5 H12) \Rightarrow (\lambda (H13: (eq C c1 (CHead c k u2))).(\lambda (H14: 
-(eq T t4 t0)).(\lambda (H15: (eq T t5 t3)).(eq_ind C (CHead c k u2) (\lambda 
-(_: C).((eq T t4 t0) \to ((eq T t5 t3) \to ((pr0 t4 t5) \to (pc3 (CHead c k 
-u1) t0 t3))))) (\lambda (H16: (eq T t4 t0)).(eq_ind T t0 (\lambda (t6: 
-T).((eq T t5 t3) \to ((pr0 t6 t5) \to (pc3 (CHead c k u1) t0 t3)))) (\lambda 
-(H17: (eq T t5 t3)).(eq_ind T t3 (\lambda (t6: T).((pr0 t0 t6) \to (pc3 
-(CHead c k u1) t0 t3))) (\lambda (H18: (pr0 t0 t3)).(pc3_pr2_r (CHead c k u1) 
-t0 t3 (pr2_free (CHead c k u1) t0 t3 H18))) t5 (sym_eq T t5 t3 H17))) t4 
-(sym_eq T t4 t0 H16))) c1 (sym_eq C c1 (CHead c k u2) H13) H14 H15 H12)))) | 
-(pr2_delta c1 d0 u0 i0 H12 t4 t5 H13 t6 H14) \Rightarrow (\lambda (H15: (eq C 
-c1 (CHead c k u2))).(\lambda (H16: (eq T t4 t0)).(\lambda (H17: (eq T t6 
-t3)).(eq_ind C (CHead c k u2) (\lambda (c2: C).((eq T t4 t0) \to ((eq T t6 
-t3) \to ((getl i0 c2 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t4 t5) \to ((subst0 
-i0 u0 t5 t6) \to (pc3 (CHead c k u1) t0 t3))))))) (\lambda (H18: (eq T t4 
-t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t6 t3) \to ((getl i0 (CHead c k u2) 
-(CHead d0 (Bind Abbr) u0)) \to ((pr0 t7 t5) \to ((subst0 i0 u0 t5 t6) \to 
-(pc3 (CHead c k u1) t0 t3)))))) (\lambda (H19: (eq T t6 t3)).(eq_ind T t3 
-(\lambda (t7: T).((getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to 
-((pr0 t0 t5) \to ((subst0 i0 u0 t5 t7) \to (pc3 (CHead c k u1) t0 t3))))) 
-(\lambda (H20: (getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda 
-(H21: (pr0 t0 t5)).(\lambda (H22: (subst0 i0 u0 t5 t3)).(nat_ind (\lambda (n: 
-nat).((getl n (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5 
-t3) \to (pc3 (CHead c k u1) t0 t3)))) (\lambda (H23: (getl O (CHead c k u2) 
-(CHead d0 (Bind Abbr) u0))).(\lambda (H24: (subst0 O u0 t5 t3)).(K_ind 
-(\lambda (k0: K).((clear (CHead c k0 u2) (CHead d0 (Bind Abbr) u0)) \to (pc3 
-(CHead c k0 u1) t0 t3))) (\lambda (b: B).(\lambda (H25: (clear (CHead c (Bind 
-b) u2) (CHead d0 (Bind Abbr) u0))).(let H26 \def (f_equal C C (\lambda (e: 
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 | 
-(CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) 
-u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H25)) in ((let H27 \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 
+u1)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1: 
+T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t) t1 t2) \to (pc3 
+(CHead c0 k t0) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: 
+K).(\lambda (H1: (pr2 (CHead c0 k t1) t0 t3)).(pc3_pr0_pr2_t t2 t1 H0 c0 t0 
+t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
+(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2: 
+(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda 
+(H3: (pr2 (CHead c0 k t1) t0 t3)).(insert_eq C (CHead c0 k t1) (\lambda (c1: 
+C).(pr2 c1 t0 t3)) (\lambda (_: C).(pc3 (CHead c0 k t) t0 t3)) (\lambda (y: 
+C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4: 
+T).(\lambda (t5: T).((eq C c1 (CHead c0 k t1)) \to (pc3 (CHead c0 k t) t4 
+t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0 
+t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t1))).(pc3_pr2_r (CHead c0 k t) t4 
+t5 (pr2_free (CHead c0 k t) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0: 
+C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0 
+(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4 
+t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C 
+c1 (CHead c0 k t1))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2 
+(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t1) H8) in (nat_ind (\lambda (n: 
+nat).((getl n (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5 
+t6) \to (pc3 (CHead c0 k t) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t1) 
+(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind 
+(\lambda (k0: K).((clear (CHead c0 k0 t1) (CHead d0 (Bind Abbr) u0)) \to (pc3 
+(CHead c0 k0 t) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0 
+(Bind b) t1) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda 
+(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 
+| (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind 
+b) t1) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t1 H12)) in ((let H14 
+\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 d0 (Bind Abbr) u0) (CHead c (Bind b) u2) 
-(clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H25)) in ((let H28 \def 
+\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t1) 
+(clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t1 H12)) in ((let H15 \def 
 (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
 [(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind 
-Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) 
-u2 H25)) in (\lambda (H29: (eq B Abbr b)).(\lambda (_: (eq C d0 c)).(let H31 
-\def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t3)) H24 u2 H28) in 
-(eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t0 t3)) (ex2_ind 
-T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(pr0 t3 t7)) (pc3 
-(CHead c (Bind Abbr) u1) t0 t3) (\lambda (x: T).(\lambda (H32: (subst0 O t2 
-t5 x)).(\lambda (H33: (pr0 t3 x)).(ex2_ind T (\lambda (t7: T).(subst0 O u1 t5 
-t7)) (\lambda (t7: T).(subst0 (S (plus i O)) u x t7)) (pc3 (CHead c (Bind 
-Abbr) u1) t0 t3) (\lambda (x0: T).(\lambda (H34: (subst0 O u1 t5 
-x0)).(\lambda (H35: (subst0 (S (plus i O)) u x x0)).(let H36 \def (f_equal 
-nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H37 
-\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n u x x0)) H35 (S 
-i) H36) in (pc3_pr2_u (CHead c (Bind Abbr) u1) x0 t0 (pr2_delta (CHead c 
-(Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t0 t5 H21 x0 H34) t3 (pc3_pr2_x 
-(CHead c (Bind Abbr) u1) x0 t3 (pr2_delta (CHead c (Bind Abbr) u1) d u (S i) 
-(getl_head (Bind Abbr) i c (CHead d (Bind Abbr) u) H8 u1) t3 x H33 x0 
-H37)))))))) (subst0_subst0_back t5 x t2 O H32 u1 u i H10))))) (pr0_subst0_fwd 
-u2 t5 t3 O H31 t2 H9)) b H29))))) H27)) H26)))) (\lambda (f: F).(\lambda 
-(H25: (clear (CHead c (Flat f) u2) (CHead d0 (Bind Abbr) 
-u0))).(clear_pc3_trans (CHead d0 (Bind Abbr) u0) t0 t3 (pc3_pr2_r (CHead d0 
-(Bind Abbr) u0) t0 t3 (pr2_delta (CHead d0 (Bind Abbr) u0) d0 u0 O (getl_refl 
-Abbr d0 u0) t0 t5 H21 t3 H24)) (CHead c (Flat f) u1) (clear_flat c (CHead d0 
-(Bind Abbr) u0) (clear_gen_flat f c (CHead d0 (Bind Abbr) u0) u2 H25) f 
-u1)))) k (getl_gen_O (CHead c k u2) (CHead d0 (Bind Abbr) u0) H23)))) 
-(\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c k u2) (CHead d0 (Bind 
-Abbr) u0)) \to ((subst0 i1 u0 t5 t3) \to (pc3 (CHead c k u1) t0 
-t3))))).(\lambda (H23: (getl (S i1) (CHead c k u2) (CHead d0 (Bind Abbr) 
-u0))).(\lambda (H24: (subst0 (S i1) u0 t5 t3)).(K_ind (\lambda (k0: K).((getl 
-(r k0 i1) c (CHead d0 (Bind Abbr) u0)) \to (pc3 (CHead c k0 u1) t0 t3))) 
-(\lambda (b: B).(\lambda (H25: (getl (r (Bind b) i1) c (CHead d0 (Bind Abbr) 
-u0))).(pc3_pr2_r (CHead c (Bind b) u1) t0 t3 (pr2_delta (CHead c (Bind b) u1) 
-d0 u0 (S i1) (getl_head (Bind b) i1 c (CHead d0 (Bind Abbr) u0) H25 u1) t0 t5 
-H21 t3 H24)))) (\lambda (f: F).(\lambda (H25: (getl (r (Flat f) i1) c (CHead 
-d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead c (Flat f) u1) t0 t3 (pr2_cflat c t0 
-t3 (pr2_delta c d0 u0 (r (Flat f) i1) H25 t0 t5 H21 t3 H24) f u1)))) k 
-(getl_gen_S k c (CHead d0 (Bind Abbr) u0) u2 i1 H23)))))) i0 H20 H22)))) t6 
-(sym_eq T t6 t3 H19))) t4 (sym_eq T t4 t0 H18))) c1 (sym_eq C c1 (CHead c k 
-u2) H15) H16 H17 H12 H13 H14))))]) in (H12 (refl_equal C (CHead c k u2)) 
-(refl_equal T t0) (refl_equal T t3)))))))))) t (sym_eq T t u1 H7))) t1 
-(sym_eq T t1 u2 H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 
-(refl_equal C c) (refl_equal T u2) (refl_equal T u1)))))).
+Abbr) u0) (CHead c0 (Bind b) t1) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) 
+u0) t1 H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let 
+H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) H11 t1 H15) in 
+(eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c0 (Bind b0) t) t4 t6)) (ex2_ind 
+T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(pr0 t6 t7)) (pc3 
+(CHead c0 (Bind Abbr) t) t4 t6) (\lambda (x: T).(\lambda (H19: (subst0 O t2 
+t5 x)).(\lambda (H20: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(subst0 O t t5 
+t7)) (\lambda (t7: T).(subst0 (S (plus i O)) u x t7)) (pc3 (CHead c0 (Bind 
+Abbr) t) t4 t6) (\lambda (x0: T).(\lambda (H21: (subst0 O t t5 x0)).(\lambda 
+(H22: (subst0 (S (plus i O)) u x x0)).(let H23 \def (f_equal nat nat S (plus 
+i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H24 \def (eq_ind nat 
+(S (plus i O)) (\lambda (n: nat).(subst0 n u x x0)) H22 (S i) H23) in 
+(pc3_pr2_u (CHead c0 (Bind Abbr) t) x0 t4 (pr2_delta (CHead c0 (Bind Abbr) t) 
+c0 t O (getl_refl Abbr c0 t) t4 t5 H6 x0 H21) t6 (pc3_pr2_x (CHead c0 (Bind 
+Abbr) t) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t) d u (S i) (getl_head (Bind 
+Abbr) i c0 (CHead d (Bind Abbr) u) H0 t) t6 x H20 x0 H24)))))))) 
+(subst0_subst0_back t5 x t2 O H19 t u i H2))))) (pr0_subst0_fwd t1 t5 t6 O 
+H18 t2 H1)) b H16))))) H14)) H13)))) (\lambda (f: F).(\lambda (H12: (clear 
+(CHead c0 (Flat f) t1) (CHead d0 (Bind Abbr) u0))).(clear_pc3_trans (CHead d0 
+(Bind Abbr) u0) t4 t6 (pc3_pr2_r (CHead d0 (Bind Abbr) u0) t4 t6 (pr2_delta 
+(CHead d0 (Bind Abbr) u0) d0 u0 O (getl_refl Abbr d0 u0) t4 t5 H6 t6 H11)) 
+(CHead c0 (Flat f) t) (clear_flat c0 (CHead d0 (Bind Abbr) u0) 
+(clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t1 H12) f t)))) k (getl_gen_O 
+(CHead c0 k t1) (CHead d0 (Bind Abbr) u0) H10)))) (\lambda (i1: nat).(\lambda 
+(_: (((getl i1 (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to ((subst0 i1 u0 
+t5 t6) \to (pc3 (CHead c0 k t) t4 t6))))).(\lambda (H10: (getl (S i1) (CHead 
+c0 k t1) (CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 (S i1) u0 t5 
+t6)).(K_ind (\lambda (k0: K).((getl (r k0 i1) c0 (CHead d0 (Bind Abbr) u0)) 
+\to (pc3 (CHead c0 k0 t) t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (r 
+(Bind b) i1) c0 (CHead d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead c0 (Bind b) t) 
+t4 t6 (pr2_delta (CHead c0 (Bind b) t) d0 u0 (S i1) (getl_head (Bind b) i1 c0 
+(CHead d0 (Bind Abbr) u0) H12 t) t4 t5 H6 t6 H11)))) (\lambda (f: F).(\lambda 
+(H12: (getl (r (Flat f) i1) c0 (CHead d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead 
+c0 (Flat f) t) t4 t6 (pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) 
+H12 t4 t5 H6 t6 H11) f t)))) k (getl_gen_S k c0 (CHead d0 (Bind Abbr) u0) t1 
+i1 H10)))))) i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u2 u1 
+H)))).
 
 theorem pc3_pr2_pr3_t:
  \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/subst1.ma
index 404f98150..bdf5cfbf5 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/subst1.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/subst1.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pc3/props.ma".
+include "LambdaDelta-1/pc3/props.ma".
 
-include "pr3/subst1.ma".
+include "LambdaDelta-1/pr3/subst1.ma".
 
 theorem pc3_gen_cabbr:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/wcpr0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/wcpr0.ma
index 9d8e8dc11..18e40a197 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/wcpr0.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/wcpr0.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pc3/props.ma".
+include "LambdaDelta-1/pc3/props.ma".
 
-include "wcpr0/getl.ma".
+include "LambdaDelta-1/wcpr0/getl.ma".
 
 theorem pc3_wcpr0__pc3_wcpr0_t_aux:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (k: K).(\forall 
@@ -27,46 +27,32 @@ theorem pc3_wcpr0__pc3_wcpr0_t_aux:
 K).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 
 (CHead c1 k u) t1 t2)).(pr3_ind (CHead c1 k u) (\lambda (t: T).(\lambda (t0: 
 T).(pc3 (CHead c2 k u) t t0))) (\lambda (t: T).(pc3_refl (CHead c2 k u) t)) 
-(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (pr2 (CHead c1 k u) t3 
-t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead c1 k u) t0 t4)).(\lambda (H3: 
-(pc3 (CHead c2 k u) t0 t4)).(pc3_t t0 (CHead c2 k u) t3 (let H4 \def (match 
-H1 in pr2 return (\lambda (c: C).(\lambda (t: T).(\lambda (t5: T).(\lambda 
-(_: (pr2 c t t5)).((eq C c (CHead c1 k u)) \to ((eq T t t3) \to ((eq T t5 t0) 
-\to (pc3 (CHead c2 k u) t3 t0)))))))) with [(pr2_free c t5 t6 H4) \Rightarrow 
-(\lambda (H5: (eq C c (CHead c1 k u))).(\lambda (H6: (eq T t5 t3)).(\lambda 
-(H7: (eq T t6 t0)).(eq_ind C (CHead c1 k u) (\lambda (_: C).((eq T t5 t3) \to 
-((eq T t6 t0) \to ((pr0 t5 t6) \to (pc3 (CHead c2 k u) t3 t0))))) (\lambda 
-(H8: (eq T t5 t3)).(eq_ind T t3 (\lambda (t: T).((eq T t6 t0) \to ((pr0 t t6) 
-\to (pc3 (CHead c2 k u) t3 t0)))) (\lambda (H9: (eq T t6 t0)).(eq_ind T t0 
-(\lambda (t: T).((pr0 t3 t) \to (pc3 (CHead c2 k u) t3 t0))) (\lambda (H10: 
-(pr0 t3 t0)).(pc3_pr2_r (CHead c2 k u) t3 t0 (pr2_free (CHead c2 k u) t3 t0 
-H10))) t6 (sym_eq T t6 t0 H9))) t5 (sym_eq T t5 t3 H8))) c (sym_eq C c (CHead 
-c1 k u) H5) H6 H7 H4)))) | (pr2_delta c d u0 i H4 t5 t6 H5 t H6) \Rightarrow 
-(\lambda (H7: (eq C c (CHead c1 k u))).(\lambda (H8: (eq T t5 t3)).(\lambda 
-(H9: (eq T t t0)).(eq_ind C (CHead c1 k u) (\lambda (c0: C).((eq T t5 t3) \to 
-((eq T t t0) \to ((getl i c0 (CHead d (Bind Abbr) u0)) \to ((pr0 t5 t6) \to 
-((subst0 i u0 t6 t) \to (pc3 (CHead c2 k u) t3 t0))))))) (\lambda (H10: (eq T 
-t5 t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t t0) \to ((getl i (CHead c1 k 
-u) (CHead d (Bind Abbr) u0)) \to ((pr0 t7 t6) \to ((subst0 i u0 t6 t) \to 
-(pc3 (CHead c2 k u) t3 t0)))))) (\lambda (H11: (eq T t t0)).(eq_ind T t0 
-(\lambda (t7: T).((getl i (CHead c1 k u) (CHead d (Bind Abbr) u0)) \to ((pr0 
-t3 t6) \to ((subst0 i u0 t6 t7) \to (pc3 (CHead c2 k u) t3 t0))))) (\lambda 
-(H12: (getl i (CHead c1 k u) (CHead d (Bind Abbr) u0))).(\lambda (H13: (pr0 
-t3 t6)).(\lambda (H14: (subst0 i u0 t6 t0)).(ex3_2_ind C T (\lambda (e2: 
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 (CHead c1 k u) t4 
+t3)).(\lambda (t5: T).(\lambda (_: (pr3 (CHead c1 k u) t3 t5)).(\lambda (H3: 
+(pc3 (CHead c2 k u) t3 t5)).(pc3_t t3 (CHead c2 k u) t4 (insert_eq C (CHead 
+c1 k u) (\lambda (c: C).(pr2 c t4 t3)) (\lambda (_: C).(pc3 (CHead c2 k u) t4 
+t3)) (\lambda (y: C).(\lambda (H4: (pr2 y t4 t3)).(pr2_ind (\lambda (c: 
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CHead c1 k u)) \to (pc3 (CHead 
+c2 k u) t t0))))) (\lambda (c: C).(\lambda (t6: T).(\lambda (t0: T).(\lambda 
+(H5: (pr0 t6 t0)).(\lambda (_: (eq C c (CHead c1 k u))).(pc3_pr2_r (CHead c2 
+k u) t6 t0 (pr2_free (CHead c2 k u) t6 t0 H5))))))) (\lambda (c: C).(\lambda 
+(d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d 
+(Bind Abbr) u0))).(\lambda (t6: T).(\lambda (t0: T).(\lambda (H6: (pr0 t6 
+t0)).(\lambda (t: T).(\lambda (H7: (subst0 i u0 t0 t)).(\lambda (H8: (eq C c 
+(CHead c1 k u))).(let H9 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead 
+d (Bind Abbr) u0))) H5 (CHead c1 k u) H8) in (ex3_2_ind C T (\lambda (e2: 
 C).(\lambda (u2: T).(getl i (CHead c2 k u) (CHead e2 (Bind Abbr) u2)))) 
 (\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: 
-T).(pr0 u0 u2))) (pc3 (CHead c2 k u) t3 t0) (\lambda (x0: C).(\lambda (x1: 
-T).(\lambda (H15: (getl i (CHead c2 k u) (CHead x0 (Bind Abbr) x1))).(\lambda 
-(_: (wcpr0 d x0)).(\lambda (H17: (pr0 u0 x1)).(ex2_ind T (\lambda (t7: 
-T).(subst0 i x1 t6 t7)) (\lambda (t7: T).(pr0 t0 t7)) (pc3 (CHead c2 k u) t3 
-t0) (\lambda (x: T).(\lambda (H18: (subst0 i x1 t6 x)).(\lambda (H19: (pr0 t0 
-x)).(pc3_pr2_u (CHead c2 k u) x t3 (pr2_delta (CHead c2 k u) x0 x1 i H15 t3 
-t6 H13 x H18) t0 (pc3_pr2_x (CHead c2 k u) x t0 (pr2_free (CHead c2 k u) t0 x 
-H19)))))) (pr0_subst0_fwd u0 t6 t0 i H14 x1 H17))))))) (wcpr0_getl (CHead c1 
-k u) (CHead c2 k u) (wcpr0_comp c1 c2 H u u (pr0_refl u) k) i d u0 (Bind 
-Abbr) H12))))) t (sym_eq T t t0 H11))) t5 (sym_eq T t5 t3 H10))) c (sym_eq C 
-c (CHead c1 k u) H7) H8 H9 H4 H5 H6))))]) in (H4 (refl_equal C (CHead c1 k 
-u)) (refl_equal T t3) (refl_equal T t0))) t4 H3))))))) t1 t2 H0)))))))).
+T).(pr0 u0 u2))) (pc3 (CHead c2 k u) t6 t) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (H10: (getl i (CHead c2 k u) (CHead x0 (Bind Abbr) x1))).(\lambda 
+(_: (wcpr0 d x0)).(\lambda (H12: (pr0 u0 x1)).(ex2_ind T (\lambda (t7: 
+T).(subst0 i x1 t0 t7)) (\lambda (t7: T).(pr0 t t7)) (pc3 (CHead c2 k u) t6 
+t) (\lambda (x: T).(\lambda (H13: (subst0 i x1 t0 x)).(\lambda (H14: (pr0 t 
+x)).(pc3_pr2_u (CHead c2 k u) x t6 (pr2_delta (CHead c2 k u) x0 x1 i H10 t6 
+t0 H6 x H13) t (pc3_pr2_x (CHead c2 k u) x t (pr2_free (CHead c2 k u) t x 
+H14)))))) (pr0_subst0_fwd u0 t0 t i H7 x1 H12))))))) (wcpr0_getl (CHead c1 k 
+u) (CHead c2 k u) (wcpr0_comp c1 c2 H u u (pr0_refl u) k) i d u0 (Bind Abbr) 
+H9)))))))))))))) y t4 t3 H4))) H1) t5 H3))))))) t1 t2 H0)))))))).
 
 theorem pc3_wcpr0_t:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1: 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/dec.ma
index 81a24cd79..cf86f1393 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/dec.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/dec.ma
@@ -14,13 +14,13 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr0/fwd.ma".
+include "LambdaDelta-1/pr0/fwd.ma".
 
-include "subst0/dec.ma".
+include "LambdaDelta-1/subst0/dec.ma".
 
-include "T/dec.ma".
+include "LambdaDelta-1/T/dec.ma".
 
-include "T/props.ma".
+include "LambdaDelta-1/T/props.ma".
 
 theorem nf0_dec:
  \forall (t1: T).(or (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/defs.ma
index a0f6bb299..156b6ff85 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "subst0/defs.ma".
+include "LambdaDelta-1/subst0/defs.ma".
 
 inductive pr0: T \to (T \to Prop) \def
 | pr0_refl: \forall (t: T).(pr0 t t)
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/fwd.ma
index fb106f9ce..e6ca5293d 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/fwd.ma
@@ -14,234 +14,131 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr0/props.ma".
-
-theorem pr0_inv_coq:
- \forall (t1: T).(\forall (t2: T).(\forall (P: ((T \to (T \to 
-Prop)))).((((pr0 t1 t2) \to (\forall (t: T).((eq T t t1) \to ((eq T t t2) \to 
-(P t1 t2)))))) \to ((((pr0 t1 t2) \to (\forall (u1: T).(\forall (u2: 
-T).(\forall (t0: T).(\forall (t3: T).(\forall (k: K).((eq T (THead k u1 t0) 
-t1) \to ((eq T (THead k u2 t3) t2) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to (P 
-t1 t2)))))))))))) \to ((((pr0 t1 t2) \to (\forall (u: T).(\forall (v1: 
-T).(\forall (v2: T).(\forall (t0: T).(\forall (t3: T).((eq T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t0)) t1) \to ((eq T (THead (Bind Abbr) v2 t3) 
-t2) \to ((pr0 v1 v2) \to ((pr0 t0 t3) \to (P t1 t2)))))))))))) \to ((((pr0 t1 
-t2) \to (\forall (b: B).(\forall (v1: T).(\forall (v2: T).(\forall (u1: 
-T).(\forall (u2: T).(\forall (t0: T).(\forall (t3: T).((eq T (THead (Flat 
-Appl) v1 (THead (Bind b) u1 t0)) t1) \to ((eq T (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t3)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 
-v2) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to (P t1 t2)))))))))))))))) \to 
-((((pr0 t1 t2) \to (\forall (u1: T).(\forall (u2: T).(\forall (t0: 
-T).(\forall (t3: T).(\forall (w: T).((eq T (THead (Bind Abbr) u1 t0) t1) \to 
-((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to 
-((subst0 O u2 t3 w) \to (P t1 t2))))))))))))) \to ((((pr0 t1 t2) \to (\forall 
-(b: B).(\forall (t0: T).(\forall (t3: T).(\forall (u: T).((eq T (THead (Bind 
-b) u (lift (S O) O t0)) t1) \to ((eq T t3 t2) \to ((not (eq B b Abst)) \to 
-((pr0 t0 t3) \to (P t1 t2))))))))))) \to ((((pr0 t1 t2) \to (\forall (t0: 
-T).(\forall (t3: T).(\forall (u: T).((eq T (THead (Flat Cast) u t0) t1) \to 
-((eq T t3 t2) \to ((pr0 t0 t3) \to (P t1 t2))))))))) \to ((pr0 t1 t2) \to (P 
-t1 t2)))))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (P: ((T \to (T \to 
-Prop)))).(\lambda (H: (((pr0 t1 t2) \to (\forall (t: T).((eq T t t1) \to ((eq 
-T t t2) \to (P t1 t2))))))).(\lambda (H0: (((pr0 t1 t2) \to (\forall (u1: 
-T).(\forall (u2: T).(\forall (t0: T).(\forall (t3: T).(\forall (k: K).((eq T 
-(THead k u1 t0) t1) \to ((eq T (THead k u2 t3) t2) \to ((pr0 u1 u2) \to ((pr0 
-t0 t3) \to (P t1 t2))))))))))))).(\lambda (H1: (((pr0 t1 t2) \to (\forall (u: 
-T).(\forall (v1: T).(\forall (v2: T).(\forall (t0: T).(\forall (t3: T).((eq T 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t1) \to ((eq T (THead (Bind 
-Abbr) v2 t3) t2) \to ((pr0 v1 v2) \to ((pr0 t0 t3) \to (P t1 
-t2))))))))))))).(\lambda (H2: (((pr0 t1 t2) \to (\forall (b: B).(\forall (v1: 
-T).(\forall (v2: T).(\forall (u1: T).(\forall (u2: T).(\forall (t0: 
-T).(\forall (t3: T).((eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t1) 
-\to ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t2) 
-\to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t0 t3) 
-\to (P t1 t2))))))))))))))))).(\lambda (H3: (((pr0 t1 t2) \to (\forall (u1: 
-T).(\forall (u2: T).(\forall (t0: T).(\forall (t3: T).(\forall (w: T).((eq T 
-(THead (Bind Abbr) u1 t0) t1) \to ((eq T (THead (Bind Abbr) u2 w) t2) \to 
-((pr0 u1 u2) \to ((pr0 t0 t3) \to ((subst0 O u2 t3 w) \to (P t1 
-t2)))))))))))))).(\lambda (H4: (((pr0 t1 t2) \to (\forall (b: B).(\forall 
-(t0: T).(\forall (t3: T).(\forall (u: T).((eq T (THead (Bind b) u (lift (S O) 
-O t0)) t1) \to ((eq T t3 t2) \to ((not (eq B b Abst)) \to ((pr0 t0 t3) \to (P 
-t1 t2)))))))))))).(\lambda (H5: (((pr0 t1 t2) \to (\forall (t0: T).(\forall 
-(t3: T).(\forall (u: T).((eq T (THead (Flat Cast) u t0) t1) \to ((eq T t3 t2) 
-\to ((pr0 t0 t3) \to (P t1 t2)))))))))).(\lambda (H6: (pr0 t1 t2)).(let H7 
-\def (match H6 in pr0 return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: 
-(pr0 t t0)).((eq T t t1) \to ((eq T t0 t2) \to (P t1 t2)))))) with [(pr0_refl 
-t) \Rightarrow (\lambda (H7: (eq T t t1)).(\lambda (H8: (eq T t t2)).(H H6 t 
-H7 H8))) | (pr0_comp u1 u2 H7 t0 t3 H8 k) \Rightarrow (\lambda (H9: (eq T 
-(THead k u1 t0) t1)).(\lambda (H10: (eq T (THead k u2 t3) t2)).(H0 H6 u1 u2 
-t0 t3 k H9 H10 H7 H8))) | (pr0_beta u v1 v2 H7 t0 t3 H8) \Rightarrow (\lambda 
-(H9: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t1)).(\lambda 
-(H10: (eq T (THead (Bind Abbr) v2 t3) t2)).(H1 H6 u v1 v2 t0 t3 H9 H10 H7 
-H8))) | (pr0_upsilon b H7 v1 v2 H8 u1 u2 H9 t0 t3 H10) \Rightarrow (\lambda 
-(H11: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t1)).(\lambda (H12: 
-(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t2)).(H2 
-H6 b v1 v2 u1 u2 t0 t3 H11 H12 H7 H8 H9 H10))) | (pr0_delta u1 u2 H7 t0 t3 H8 
-w H9) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr) u1 t0) 
-t1)).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t2)).(H3 H6 u1 u2 t0 t3 w 
-H10 H11 H7 H8 H9))) | (pr0_zeta b H7 t0 t3 H8 u) \Rightarrow (\lambda (H9: 
-(eq T (THead (Bind b) u (lift (S O) O t0)) t1)).(\lambda (H10: (eq T t3 
-t2)).(H4 H6 b t0 t3 u H9 H10 H7 H8))) | (pr0_epsilon t0 t3 H7 u) \Rightarrow 
-(\lambda (H8: (eq T (THead (Flat Cast) u t0) t1)).(\lambda (H9: (eq T t3 
-t2)).(H5 H6 t0 t3 u H8 H9 H7)))]) in (H7 (refl_equal T t1) (refl_equal T 
-t2))))))))))))).
+include "LambdaDelta-1/pr0/props.ma".
 
 theorem 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)).(pr0_inv_coq (TSort n) x (\lambda (t: T).(\lambda (t0: T).(eq T t0 t))) 
-(\lambda (H0: (pr0 (TSort n) x)).(\lambda (t: T).(\lambda (H1: (eq T t (TSort 
-n))).(\lambda (H2: (eq T t x)).(let H3 \def (eq_ind T t (\lambda (t0: T).(eq 
-T t0 (TSort n))) H1 x H2) in (let H4 \def (eq_ind T x (\lambda (t0: T).(pr0 
-(TSort n) t0)) H0 (TSort n) H3) in (let H5 \def (eq_ind T x (\lambda (t0: 
-T).(pr0 (TSort n) t0)) H (TSort n) H3) in (eq_ind_r T (TSort n) (\lambda (t0: 
-T).(eq T t0 (TSort n))) (refl_equal T (TSort n)) x H3)))))))) (\lambda (H0: 
-(pr0 (TSort n) x)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H2: (eq T (THead k u1 t0) 
-(TSort n))).(\lambda (H3: (eq T (THead k u2 t3) x)).(\lambda (_: (pr0 u1 
-u2)).(\lambda (_: (pr0 t0 t3)).(let H5 \def (eq_ind_r T x (\lambda (t: 
-T).(pr0 (TSort n) t)) H0 (THead k u2 t3) H3) in (let H6 \def (eq_ind_r T x 
-(\lambda (t: T).(pr0 (TSort n) t)) H (THead k u2 t3) H3) in (eq_ind T (THead 
-k u2 t3) (\lambda (t: T).(eq T t (TSort n))) (let H7 \def (eq_ind T (THead k 
-u1 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+ \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TSort n) x)).(insert_eq 
+T (TSort n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda 
+(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: 
+T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T 
+t (TSort n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TSort n) H1) in 
+(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TSort n)) 
+t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 
+t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TSort n))).(let 
+H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in 
+(False_ind (eq T (THead k u2 t2) (THead k u1 t1)) H6)))))))))))) (\lambda (u: 
+T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: 
+(((eq T v1 (TSort n)) \to (eq T v2 v1)))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 
+t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) 
+(TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u 
+t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TSort n) H2) in (False_ind (eq T (THead k u2 t3) 
-(TSort n)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (TSort n) x)).(\lambda 
-(u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (TSort 
-n))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t3) x)).(\lambda (_: (pr0 v1 
-v2)).(\lambda (_: (pr0 t0 t3)).(let H5 \def (eq_ind_r T x (\lambda (t: 
-T).(pr0 (TSort n) t)) H0 (THead (Bind Abbr) v2 t3) H3) in (let H6 \def 
-(eq_ind_r T x (\lambda (t: T).(pr0 (TSort n) t)) H (THead (Bind Abbr) v2 t3) 
-H3) in (eq_ind T (THead (Bind Abbr) v2 t3) (\lambda (t: T).(eq T t (TSort 
-n))) (let H7 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) 
-(\lambda (ee: T).(match ee 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 t3) (TSort 
-n)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (TSort n) x)).(\lambda (b: 
-B).(\lambda (v1: T).(\lambda (v2: T).(\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H4: (eq T (THead (Flat Appl) 
-v1 (THead (Bind b) u1 t0)) (TSort n))).(\lambda (H5: (eq T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t3)) x)).(\lambda (_: (not (eq B b 
-Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 
-t0 t3)).(let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TSort n) t)) H0 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (let H8 
-\def (eq_ind_r T x (\lambda (t: T).(pr0 (TSort n) t)) H (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (eq_ind T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t3)) (\lambda (t: T).(eq T t (TSort n))) 
-(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda 
-(ee: T).(match ee 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) t3)) (TSort n)) H9)) x H5))))))))))))))))) (\lambda 
-(H0: (pr0 (TSort n) x)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H3: (eq T (THead (Bind Abbr) u1 
-t0) (TSort n))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).(\lambda (_: 
-(pr0 u1 u2)).(\lambda (_: (pr0 t0 t3)).(\lambda (_: (subst0 O u2 t3 w)).(let 
-H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TSort n) t)) H0 (THead (Bind 
-Abbr) u2 w) H4) in (let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TSort n) 
-t)) H (THead (Bind Abbr) u2 w) H4) in (eq_ind T (THead (Bind Abbr) u2 w) 
-(\lambda (t: T).(eq T t (TSort n))) (let H8 \def (eq_ind T (THead (Bind Abbr) 
-u1 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+\Rightarrow True])) I (TSort n) H5) in (False_ind (eq T (THead (Bind Abbr) v2 
+t2) (THead (Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda 
+(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2 
+v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 
+t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) 
+(TSort n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 
+t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TSort n) H3) in (False_ind (eq T (THead (Bind Abbr) u2 
-w) (TSort n)) H8)) x H4)))))))))))))) (\lambda (_: (pr0 (TSort n) 
-x)).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (u: 
-T).(\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (TSort 
-n))).(\lambda (H3: (eq T t3 x)).(\lambda (_: (not (eq B b Abst))).(\lambda 
-(H4: (pr0 t0 t3)).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr0 t0 t)) H4 x 
-H3) in (let H6 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda 
-(ee: T).(match ee 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 x (TSort n)) H6)))))))))))) 
-(\lambda (_: (pr0 (TSort n) x)).(\lambda (t0: T).(\lambda (t3: T).(\lambda 
-(u: T).(\lambda (H1: (eq T (THead (Flat Cast) u t0) (TSort n))).(\lambda (H2: 
-(eq T t3 x)).(\lambda (H3: (pr0 t0 t3)).(let H4 \def (eq_ind T t3 (\lambda 
-(t: T).(pr0 t0 t)) H3 x H2) in (let H5 \def (eq_ind T (THead (Flat Cast) u 
-t0) (\lambda (ee: T).(match ee 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 x (TSort n)) 
-H5)))))))))) H))).
+\Rightarrow True])) I (TSort n) H8) in (False_ind (eq T (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) v1 (THead (Bind 
+b) u1 t1))) H9))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda 
+(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to (eq T u2 
+u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda 
+(_: (((eq T t1 (TSort n)) \to (eq T t2 t1)))).(\lambda (w: T).(\lambda (_: 
+(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t1) (TSort 
+n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (ee: T).(match 
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) 
+H6) in (False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1)) 
+H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 
+(TSort n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead 
+(Bind b) u (lift (S O) O t1)) (TSort n))).(let H5 \def (eq_ind T (THead (Bind 
+b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead _ _ _) \Rightarrow True])) I (TSort n) H4) in (False_ind (eq T t2 
+(THead (Bind b) u (lift (S O) O t1))) H5)))))))))) (\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq 
+T t2 t1)))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) 
+(TSort n))).(let H4 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (ee: 
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
+(TSort n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) H4)))))))) y x 
+H0))) H))).
 
 theorem 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)).(pr0_inv_coq (TLRef n) x (\lambda (t: T).(\lambda (t0: T).(eq T t0 t))) 
-(\lambda (H0: (pr0 (TLRef n) x)).(\lambda (t: T).(\lambda (H1: (eq T t (TLRef 
-n))).(\lambda (H2: (eq T t x)).(let H3 \def (eq_ind T t (\lambda (t0: T).(eq 
-T t0 (TLRef n))) H1 x H2) in (let H4 \def (eq_ind T x (\lambda (t0: T).(pr0 
-(TLRef n) t0)) H0 (TLRef n) H3) in (let H5 \def (eq_ind T x (\lambda (t0: 
-T).(pr0 (TLRef n) t0)) H (TLRef n) H3) in (eq_ind_r T (TLRef n) (\lambda (t0: 
-T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) x H3)))))))) (\lambda (H0: 
-(pr0 (TLRef n) x)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H2: (eq T (THead k u1 t0) 
-(TLRef n))).(\lambda (H3: (eq T (THead k u2 t3) x)).(\lambda (_: (pr0 u1 
-u2)).(\lambda (_: (pr0 t0 t3)).(let H5 \def (eq_ind_r T x (\lambda (t: 
-T).(pr0 (TLRef n) t)) H0 (THead k u2 t3) H3) in (let H6 \def (eq_ind_r T x 
-(\lambda (t: T).(pr0 (TLRef n) t)) H (THead k u2 t3) H3) in (eq_ind T (THead 
-k u2 t3) (\lambda (t: T).(eq T t (TLRef n))) (let H7 \def (eq_ind T (THead k 
-u1 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+ \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TLRef n) x)).(insert_eq 
+T (TLRef n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda 
+(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: 
+T).((eq T t (TLRef n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T 
+t (TLRef n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TLRef n) H1) in 
+(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TLRef n)) 
+t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 
+t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TLRef n))).(let 
+H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H5) in 
+(False_ind (eq T (THead k u2 t2) (THead k u1 t1)) H6)))))))))))) (\lambda (u: 
+T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: 
+(((eq T v1 (TLRef n)) \to (eq T v2 v1)))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 
+t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) 
+(TLRef n))).(let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u 
+t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TLRef n) H2) in (False_ind (eq T (THead k u2 t3) 
-(TLRef n)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (TLRef n) x)).(\lambda 
-(u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (TLRef 
-n))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t3) x)).(\lambda (_: (pr0 v1 
-v2)).(\lambda (_: (pr0 t0 t3)).(let H5 \def (eq_ind_r T x (\lambda (t: 
-T).(pr0 (TLRef n) t)) H0 (THead (Bind Abbr) v2 t3) H3) in (let H6 \def 
-(eq_ind_r T x (\lambda (t: T).(pr0 (TLRef n) t)) H (THead (Bind Abbr) v2 t3) 
-H3) in (eq_ind T (THead (Bind Abbr) v2 t3) (\lambda (t: T).(eq T t (TLRef 
-n))) (let H7 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) 
-(\lambda (ee: T).(match ee 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 t3) (TLRef 
-n)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (TLRef n) x)).(\lambda (b: 
-B).(\lambda (v1: T).(\lambda (v2: T).(\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H4: (eq T (THead (Flat Appl) 
-v1 (THead (Bind b) u1 t0)) (TLRef n))).(\lambda (H5: (eq T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t3)) x)).(\lambda (_: (not (eq B b 
-Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 
-t0 t3)).(let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TLRef n) t)) H0 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (let H8 
-\def (eq_ind_r T x (\lambda (t: T).(pr0 (TLRef n) t)) H (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (eq_ind T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t3)) (\lambda (t: T).(eq T t (TLRef n))) 
-(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda 
-(ee: T).(match ee 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) t3)) (TLRef n)) H9)) x H5))))))))))))))))) (\lambda 
-(H0: (pr0 (TLRef n) x)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H3: (eq T (THead (Bind Abbr) u1 
-t0) (TLRef n))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).(\lambda (_: 
-(pr0 u1 u2)).(\lambda (_: (pr0 t0 t3)).(\lambda (_: (subst0 O u2 t3 w)).(let 
-H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TLRef n) t)) H0 (THead (Bind 
-Abbr) u2 w) H4) in (let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TLRef n) 
-t)) H (THead (Bind Abbr) u2 w) H4) in (eq_ind T (THead (Bind Abbr) u2 w) 
-(\lambda (t: T).(eq T t (TLRef n))) (let H8 \def (eq_ind T (THead (Bind Abbr) 
-u1 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+\Rightarrow True])) I (TLRef n) H5) in (False_ind (eq T (THead (Bind Abbr) v2 
+t2) (THead (Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda 
+(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TLRef n)) \to (eq T v2 
+v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 
+t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) 
+(TLRef n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 
+t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TLRef n) H3) in (False_ind (eq T (THead (Bind Abbr) u2 
-w) (TLRef n)) H8)) x H4)))))))))))))) (\lambda (_: (pr0 (TLRef n) 
-x)).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (u: 
-T).(\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (TLRef 
-n))).(\lambda (H3: (eq T t3 x)).(\lambda (_: (not (eq B b Abst))).(\lambda 
-(H4: (pr0 t0 t3)).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr0 t0 t)) H4 x 
-H3) in (let H6 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda 
-(ee: T).(match ee 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 x (TLRef n)) H6)))))))))))) 
-(\lambda (_: (pr0 (TLRef n) x)).(\lambda (t0: T).(\lambda (t3: T).(\lambda 
-(u: T).(\lambda (H1: (eq T (THead (Flat Cast) u t0) (TLRef n))).(\lambda (H2: 
-(eq T t3 x)).(\lambda (H3: (pr0 t0 t3)).(let H4 \def (eq_ind T t3 (\lambda 
-(t: T).(pr0 t0 t)) H3 x H2) in (let H5 \def (eq_ind T (THead (Flat Cast) u 
-t0) (\lambda (ee: T).(match ee 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 x (TLRef n)) 
-H5)))))))))) H))).
+\Rightarrow True])) I (TLRef n) H8) in (False_ind (eq T (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) v1 (THead (Bind 
+b) u1 t1))) H9))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda 
+(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (eq T u2 
+u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda 
+(_: (((eq T t1 (TLRef n)) \to (eq T t2 t1)))).(\lambda (w: T).(\lambda (_: 
+(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t1) (TLRef 
+n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (ee: T).(match 
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) 
+H6) in (False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1)) 
+H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 
+(TLRef n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead 
+(Bind b) u (lift (S O) O t1)) (TLRef n))).(let H5 \def (eq_ind T (THead (Bind 
+b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead _ _ _) \Rightarrow True])) I (TLRef n) H4) in (False_ind (eq T t2 
+(THead (Bind b) u (lift (S O) O t1))) H5)))))))))) (\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq 
+T t2 t1)))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) 
+(TLRef n))).(let H4 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (ee: 
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
+(TLRef n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) H4)))))))) y x 
+H0))) H))).
 
 theorem pr0_gen_abst:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abst) u1 
@@ -250,165 +147,170 @@ 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)).(pr0_inv_coq (THead (Bind Abst) u1 t1) x (\lambda (_: 
-T).(\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)))))) (\lambda (H0: (pr0 (THead 
-(Bind Abst) u1 t1) x)).(\lambda (t: T).(\lambda (H1: (eq T t (THead (Bind 
-Abst) u1 t1))).(\lambda (H2: (eq T t x)).(let H3 \def (eq_ind T t (\lambda 
-(t0: T).(eq T t0 (THead (Bind Abst) u1 t1))) H1 x H2) in (let H4 \def (eq_ind 
-T x (\lambda (t0: T).(pr0 (THead (Bind Abst) u1 t1) t0)) H0 (THead (Bind 
-Abst) u1 t1) H3) in (let H5 \def (eq_ind T x (\lambda (t0: T).(pr0 (THead 
-(Bind Abst) u1 t1) t0)) H (THead (Bind Abst) u1 t1) H3) in (eq_ind_r T (THead 
-(Bind Abst) u1 t1) (\lambda (t0: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+(Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 t1) (\lambda (t: 
+T).(pr0 t x)) (\lambda (_: T).(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 (y: 
+T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T 
+t (THead (Bind Abst) u1 t1)) \to (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 H3)))))))) (\lambda 
-(H0: (pr0 (THead (Bind Abst) u1 t1) x)).(\lambda (u0: T).(\lambda (u2: 
-T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H2: (eq T 
-(THead k u0 t0) (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq T (THead k u2 
-t3) x)).(\lambda (H1: (pr0 u0 u2)).(\lambda (H4: (pr0 t0 t3)).(let H5 \def 
-(eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t1) t)) H0 (THead k 
-u2 t3) H3) in (let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind 
-Abst) u1 t1) t)) H (THead k u2 t3) H3) in (eq_ind T (THead k u2 t3) (\lambda 
-(t: T).(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Bind 
-Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 t1 t2))))) (let H7 \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 ((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 _ t _) \Rightarrow t])) (THead k u0 t0) 
-(THead (Bind Abst) u1 t1) H2) in ((let H9 \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 (\lambda (H10: (eq T u0 u1)).(\lambda (H11: 
-(eq K k (Bind Abst))).(let H12 \def (eq_ind K k (\lambda (k0: K).(pr0 (THead 
-(Bind Abst) u1 t1) (THead k0 u2 t3))) H6 (Bind Abst) H11) in (let H13 \def 
-(eq_ind K k (\lambda (k0: K).(pr0 (THead (Bind Abst) u1 t1) (THead k0 u2 
-t3))) H5 (Bind Abst) H11) in (eq_ind_r K (Bind Abst) (\lambda (k0: K).(ex3_2 
-T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k0 u2 t3) (THead (Bind 
-Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 t1 t2))))) (let H14 \def (eq_ind T t0 (\lambda (t: 
-T).(pr0 t t3)) H4 t1 H9) in (let H15 \def (eq_ind T u0 (\lambda (t: T).(pr0 t 
-u2)) H1 u1 H10) in (ex3_2_intro T T (\lambda (u3: T).(\lambda (t2: T).(eq T 
-(THead (Bind Abst) u2 t3) (THead (Bind Abst) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
-t2))) u2 t3 (refl_equal T (THead (Bind Abst) u2 t3)) H15 H14))) k H11)))))) 
-H8)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (THead (Bind Abst) u1 t1) 
-x)).(\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t0: 
-T).(\lambda (t3: T).(\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 t3) x)).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t0 t3)).(let H5 
-\def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t1) t)) H0 
-(THead (Bind Abbr) v2 t3) H3) in (let H6 \def (eq_ind_r T x (\lambda (t: 
-T).(pr0 (THead (Bind Abst) u1 t1) t)) H (THead (Bind Abbr) v2 t3) H3) in 
-(eq_ind T (THead (Bind Abbr) v2 t3) (\lambda (t: T).(ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T t (THead (Bind Abst) u2 t2)))) (\lambda (u2: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))))) (\lambda 
+(t: T).(\lambda (H1: (eq T t (THead (Bind Abst) u1 t1))).(let H2 \def 
+(f_equal T T (\lambda (e: T).e) t (THead (Bind Abst) u1 t1) H1) in (eq_ind_r 
+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))))) (let H7 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t0)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1 
-t1) H2) in (False_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T 
-(THead (Bind Abbr) v2 t3) (THead (Bind Abst) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
-t2)))) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (THead (Bind Abst) u1 t1) 
-x)).(\lambda (b: B).(\lambda (v1: T).(\lambda (v2: T).(\lambda (u0: 
-T).(\lambda (u2: T).(\lambda (t0: T).(\lambda (t3: T).(\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) t3)) x)).(\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 
-v2)).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (pr0 t0 t3)).(let H7 \def 
-(eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t1) t)) H0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (let H8 \def 
-(eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t1) t)) H (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (eq_ind T (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (\lambda (t: T).(ex3_2 
-T T (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Bind Abst) u3 t2)))) 
+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)) t 
+H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda 
+(H2: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
+t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 
+t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3))))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) 
+(THead (Bind Abst) u1 t1))).(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) H5) in ((let H7 \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) H5) in ((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 k u0 t0) (THead (Bind 
+Abst) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind 
+Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abst) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead 
+(Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 
+(THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))) H4 t1 H8) in (let H12 \def 
+(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T 
+u0 (\lambda (t: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Bind Abst) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3)))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: 
+T).(pr0 t u2)) H1 u1 H9) in (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)) H14 H12))))) k H10)))) 
+H7)) H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: 
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 
+t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind 
+Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 
+t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (H5: (eq T (THead (Flat Appl) 
+v1 (THead (Bind Abst) u t0)) (THead (Bind Abst) u1 t1))).(let H6 \def (eq_ind 
+T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee 
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef 
+_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return 
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
+True])])) I (THead (Bind Abst) u1 t1) H5) in (False_ind (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abst) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) H6)))))))))))) (\lambda (b: B).(\lambda 
+(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 
+v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abst) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: 
+T).(pr0 t1 t2))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 
+u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) 
 (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
-T).(pr0 t1 t2))))) (let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind 
-b) u0 t0)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1 
-t1) H4) in (False_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Bind 
-Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 t1 t2)))) H9)) x H5))))))))))))))))) (\lambda (H0: 
-(pr0 (THead (Bind Abst) u1 t1) x)).(\lambda (u0: T).(\lambda (u2: T).(\lambda 
-(t0: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H3: (eq T (THead (Bind 
-Abbr) u0 t0) (THead (Bind Abst) u1 t1))).(\lambda (H4: (eq T (THead (Bind 
-Abbr) u2 w) x)).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (pr0 t0 t3)).(\lambda 
-(_: (subst0 O u2 t3 w)).(let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 
-(THead (Bind Abst) u1 t1) t)) H0 (THead (Bind Abbr) u2 w) H4) in (let H7 \def 
-(eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t1) t)) H (THead 
-(Bind Abbr) u2 w) H4) in (eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: 
-T).(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Bind Abst) u3 
-t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 t1 t2))))) (let H8 \def (eq_ind T (THead (Bind Abbr) 
-u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind 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 (ex3_2 T 
-T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u2 w) (THead 
-(Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) H8)) x H4)))))))))))))) (\lambda (H0: 
-(pr0 (THead (Bind Abst) u1 t1) x)).(\lambda (b: B).(\lambda (t0: T).(\lambda 
-(t3: T).(\lambda (u: T).(\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O 
-t0)) (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq T t3 x)).(\lambda (H1: 
-(not (eq B b Abst))).(\lambda (H4: (pr0 t0 t3)).(let H5 \def (eq_ind T t3 
-(\lambda (t: T).(pr0 t0 t)) H4 x H3) 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 
+T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (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 (THead (Flat Appl) v1 (THead (Bind 
+b) u0 t0)) (THead (Bind Abst) u1 t1))).(let H9 \def (eq_ind T (THead (Flat 
+Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
+True])])) I (THead (Bind Abst) u1 t1) H8) in (False_ind (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 (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) 
+H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 
+u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3))))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 
+w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abst) u1 
+t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (ee: 
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K 
+return (\lambda (_: K).Prop) with [(Bind 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) H6) in (False_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
+T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abst) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b 
+Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
+(H3: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3))))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O 
+t0)) (THead (Bind Abst) u1 t1))).(let H5 \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 
-((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
+b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1) H4) in 
+((let H6 \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 H8 \def (f_equal T T (\lambda (e: T).(match e in T return 
+t1) H4) in ((let H7 \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 t2) \Rightarrow 
-(THead k (lref_map f d u0) (lref_map f (s k d) t2))]) in lref_map) (\lambda 
+\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 t2) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t2))]) in 
+[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 (\lambda (_: (eq T u u1)).(\lambda (H10: (eq B b Abst)).(let H11 
-\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abst H10) in (let 
-H12 \def (eq_ind_r T t1 (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t) x)) H0 
-(lift (S O) O t0) H8) in (let H13 \def (eq_ind_r T t1 (\lambda (t: T).(pr0 
-(THead (Bind Abst) u1 t) x)) H (lift (S O) O t0) H8) in (eq_ind T (lift (S O) 
-O t0) (\lambda (t: T).(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 t t2))))) (let H14 \def (match (H11 
-(refl_equal B Abst)) in False return (\lambda (_: False).(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 (lift 
-(S O) O t0) t2))))) with []) in H14) t1 H8))))))) H7)) H6)))))))))))) 
-(\lambda (_: (pr0 (THead (Bind Abst) u1 t1) x)).(\lambda (t0: T).(\lambda 
-(t3: T).(\lambda (u: T).(\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead 
-(Bind Abst) u1 t1))).(\lambda (H2: (eq T t3 x)).(\lambda (H3: (pr0 t0 
-t3)).(let H4 \def (eq_ind T t3 (\lambda (t: T).(pr0 t0 t)) H3 x H2) in (let 
-H5 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (ee: T).(match ee in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind Abst) u1 t1) H1) in (False_ind (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)))) H5)))))))))) H)))).
+t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abst)).(let H10 
+\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abst H9) in (let 
+H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abst) u1 t)) 
+\to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t t3)))))) H3 (lift (S O) O t0) H7) in (eq_ind T 
+(lift (S O) O t0) (\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3))))) (let H12 
+\def (match (H10 (refl_equal B Abst)) in False return (\lambda (_: 
+False).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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))))) with []) in H12) t1 
+H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_: 
+(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u 
+t0) (THead (Bind Abst) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u 
+t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort 
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1 
+t1) H3) in (False_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
+(THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) H4)))))))) y x H0))) H)))).
 
 theorem pr0_gen_appl:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Appl) u1 
@@ -433,12 +335,32 @@ 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)).(pr0_inv_coq (THead (Flat Appl) u1 t1) x (\lambda (_: 
-T).(\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: 
+(Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 t1) (\lambda (t: 
+T).(pr0 t x)) (\lambda (_: T).(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 (y: 
+T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T 
+t (THead (Flat Appl) u1 t1)) \to (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)))))) 
@@ -452,436 +374,714 @@ t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2)))))))))
 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 (H0: (pr0 
-(THead (Flat Appl) u1 t1) x)).(\lambda (t: T).(\lambda (H1: (eq T t (THead 
-(Flat Appl) u1 t1))).(\lambda (H2: (eq T t x)).(let H3 \def (eq_ind T t 
-(\lambda (t0: T).(eq T t0 (THead (Flat Appl) u1 t1))) H1 x H2) in (let H4 
-\def (eq_ind T x (\lambda (t0: T).(pr0 (THead (Flat Appl) u1 t1) t0)) H0 
-(THead (Flat Appl) u1 t1) H3) in (let H5 \def (eq_ind T x (\lambda (t0: 
-T).(pr0 (THead (Flat Appl) u1 t1) t0)) H (THead (Flat Appl) u1 t1) H3) in 
-(eq_ind_r 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).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))))) (\lambda (t: 
+T).(\lambda (H1: (eq T t (THead (Flat Appl) u1 t1))).(let H2 \def (f_equal T 
+T (\lambda (e: T).e) t (THead (Flat Appl) u1 t1) H1) in (eq_ind_r 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))) t H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 
+u2)).(\lambda (H2: (((eq T u0 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\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 (_: T).(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind 
+Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr0 u1 u3))))) (\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 (_: 
+(u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind b) v2 (THead 
+(Flat Appl) (lift (S O) O u3) t2))))))))) (\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 
-(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 H3)))))))) (\lambda (H0: (pr0 (THead 
-(Flat Appl) u1 t1) x)).(\lambda (u0: T).(\lambda (u2: T).(\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H2: (eq T (THead k u0 t0) 
-(THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T (THead k u2 t3) x)).(\lambda 
-(H1: (pr0 u0 u2)).(\lambda (H4: (pr0 t0 t3)).(let H5 \def (eq_ind_r T x 
-(\lambda (t: T).(pr0 (THead (Flat Appl) u1 t1) t)) H0 (THead k u2 t3) H3) in 
-(let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t1) t)) 
-H (THead k u2 t3) H3) in (eq_ind T (THead k u2 t3) (\lambda (t: T).(or3 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Flat Appl) u3 
-t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda 
+(H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1 t1)) \to 
+(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 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 (t2: 
-T).(eq T t (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: 
+T).(eq T 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 (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T 
+(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 (t2: T).(eq T t (THead (Bind b) 
-v2 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\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 (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T 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)))))))))))).(\lambda (k: K).(\lambda (H5: (eq 
+T (THead k u0 t0) (THead (Flat Appl) u1 t1))).(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) H5) in ((let H7 \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) H5) in ((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 k u0 t0) (THead (Flat Appl) u1 t1) H5) in (\lambda (H9: (eq T u0 
+u1)).(\lambda (H10: (eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda 
+(k0: K).(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 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 k0 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 k0 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 
-(t2: T).(pr0 z1 t2)))))))))) (let H7 \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 ((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 _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Flat 
-Appl) u1 t1) H2) in ((let H9 \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 (\lambda (H10: (eq T u0 u1)).(\lambda (H11: (eq K k (Flat 
-Appl))).(let H12 \def (eq_ind K k (\lambda (k0: K).(pr0 (THead (Flat Appl) u1 
-t1) (THead k0 u2 t3))) H6 (Flat Appl) H11) in (let H13 \def (eq_ind K k 
-(\lambda (k0: K).(pr0 (THead (Flat Appl) u1 t1) (THead k0 u2 t3))) H5 (Flat 
-Appl) H11) in (eq_ind_r K (Flat Appl) (\lambda (k0: K).(or3 (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T (THead k0 u2 t3) (THead (Flat Appl) 
-u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\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 (u3: T).(\lambda (t2: 
-T).(eq T (THead k0 u2 t3) (THead (Bind Abbr) u3 t2)))))) (\lambda (_: 
+(t3: T).(pr0 z1 t3)))))))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq 
+T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda 
+(t3: T).(eq T 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 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 (t2: T).(pr0 z1 t2)))))) 
+(_: 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 (t2: T).(eq T 
-(THead k0 u2 t3) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) 
-t2))))))))) (\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 (t2: T).(pr0 z1 t2)))))))))) (let H14 \def 
-(eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H4 t1 H9) in (let H15 \def (eq_ind T 
-u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H10) in (or3_intro0 (ex3_2 T T (\lambda 
-(u3: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u2 t3) (THead (Flat Appl) 
-u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T 
+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))))))))))) H4 t1 H8) in (let 
+H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def 
+(eq_ind T u0 (\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (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 (t2: 
-T).(eq T (THead (Flat Appl) u2 t3) (THead (Bind Abbr) u3 t2)))))) (\lambda 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: 
+T).(eq T u2 (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 u2 (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))))))))))) H2 u1 H9) in (let H14 \def (eq_ind 
+T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in (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 (t2: T).(pr0 z1 
-t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+(\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 
-(t2: T).(eq T (THead (Flat Appl) u2 t3) (THead (Bind b) v2 (THead (Flat Appl) 
-(lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_: 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 (t2: T).(pr0 z1 
-t2)))))))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead 
-(Flat Appl) u2 t3) (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u2 t3 
-(refl_equal T (THead (Flat Appl) u2 t3)) H15 H14)))) k H11)))))) H8)) H7)) x 
-H3))))))))))))) (\lambda (H0: (pr0 (THead (Flat Appl) u1 t1) x)).(\lambda (u: 
-T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t3: 
-T).(\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 t3) 
-x)).(\lambda (H1: (pr0 v1 v2)).(\lambda (H4: (pr0 t0 t3)).(let H5 \def 
-(eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t1) t)) H0 (THead 
-(Bind Abbr) v2 t3) H3) in (let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 
-(THead (Flat Appl) u1 t1) t)) H (THead (Bind Abbr) v2 t3) H3) in (eq_ind T 
-(THead (Bind Abbr) v2 t3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T t (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 t (THead (Bind 
-Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\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)) H14 H12)))))) k H10)))) H7)) 
+H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda 
+(H1: (pr0 v1 v2)).(\lambda (H2: (((eq T v1 (THead (Flat Appl) u1 t1)) \to 
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (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 v2 (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 (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind 
+b) v3 (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 (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1 
+t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 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 t1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T 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)))))))))))).(\lambda (H5: (eq T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1))).(let H6 \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) H5) in ((let H7 \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) H5) in (\lambda (H8: (eq T v1 u1)).(let H9 \def (eq_ind T v1 
+(\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T v2 (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 v2 (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 (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: 
+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 (t2: T).(eq T t (THead (Bind b) v3 (THead 
-(Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: 
+(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (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 
-(t2: T).(pr0 z1 t2)))))))))) (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 Abst) u t0)) (THead (Flat Appl) u1 t1) H2) in ((let H8 \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 (\lambda (H9: (eq T v1 
-u1)).(let H10 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H1 u1 H9) in (let 
-H11 \def (eq_ind_r T t1 (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t) (THead 
-(Bind Abbr) v2 t3))) H6 (THead (Bind Abst) u t0) H8) in (let H12 \def 
-(eq_ind_r T t1 (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t) (THead (Bind 
-Abbr) v2 t3))) H5 (THead (Bind Abst) u t0) H8) in (eq_ind T (THead (Bind 
-Abst) u t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T (THead (Bind Abbr) v2 t3) (THead (Flat Appl) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t 
-t2)))) (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 (t2: T).(eq T (THead (Bind Abbr) 
-v2 t3) (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: 
+(t3: T).(pr0 z1 t3))))))))))) H2 u1 H8) in (let H10 \def (eq_ind T v1 
+(\lambda (t: T).(pr0 t v2)) H1 u1 H8) in (let H11 \def (eq_ind_r T t1 
+(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 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 t (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T 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))))))))))) H4 (THead (Bind Abst) u t0) H7) in (let H12 
+\def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1 t)) \to 
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T v2 (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 v2 (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 (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T 
+(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 (t2: T).(eq T (THead (Bind 
-Abbr) v2 t3) (THead (Bind b) v3 (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 (v3: T).(\lambda (_: T).(pr0 
-y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (or3_intro1 (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) v2 t3) (THead 
-(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t2: T).(pr0 (THead (Bind Abst) u t0) t2)))) (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 (t2: T).(eq T (THead (Bind Abbr) 
-v2 t3) (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 
-(THead (Bind Abst) u t0) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda 
-(t2: T).(eq T (THead (Bind Abbr) v2 t3) (THead (Bind b) v3 (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 
-(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 
-t2)))))))) (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 (t2: 
-T).(eq T (THead (Bind Abbr) v2 t3) (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))))) u t0 v2 t3 (refl_equal T (THead (Bind Abst) u t0)) (refl_equal T 
-(THead (Bind Abbr) v2 t3)) H10 H4)) t1 H8)))))) H7)) x H3))))))))))))) 
-(\lambda (H0: (pr0 (THead (Flat Appl) u1 t1) x)).(\lambda (b: B).(\lambda 
-(v1: T).(\lambda (v2: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (t0: 
-T).(\lambda (t3: T).(\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) t3)) x)).(\lambda (H1: (not (eq B b 
-Abst))).(\lambda (H2: (pr0 v1 v2)).(\lambda (H3: (pr0 u0 u2)).(\lambda (H6: 
-(pr0 t0 t3)).(let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat 
-Appl) u1 t1) t)) H0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t3)) H5) in (let H8 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat 
-Appl) u1 t1) t)) H (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t3)) H5) in (eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t3)) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T 
-t (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
-(\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 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (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))))))))))) H9 (THead (Bind Abst) u t0) H7) in 
+(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).(or3 (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 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 (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 t (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)))))))))) (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)) H10 H3)) t1 H7))))))) H6)))))))))))) (\lambda (b: B).(\lambda (H1: (not 
+(eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H2: (pr0 v1 
+v2)).(\lambda (H3: (((eq T v1 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (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 v2 (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 (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 (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind b0) v3 (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 (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(t2: T).(pr0 z1 t2)))))))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda 
+(H4: (pr0 u0 u2)).(\lambda (H5: (((eq T u0 (THead (Flat Appl) u1 t1)) \to 
+(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) 
+u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\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 (u3: T).(\lambda (t2: 
+T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 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).(eq T t1 
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t2: T).(eq T u2 (THead (Bind 
+b0) v3 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\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 (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (H6: (pr0 t0 t2)).(\lambda (H7: (((eq T t0 (THead (Flat Appl) u1 
+t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 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 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 t1 
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T 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)))))))))))).(\lambda (H8: (eq T (THead (Flat 
+Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1))).(let H9 \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) H8) in ((let H10 \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) 
+H8) in (\lambda (H11: (eq T v1 u1)).(let H12 \def (eq_ind T v1 (\lambda (t: 
+T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T v2 (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 v2 (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 v2 (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))))))))))) H3 u1 H11) in (let H13 \def (eq_ind T v1 
+(\lambda (t: T).(pr0 t v2)) H2 u1 H11) in (let H14 \def (eq_ind_r T t1 
+(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (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 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 t (THead (Bind 
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T 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))))))))))) H7 (THead (Bind b) u0 t0) H10) in (let H15 \def 
+(eq_ind_r T t1 (\lambda (t: T).((eq T u0 (THead (Flat Appl) u1 t)) \to (or3 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (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 u2 (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 u2 (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))))))))))) H5 (THead (Bind b) u0 t0) H10) in 
+(let H16 \def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1 
+t)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T v2 (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 
-(t2: T).(eq T t (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: 
+(t3: T).(eq T v2 (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 (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T 
+(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 
+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 (t2: T).(eq T t (THead (Bind 
-b0) v3 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: 
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (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 (t2: T).(pr0 z1 t2)))))))))) (let H9 \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 ((let H10 
-\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 (\lambda (H11: (eq T 
-v1 u1)).(let H12 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H2 u1 H11) in 
-(let H13 \def (eq_ind_r T t1 (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t) 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)))) H8 (THead 
-(Bind b) u0 t0) H10) in (let H14 \def (eq_ind_r T t1 (\lambda (t: T).(pr0 
-(THead (Flat Appl) u1 t) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t3)))) H7 (THead (Bind b) u0 t0) H10) in (eq_ind T (THead (Bind b) u0 t0) 
-(\lambda (t: T).(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Flat 
-Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 t t2)))) (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 (t2: 
-T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead 
-(Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (t2: T).(pr0 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).(eq T t (THead (Bind 
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (v3: T).(\lambda (t2: T).(eq T (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t3)) (THead (Bind b0) v3 (THead (Flat Appl) 
-(lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) 
+T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H12 (THead (Bind b) u0 t0) H10) in 
+(eq_ind T (THead (Bind b) u0 t0) (\lambda (t: T).(or3 (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 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 (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 t 
+(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 (t2: T).(pr0 z1 
-t2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Flat 
-Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 (THead (Bind b) u0 t0) t2)))) (ex4_4 T T T T 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
+t3)))))))))) (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 (t2: T).(eq T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Bind Abbr) u3 t2)))))) 
+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 (t2: 
-T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: 
+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 (t2: T).(eq T (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Bind b0) v3 (THead (Flat 
-Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: 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 (t2: T).(pr0 z1 
-t2)))))))) (ex6_6_intro B T T T T T (\lambda (b0: B).(\lambda (_: T).(\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 (t2: T).(eq T (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t3)) (THead (Bind b0) v3 (THead (Flat Appl) 
-(lift (S O) O u3) t2))))))))) (\lambda (_: B).(\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 (t2: T).(pr0 z1 
-t2))))))) b u0 t0 v2 u2 t3 H1 (refl_equal T (THead (Bind b) u0 t0)) 
-(refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3))) 
-H12 H3 H6)) t1 H10)))))) H9)) x H5))))))))))))))))) (\lambda (H0: (pr0 (THead 
-(Flat Appl) u1 t1) x)).(\lambda (u0: T).(\lambda (u2: T).(\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H3: (eq T (THead (Bind Abbr) u0 
-t0) (THead (Flat Appl) u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) 
-x)).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (pr0 t0 t3)).(\lambda (_: (subst0 
-O u2 t3 w)).(let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat 
-Appl) u1 t1) t)) H0 (THead (Bind Abbr) u2 w) H4) in (let H7 \def (eq_ind_r T 
-x (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t1) t)) H (THead (Bind Abbr) u2 
-w) H4) in (eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).(or3 (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Flat Appl) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\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 (u3: T).(\lambda (t2: T).(eq T t (THead (Bind 
-Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr0 u1 u3))))) (\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 
-(u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T t (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O u3) t2))))))))) (\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 
-(t2: T).(pr0 z1 t2)))))))))) (let H8 \def (eq_ind T (THead (Bind Abbr) u0 t0) 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
+t3))))))) b u0 t0 v2 u2 t2 H1 (refl_equal T (THead (Bind b) u0 t0)) 
+(refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))) 
+H13 H4 H6)) t1 H10)))))))) H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: 
+T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u1 
+t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
+(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\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 (u3: T).(\lambda 
+(t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\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 (u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind 
+b) v2 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\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 (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 
+t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 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 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 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)))))))))))).(\lambda (w: T).(\lambda (_: 
+(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead 
+(Flat Appl) u1 t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) 
 (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
 (match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | 
-(Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 t1) H3) in (False_ind 
-(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u2 
-w) (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda 
+(Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 t1) H6) in (False_ind 
+(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 
+w) (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 (t2: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 
-t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: 
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (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 
-(t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\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 (t2: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind b) v2 
-(THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_: 
+(v2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (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 
-(t2: T).(pr0 z1 t2))))))))) H8)) x H4)))))))))))))) (\lambda (_: (pr0 (THead 
-(Flat Appl) u1 t1) x)).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (u: T).(\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) 
-(THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T t3 x)).(\lambda (_: (not (eq 
-B b Abst))).(\lambda (H4: (pr0 t0 t3)).(let H5 \def (eq_ind T t3 (\lambda (t: 
-T).(pr0 t0 t)) H4 x H3) in (let H6 \def (eq_ind T (THead (Bind b) u (lift (S 
-O) O t0)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 
-t1) H2) in (False_ind (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: 
+(t3: T).(pr0 z1 t3))))))))) H7))))))))))))) (\lambda (b: B).(\lambda (_: (not 
+(eq B b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
+t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 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 (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 t2 (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)))))))))))).(\lambda (u: T).(\lambda (H4: (eq T (THead 
+(Bind b) u (lift (S O) O t0)) (THead (Flat Appl) u1 t1))).(let H5 \def 
+(eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (ee: T).(match ee in 
+T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
+(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
+False])])) I (THead (Flat Appl) u1 t1) H4) in (False_ind (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 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 (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 t2 (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))))))))) H5)))))))))) (\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 
+t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 
-(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: 
+(t3: T).(eq T 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 (t2: T).(pr0 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: 
+(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 b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind 
-b0) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (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 (t2: T).(pr0 z1 t2))))))))) H6)))))))))))) (\lambda (_: (pr0 
-(THead (Flat Appl) u1 t1) x)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (u: 
-T).(\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Flat Appl) u1 
-t1))).(\lambda (H2: (eq T t3 x)).(\lambda (H3: (pr0 t0 t3)).(let H4 \def 
-(eq_ind T t3 (\lambda (t: T).(pr0 t0 t)) H3 x H2) in (let H5 \def (eq_ind T 
+T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: T).(\lambda (H3: (eq 
+T (THead (Flat Cast) u t0) (THead (Flat Appl) u1 t1))).(let H4 \def (eq_ind T 
 (THead (Flat Cast) u t0) (\lambda (ee: T).(match ee in T return (\lambda (_: 
 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
 (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
 [(Bind _) \Rightarrow 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 (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 
+True])])])) I (THead (Flat Appl) u1 t1) H3) in (False_ind (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 (t2: T).(eq T x (THead (Bind 
-Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\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).(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: 
+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 (t2: T).(eq T x (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: 
+(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (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 
-(t2: T).(pr0 z1 t2))))))))) H5)))))))))) H)))).
+(t3: T).(pr0 z1 t3))))))))) H4)))))))) y x H0))) H)))).
 
 theorem pr0_gen_cast:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Cast) u1 
@@ -890,154 +1090,164 @@ t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
 (_: 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)).(pr0_inv_coq (THead (Flat Cast) u1 t1) x (\lambda (_: 
-T).(\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)))) (\lambda (H0: 
-(pr0 (THead (Flat Cast) u1 t1) x)).(\lambda (t: T).(\lambda (H1: (eq T t 
-(THead (Flat Cast) u1 t1))).(\lambda (H2: (eq T t x)).(let H3 \def (eq_ind T 
-t (\lambda (t0: T).(eq T t0 (THead (Flat Cast) u1 t1))) H1 x H2) in (let H4 
-\def (eq_ind T x (\lambda (t0: T).(pr0 (THead (Flat Cast) u1 t1) t0)) H0 
-(THead (Flat Cast) u1 t1) H3) in (let H5 \def (eq_ind T x (\lambda (t0: 
-T).(pr0 (THead (Flat Cast) u1 t1) t0)) H (THead (Flat Cast) u1 t1) H3) in 
-(eq_ind_r 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)) 
+(Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 t1) (\lambda (t: 
+T).(pr0 t x)) (\lambda (_: T).(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 (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda 
+(t0: T).((eq T t (THead (Flat Cast) u1 t1)) \to (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))))) (\lambda (t: T).(\lambda (H1: (eq T t (THead (Flat 
+Cast) u1 t1))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (THead (Flat 
+Cast) u1 t1) H1) in (eq_ind_r 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 H3)))))))) 
-(\lambda (H0: (pr0 (THead (Flat Cast) u1 t1) x)).(\lambda (u0: T).(\lambda 
-(u2: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H2: (eq T 
-(THead k u0 t0) (THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T (THead k u2 
-t3) x)).(\lambda (H1: (pr0 u0 u2)).(\lambda (H4: (pr0 t0 t3)).(let H5 \def 
-(eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Cast) u1 t1) t)) H0 (THead k 
-u2 t3) H3) in (let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat 
-Cast) u1 t1) t)) H (THead k u2 t3) H3) in (eq_ind T (THead k u2 t3) (\lambda 
-(t: T).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Flat 
-Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t))) (let H7 \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 ((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 _ t _) \Rightarrow t])) 
-(THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in ((let H9 \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 (\lambda (H10: (eq T u0 
-u1)).(\lambda (H11: (eq K k (Flat Cast))).(let H12 \def (eq_ind K k (\lambda 
-(k0: K).(pr0 (THead (Flat Cast) u1 t1) (THead k0 u2 t3))) H6 (Flat Cast) H11) 
-in (let H13 \def (eq_ind K k (\lambda (k0: K).(pr0 (THead (Flat Cast) u1 t1) 
-(THead k0 u2 t3))) H5 (Flat Cast) H11) in (eq_ind_r K (Flat Cast) (\lambda 
-(k0: K).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k0 u2 
-t3) (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead k0 u2 
-t3)))) (let H14 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H4 t1 H9) in 
-(let H15 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H10) in 
-(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Flat 
-Cast) u2 t3) (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
+(THead (Flat Cast) u1 t1)) (pr0_refl u1) (pr0_refl t1))) t H2)))) (\lambda 
+(u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 
+(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
+T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
 T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
-(THead (Flat Cast) u2 t3)) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t2: 
-T).(eq T (THead (Flat Cast) u2 t3) (THead (Flat Cast) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
-t2))) u2 t3 (refl_equal T (THead (Flat Cast) u2 t3)) H15 H14)))) k H11)))))) 
-H8)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (THead (Flat Cast) u1 t1) 
-x)).(\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t0: 
-T).(\lambda (t3: T).(\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 t3) x)).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t0 t3)).(let H5 
-\def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Cast) u1 t1) t)) H0 
-(THead (Bind Abbr) v2 t3) H3) in (let H6 \def (eq_ind_r T x (\lambda (t: 
-T).(pr0 (THead (Flat Cast) u1 t1) t)) H (THead (Bind Abbr) v2 t3) H3) in 
-(eq_ind T (THead (Bind Abbr) v2 t3) (\lambda (t: T).(or (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t2: T).(eq T t (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
+u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 t2)).(\lambda 
+(H4: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 t2))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) 
+(THead (Flat Cast) u1 t1))).(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) H5) in ((let H7 \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) H5) in ((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 k u0 t0) (THead (Flat 
+Cast) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Flat 
+Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T (THead k0 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 k0 u2 t2)))) (let H11 \def (eq_ind T t0 
+(\lambda (t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3)))) (pr0 t1 t2)))) H4 t1 H8) in (let H12 \def (eq_ind T t0 
+(\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda 
+(t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T u2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 u2)))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: 
+T).(pr0 t u2)) H1 u1 H9) in (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)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda 
+(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 
+(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
+v2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
+(_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 t2))))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind 
+Abst) u t0)) (THead (Flat Cast) u1 t1))).(let H6 \def (eq_ind T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
+(_: K).Prop) with [(Bind _) \Rightarrow 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) H5) in (False_ind (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) 
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind Abbr) v2 
+t2))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b 
+Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda 
+(_: (((eq T v1 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
-t2)))) (pr0 t1 t))) (let H7 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t0)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
-_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow 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 (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T (THead (Bind Abbr) v2 t3) (THead (Flat Cast) u2 t2)))) (\lambda 
-(u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 
-t1 t2)))) (pr0 t1 (THead (Bind Abbr) v2 t3))) H7)) x H3))))))))))))) (\lambda 
-(H0: (pr0 (THead (Flat Cast) u1 t1) x)).(\lambda (b: B).(\lambda (v1: 
-T).(\lambda (v2: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (t0: 
-T).(\lambda (t3: T).(\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) t3)) x)).(\lambda (_: (not (eq B b 
-Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (pr0 
-t0 t3)).(let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Cast) u1 
-t1) t)) H0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) 
-in (let H8 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Cast) u1 t1) 
-t)) H (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in 
-(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) 
-(\lambda (t: T).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t 
-(THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
-(\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t))) (let H9 \def 
-(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: 
+t2)))) (pr0 t1 v2))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 
+u2)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(pr0 t1 t2)))) (pr0 t1 u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda 
+(_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (H8: (eq T (THead 
+(Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Cast) u1 t1))).(let H9 
+\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: 
 T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
 False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K 
 return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow 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 (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Flat Cast) u3 
-t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t3)))) H9)) x H5))))))))))))))))) (\lambda (H0: (pr0 
-(THead (Flat Cast) u1 t1) x)).(\lambda (u0: T).(\lambda (u2: T).(\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H3: (eq T (THead (Bind Abbr) u0 
-t0) (THead (Flat Cast) u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) 
-x)).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (pr0 t0 t3)).(\lambda (_: (subst0 
-O u2 t3 w)).(let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat 
-Cast) u1 t1) t)) H0 (THead (Bind Abbr) u2 w) H4) in (let H7 \def (eq_ind_r T 
-x (\lambda (t: T).(pr0 (THead (Flat Cast) u1 t1) t)) H (THead (Bind Abbr) u2 
-w) H4) in (eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).(or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Flat Cast) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
-T).(pr0 t1 t2)))) (pr0 t1 t))) (let H8 \def (eq_ind T (THead (Bind Abbr) u0 
-t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1 
-t1) H3) in (False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T 
-(THead (Bind Abbr) u2 w) (THead (Flat Cast) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
-t2)))) (pr0 t1 (THead (Bind Abbr) u2 w))) H8)) x H4)))))))))))))) (\lambda 
-(_: (pr0 (THead (Flat Cast) u1 t1) x)).(\lambda (b: B).(\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (u: T).(\lambda (H2: (eq T (THead (Bind b) u 
-(lift (S O) O t0)) (THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T t3 
-x)).(\lambda (_: (not (eq B b Abst))).(\lambda (H4: (pr0 t0 t3)).(let H5 \def 
-(eq_ind T t3 (\lambda (t: T).(pr0 t0 t)) H4 x H3) in (let H6 \def (eq_ind T 
-(THead (Bind b) u (lift (S O) O t0)) (\lambda (ee: T).(match ee in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
-False])])) I (THead (Flat Cast) u1 t1) H2) in (False_ind (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)) H6)))))))))))) (\lambda (_: (pr0 (THead (Flat 
-Cast) u1 t1) x)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (u: T).(\lambda 
-(H1: (eq T (THead (Flat Cast) u t0) (THead (Flat Cast) u1 t1))).(\lambda (H2: 
-(eq T t3 x)).(\lambda (H3: (pr0 t0 t3)).(let H4 \def (eq_ind T t3 (\lambda 
-(t: T).(pr0 t0 t)) H3 x 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 (Flat Cast) u 
-t0) (THead (Flat Cast) u1 t1) H1) in ((let H6 \def (f_equal T T (\lambda (e: 
+True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H8) in 
+(False_ind (or (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 Cast) u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t2)))) H9))))))))))))))))) (\lambda (u0: T).(\lambda 
+(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Cast) 
+u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
+(Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 u2))))).(\lambda (t0: 
+T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead 
+(Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq 
+T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
+u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 
+t2))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T 
+(THead (Bind Abbr) u0 t0) (THead (Flat Cast) u1 t1))).(let H7 \def (eq_ind T 
+(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat 
+Cast) u1 t1) H6) in (False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
+T).(eq T (THead (Bind Abbr) u2 w) (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 (THead (Bind Abbr) u2 w))) H7))))))))))))) (\lambda (b: 
+B).(\lambda (_: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u1 
+t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (u: 
+T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat 
+Cast) u1 t1))).(let H5 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) 
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | 
+(Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1 t1) H4) in (False_ind 
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2)) H5)))))))))) (\lambda (t0: 
+T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (((eq T t0 
+(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 
+t2))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead 
+(Flat Cast) u1 t1))).(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) H3) in ((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 (Flat Cast) 
-u t0) (THead (Flat Cast) u1 t1) H1) in (\lambda (_: (eq T u u1)).(let H8 \def 
-(eq_ind T t0 (\lambda (t: T).(pr0 t x)) H4 t1 H6) in (or_intror (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) H8)))) H5)))))))))) H)))).
+u t0) (THead (Flat Cast) u1 t1) H3) in (\lambda (_: (eq T u u1)).(let H7 \def 
+(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2)))) H2 t1 H5) in (let H8 \def 
+(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 t1 H5) in (or_intror (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3)))) (pr0 t1 t2) H8))))) H4)))))))) y x H0))) H)))).
 
 theorem pr0_gen_abbr:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abbr) u1 
@@ -1047,214 +1257,241 @@ t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
 (\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: 
+(Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 t1) (\lambda (t: 
+T).(pr0 t x)) (\lambda (_: T).(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)))) 
+O) O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: 
+T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (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).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
+T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: 
+T).(\lambda (H1: (eq T t (THead (Bind Abbr) u1 t1))).(let H2 \def (f_equal T 
+T (\lambda (e: T).e) t (THead (Bind Abbr) u1 t1) H1) in (eq_ind_r 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 (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 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)))))) 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 
+t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 
+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 (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t2)))))) u1 t1 (refl_equal T (THead (Bind 
+Abbr) u1 t1)) (pr0_refl u1) (or_introl (pr0 t1 t1) (ex2 T (\lambda (y0: 
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u1 y0 t1))) (pr0_refl t1)))) t 
+H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda 
+(H2: (((eq T u0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 
+t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 
+t2))))))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 
+t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 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 (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O 
+t2)))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) (THead (Bind 
+Abbr) u1 t1))).(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) 
+H5) in ((let H7 \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) 
+H5) in ((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 k u0 t0) (THead (Bind Abbr) u1 t1) 
+H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind 
+Abbr))).(eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T (THead k0 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 (y0: T).(pr0 t1 y0)) (\lambda (y0: 
+T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead k0 u2 t2))))) (let 
+H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to 
+(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 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 (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2))))) H4 
+t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in 
+(let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 
+t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (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 (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O u2))))) H2 
+u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in 
+(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 (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 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))))))) (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 
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 t2 (refl_equal T (THead (Bind 
+Abbr) u2 t2)) H14 (or_introl (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t2))) H12))))))) k H10)))) H7)) 
+H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: 
+(pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (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 (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O 
+v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
+(_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (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 (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 
+t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Abbr) u1 t1))).(let H6 \def 
+(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: 
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K 
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) 
+\Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H5) in (False_ind (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) 
 (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 
+(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: 
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S 
+O) O (THead (Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: 
+(not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 
+v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (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 (y0: T).(pr0 t1 y0)) (\lambda (y0: 
+T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: 
+T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead 
+(Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
+T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
+u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: 
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 t1 (lift (S 
+O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T 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 (y0: T).(pr0 t1 y0)) (\lambda (y0: 
+T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq 
+T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Abbr) u1 
+t1))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) 
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | 
+(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H8) in (False_ind 
+(or (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 (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 (y0: T).(pr0 t1 y0)) (\lambda (y0: 
+T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t2))))) H9))))))))))))))))) (\lambda (u0: 
+T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 
+(THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
+T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T 
+(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 
+t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: 
+(pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 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 (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O 
+t2)))))).(\lambda (w: T).(\lambda (H5: (subst0 O u2 t2 w)).(\lambda (H6: (eq 
+T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1))).(let H7 \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) H6) in ((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 (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H3) in 
+\Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H6) in 
+(\lambda (H9: (eq T u0 u1)).(let H10 \def (eq_ind T t0 (\lambda (t: T).((eq T 
+t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda 
+(t3: T).(eq T 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 (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 
+t1 (lift (S O) O t2))))) H4 t1 H8) in (let H11 \def (eq_ind T t0 (\lambda (t: 
+T).(pr0 t t2)) H3 t1 H8) in (let H12 \def (eq_ind T u0 (\lambda (t: T).((eq T 
+t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda 
+(t3: T).(eq T u2 (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 (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 
+t1 (lift (S O) O u2))))) H2 u1 H9) in (let H13 \def (eq_ind T u0 (\lambda (t: 
+T).(pr0 t u2)) H1 u1 H9) in (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 (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O 
+u3 y0 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 (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 w (refl_equal T (THead (Bind 
+Abbr) u2 w)) H13 (or_intror (pr0 t1 w) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 w))) (ex_intro2 T (\lambda (y0: T).(pr0 t1 
+y0)) (\lambda (y0: T).(subst0 O u2 y0 w)) t2 H11 H5)))))))))) H7))))))))))))) 
+(\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda (t0: T).(\lambda 
+(t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (H3: (((eq T t0 (THead (Bind 
+Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
+(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 (y0: 
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S 
+O) O t2)))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S 
+O) O t0)) (THead (Bind Abbr) u1 t1))).(let H5 \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) H4) 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) 
+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 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)))))).
+t1) H4) in ((let H7 \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) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abbr)).(let H10 
+\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abbr H9) in (let 
+H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abbr) u1 t)) 
+\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 (y0: T).(pr0 t y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2))))) H3 
+(lift (S O) O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 (y0: T).(pr0 t y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2)))) 
+(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 (y0: 
+T).(pr0 (lift (S O) O t0) y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) 
+(pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2 H2 (S O) O)) t1 
+H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_: 
+(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O 
+t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead 
+(Bind Abbr) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda 
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | 
+(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H3) in (False_ind 
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2))) 
+H4)))))))) y x H0))) H)))).
 
 theorem pr0_gen_void:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Void) u1 
@@ -1263,109 +1500,148 @@ t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
 (_: 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: 
+(Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 t1) (\lambda (t: 
+T).(pr0 t x)) (\lambda (_: T).(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 
+O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: 
+T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (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 
+T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: T).(\lambda 
+(H1: (eq T t (THead (Bind Void) u1 t1))).(let H2 \def (f_equal T T (\lambda 
+(e: T).e) t (THead (Bind Void) u1 t1) H1) in (eq_ind_r 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))) 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 
+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))) t H2)))) (\lambda (u0: T).(\lambda (u2: 
+T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 (THead (Bind Void) u1 
+t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
+(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O 
+u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 
+t2)).(\lambda (H4: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T 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 t2)))))).(\lambda (k: K).(\lambda 
+(H5: (eq T (THead k u0 t0) (THead (Bind Void) u1 t1))).(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) H5) in ((let H7 \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 (_: 
+(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in ((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 k u0 t0) (THead (Bind Void) u1 t1) H5) in (\lambda (H9: (eq T u0 
+u1)).(\lambda (H10: (eq K k (Bind Void))).(eq_ind_r K (Bind Void) (\lambda 
+(k0: K).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 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 k0 u2 t2))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t 
+(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
+T).(eq T 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 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 
+(lift (S O) O t2))))) H4 t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: 
+T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T 
+t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda 
+(t3: T).(eq T u2 (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 u2))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: 
+T).(pr0 t u2)) H1 u1 H9) in (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 
+u2 t2)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda 
+(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 
+(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T v2 (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 v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 t2)))))).(\lambda (H5: (eq T (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Void) u1 t1))).(let H6 
+\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: 
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K 
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) 
+\Rightarrow True])])) I (THead (Bind Void) u1 t1) H5) in (False_ind (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) 
 (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 (_: 
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead 
+(Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B 
+b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 
+v2)).(\lambda (_: (((eq T v1 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (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 v2)))))).(\lambda (u0: T).(\lambda 
+(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) 
+u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
+(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O 
+u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
+(_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T 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 t2)))))).(\lambda (H8: (eq T (THead (Flat Appl) 
+v1 (THead (Bind b) u0 t0)) (THead (Bind Void) u1 t1))).(let H9 \def (eq_ind T 
+(THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee in T 
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
+True])])) I (THead (Bind Void) u1 t1) H8) in (False_ind (or (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 (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 b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))))) 
+H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 
+u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Void) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda 
+(t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) 
+u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 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 
+t2)))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T 
+(THead (Bind Abbr) u0 t0) (THead (Bind Void) u1 t1))).(let H7 \def (eq_ind T 
+(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: 
 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
 (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
-[(Bind _) \Rightarrow 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 _) 
+[(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) H6) in (False_ind (or 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) 
+(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 Abbr) u2 w)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B 
+b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 
+t2)).(\lambda (H3: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 t2)))))).(\lambda (u: T).(\lambda 
+(H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 
+t1))).(let H5 \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) H4) in ((let H6 \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) H4) in ((let H7 \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)])) 
@@ -1377,50 +1653,34 @@ t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (TLRef _)
 | (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)))) 
+t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Void)).(let H10 
+\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Void H9) in (let 
+H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Void) u1 t)) 
+\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 t2))))) H3 (lift (S O) 
+O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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)))))).
+T).(pr0 t t3)))) (pr0 t (lift (S O) O t2)))) (or_intror (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T t2 (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 t2)) (pr0_lift t0 t2 
+H2 (S O) O)) t1 H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 
+t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 
+t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead 
+(Bind Void) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda 
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | 
+(Flat _) \Rightarrow True])])) I (THead (Bind Void) u1 t1) H3) in (False_ind 
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 t2))) H4)))))))) y x 
+H0))) H)))).
 
 theorem pr0_gen_lift:
  \forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr0 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/pr0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/pr0.ma
index 207570e19..57e9f7ef1 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/pr0.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/pr0.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr0/fwd.ma".
+include "LambdaDelta-1/pr0/fwd.ma".
 
-include "lift/tlt.ma".
+include "LambdaDelta-1/lift/tlt.ma".
 
 theorem pr0_confluence__pr0_cong_upsilon_refl:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: 
@@ -451,102 +451,82 @@ k0 u3 t6) t2)).(eq_ind T (THead k0 u3 t6) (\lambda (t7: T).((pr0 u0 u3) \to
 ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda 
 (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u0 u3)).(\lambda (H15: (pr0 t5 
 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) 
-H4 (THead k0 u0 t5) H11) in (let H17 \def (match H16 in eq return (\lambda 
-(t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead k0 u0 t5)) \to (ex2 T 
-(\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 (THead k0 u3 
-t6) t8)))))) with [refl_equal \Rightarrow (\lambda (H17: (eq T (THead k u1 
-t3) (THead k0 u0 t5))).(let H18 \def (f_equal T T (\lambda (e: T).(match e in 
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) 
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead k0 
-u0 t5) H17) in ((let H19 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
-\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead k0 
-u0 t5) H17) in ((let H20 \def (f_equal T K (\lambda (e: T).(match e in T 
-return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
+H4 (THead k0 u0 t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match 
+e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
 \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u1 t3) (THead k0 u0 
-t5) H17) in (eq_ind K k0 (\lambda (k1: K).((eq T u1 u0) \to ((eq T t3 t5) \to 
-(ex2 T (\lambda (t7: T).(pr0 (THead k1 u2 t4) t7)) (\lambda (t7: T).(pr0 
-(THead k0 u3 t6) t7)))))) (\lambda (H21: (eq T u1 u0)).(eq_ind T u0 (\lambda 
-(_: T).((eq T t3 t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead k0 u2 t4) t8)) 
-(\lambda (t8: T).(pr0 (THead k0 u3 t6) t8))))) (\lambda (H22: (eq T t3 
-t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead k0 u2 
-t4) t8)) (\lambda (t8: T).(pr0 (THead k0 u3 t6) t8)))) (let H23 \def 
-(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: 
-T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: 
-T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead k0 u0 t5) 
-H11) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 t5 H22) 
-in (let H25 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u0 H21) in 
-(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) (ex2 T 
-(\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 
-u3 t6) t7))) (\lambda (x: T).(\lambda (H26: (pr0 u2 x)).(\lambda (H27: (pr0 
-u3 x)).(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) 
-(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 
-(THead k0 u3 t6) t7))) (\lambda (x0: T).(\lambda (H28: (pr0 t4 x0)).(\lambda 
-(H29: (pr0 t6 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) 
-(\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)) (THead k0 x x0) (pr0_comp u2 x 
-H26 t4 x0 H28 k0) (pr0_comp u3 x H27 t6 x0 H29 k0))))) (H23 t5 (tlt_head_dx 
-k0 u0 t5) t4 H24 t6 H15))))) (H23 u0 (tlt_head_sx k0 u0 t5) u2 H25 u3 
-H14))))) t3 (sym_eq T t3 t5 H22))) u1 (sym_eq T u1 u0 H21))) k (sym_eq K k k0 
-H20))) H19)) H18)))]) in (H17 (refl_equal T (THead k0 u0 t5))))))) t2 H13)) t 
-H11 H12 H9 H10))) | (pr0_beta u v1 v2 H9 t5 t6 H10) \Rightarrow (\lambda 
-(H11: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) t)).(\lambda 
-(H12: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Flat Appl) v1 
-(THead (Bind Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t6) 
-t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
-k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead 
-(Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: 
-T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 
-t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 v1 
-v2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: 
-T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t5)) H11) in (let H17 \def (match H16 in eq return (\lambda (t7: T).(\lambda 
-(_: (eq ? ? t7)).((eq T t7 (THead (Flat Appl) v1 (THead (Bind Abst) u t5))) 
-\to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 
-(THead (Bind Abbr) v2 t6) t8)))))) with [refl_equal \Rightarrow (\lambda 
-(H17: (eq T (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t5)))).(let H18 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 
-| (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 
-(THead (Bind Abst) u t5)) H17) in ((let H19 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | 
-(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H17) in ((let H20 \def 
-(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with 
-[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
-\Rightarrow k0])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t5)) H17) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T u1 v1) \to ((eq T 
-t3 (THead (Bind Abst) u t5)) \to (ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 
-t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7)))))) (\lambda 
-(H21: (eq T u1 v1)).(eq_ind T v1 (\lambda (_: T).((eq T t3 (THead (Bind Abst) 
-u t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 t4) t8)) 
-(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t6) t8))))) (\lambda (H22: (eq T 
-t3 (THead (Bind Abst) u t5))).(eq_ind T (THead (Bind Abst) u t5) (\lambda (_: 
-T).(ex2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 t4) t8)) (\lambda (t8: 
-T).(pr0 (THead (Bind Abbr) v2 t6) t8)))) (let H23 \def (eq_ind_r T t (\lambda 
-(t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to 
-(\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t5)) H11) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) 
-H8 (THead (Bind Abst) u t5) H22) in (let H25 \def (match H24 in pr0 return 
-(\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead 
-(Bind Abst) u t5)) \to ((eq T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead 
-(Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) 
-t9)))))))) with [(pr0_refl t7) \Rightarrow (\lambda (H25: (eq T t7 (THead 
-(Bind Abst) u t5))).(\lambda (H26: (eq T t7 t4)).(eq_ind T (THead (Bind Abst) 
-u t5) (\lambda (t8: T).((eq T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead 
-(Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) 
-t9))))) (\lambda (H27: (eq T (THead (Bind Abst) u t5) t4)).(eq_ind T (THead 
-(Bind Abst) u t5) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat 
-Appl) u2 t8) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))) (let 
-H28 \def (eq_ind T u1 (\lambda (t8: T).(pr0 t8 u2)) H7 v1 H21) in (ex2_ind T 
-(\lambda (t8: T).(pr0 u2 t8)) (\lambda (t8: T).(pr0 v2 t8)) (ex2 T (\lambda 
-(t8: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda 
-(t8: T).(pr0 (THead (Bind Abbr) v2 t6) t8))) (\lambda (x: T).(\lambda (H29: 
-(pr0 u2 x)).(\lambda (H30: (pr0 v2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 
+t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match e in T return 
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 
+| (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in 
+((let H19 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
+T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ 
+t7) \Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in (\lambda (H20: 
+(eq T u1 u0)).(\lambda (H21: (eq K k k0)).(let H22 \def (eq_ind_r T t 
+(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) 
+\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead k0 u0 t5) H11) in (eq_ind_r 
+K k0 (\lambda (k1: K).(ex2 T (\lambda (t7: T).(pr0 (THead k1 u2 t4) t7)) 
+(\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) (let H23 \def (eq_ind T u1 
+(\lambda (t7: T).(pr0 t7 u2)) H7 u0 H20) in (let H24 \def (eq_ind T t3 
+(\lambda (t7: T).(pr0 t7 t4)) H8 t5 H19) in (ex2_ind T (\lambda (t7: T).(pr0 
+t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead k0 
+u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7))) (\lambda (x: 
+T).(\lambda (H25: (pr0 t4 x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T (\lambda 
+(t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: 
+T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7))) 
+(\lambda (x0: T).(\lambda (H27: (pr0 u2 x0)).(\lambda (H28: (pr0 u3 
+x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: 
+T).(pr0 (THead k0 u3 t6) t7)) (THead k0 x0 x) (pr0_comp u2 x0 H27 t4 x H25 
+k0) (pr0_comp u3 x0 H28 t6 x H26 k0))))) (H22 u0 (tlt_head_sx k0 u0 t5) u2 
+H23 u3 H14))))) (H22 t5 (tlt_head_dx k0 u0 t5) t4 H24 t6 H15)))) k H21))))) 
+H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u v1 v2 H9 t5 t6 H10) 
+\Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u 
+t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind 
+Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda 
+(H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind Abbr) v2 
+t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: 
+(pr0 v1 v2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda 
+(t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat Appl) v1 (THead (Bind 
+Abst) u t5)) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e in T 
+return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let H18 \def (f_equal T T 
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) 
+(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let 
+H19 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
+with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) 
+\Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u 
+t5)) H16) in (\lambda (H20: (eq T u1 v1)).(\lambda (H21: (eq K k (Flat 
+Appl))).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v 
+t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to 
+(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
+t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H11) in 
+(eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead 
+k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7)))) (let 
+H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 v1 H20) in (let H24 
+\def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (THead (Bind Abst) u t5) 
+H19) in (let H25 \def (match H24 in pr0 return (\lambda (t7: T).(\lambda (t8: 
+T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead (Bind Abst) u t5)) \to ((eq T 
+t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) 
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))))) with [(pr0_refl 
+t7) \Rightarrow (\lambda (H25: (eq T t7 (THead (Bind Abst) u t5))).(\lambda 
+(H26: (eq T t7 t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda (t8: T).((eq 
+T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) 
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) (\lambda (H27: (eq T 
+(THead (Bind Abst) u t5) t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda 
+(t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)) (\lambda 
+(t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))) (ex2_ind T (\lambda (t8: 
+T).(pr0 u2 t8)) (\lambda (t8: T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0 
 (THead (Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 
-(THead (Bind Abbr) v2 t6) t8)) (THead (Bind Abbr) x t6) (pr0_beta u u2 x H29 
-t5 t6 H15) (pr0_comp v2 x H30 t6 t6 (pr0_refl t6) (Bind Abbr)))))) (H23 v1 
-(tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 H28 v2 H14))) t4 
+(THead (Bind Abbr) v2 t6) t8))) (\lambda (x: T).(\lambda (H28: (pr0 u2 
+x)).(\lambda (H29: (pr0 v2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 (THead 
+(Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 (THead 
+(Bind Abbr) v2 t6) t8)) (THead (Bind Abbr) x t6) (pr0_beta u u2 x H28 t5 t6 
+H15) (pr0_comp v2 x H29 t6 t6 (pr0_refl t6) (Bind Abbr)))))) (H22 v1 
+(tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 H23 v2 H14)) t4 
 H27)) t7 (sym_eq T t7 (THead (Bind Abst) u t5) H25) H26))) | (pr0_comp u0 u3 
 H25 t7 t8 H26 k0) \Rightarrow (\lambda (H27: (eq T (THead k0 u0 t7) (THead 
 (Bind Abst) u t5))).(\lambda (H28: (eq T (THead k0 u3 t8) t4)).((let H29 \def 
@@ -573,22 +553,21 @@ t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10))))))))
 t4)).(eq_ind T (THead (Bind Abst) u3 t8) (\lambda (t9: T).((pr0 u u3) \to 
 ((pr0 t5 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) 
 t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))) (\lambda (_: 
-(pr0 u u3)).(\lambda (H36: (pr0 t5 t8)).(let H37 \def (eq_ind T u1 (\lambda 
-(t9: T).(pr0 t9 u2)) H7 v1 H21) in (ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) 
-(\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) 
-u2 (THead (Bind Abst) u3 t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) 
-v2 t6) t9))) (\lambda (x: T).(\lambda (H38: (pr0 u2 x)).(\lambda (H39: (pr0 
-v2 x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9)) 
-(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) 
-t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))) (\lambda (x0: 
-T).(\lambda (H40: (pr0 t8 x0)).(\lambda (H41: (pr0 t6 x0)).(ex_intro2 T 
+(pr0 u u3)).(\lambda (H36: (pr0 t5 t8)).(ex2_ind T (\lambda (t9: T).(pr0 t8 
+t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat 
+Appl) u2 (THead (Bind Abst) u3 t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind 
+Abbr) v2 t6) t9))) (\lambda (x: T).(\lambda (H37: (pr0 t8 x)).(\lambda (H38: 
+(pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 
+t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 
+t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))) (\lambda (x0: 
+T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: (pr0 v2 x0)).(ex_intro2 T 
 (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9)) 
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)) (THead (Bind Abbr) x x0) 
-(pr0_beta u3 u2 x H38 t8 x0 H40) (pr0_comp v2 x H39 t6 x0 H41 (Bind 
-Abbr)))))) (H23 t5 (tlt_trans (THead (Bind Abst) u t5) t5 (THead (Flat Appl) 
-v1 (THead (Bind Abst) u t5)) (tlt_head_dx (Bind Abst) u t5) (tlt_head_dx 
-(Flat Appl) v1 (THead (Bind Abst) u t5))) t8 H36 t6 H15))))) (H23 v1 
-(tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 H37 v2 H14))))) t4 
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)) (THead (Bind Abbr) x0 x) 
+(pr0_beta u3 u2 x0 H39 t8 x H37) (pr0_comp v2 x0 H40 t6 x H38 (Bind 
+Abbr)))))) (H22 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 
+H23 v2 H14))))) (H22 t5 (tlt_trans (THead (Bind Abst) u t5) t5 (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t5)) (tlt_head_dx (Bind Abst) u t5) 
+(tlt_head_dx (Flat Appl) v1 (THead (Bind Abst) u t5))) t8 H36 t6 H15)))) t4 
 H34)) t7 (sym_eq T t7 t5 H33))) u0 (sym_eq T u0 u H32))) k0 (sym_eq K k0 
 (Bind Abst) H31))) H30)) H29)) H28 H25 H26))) | (pr0_beta u0 v0 v3 H25 t7 t8 
 H26) \Rightarrow (\lambda (H27: (eq T (THead (Flat Appl) v0 (THead (Bind 
@@ -677,54 +656,41 @@ K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
 t4) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 
 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) H28)) H27 
 H25)))]) in (H25 (refl_equal T (THead (Bind Abst) u t5)) (refl_equal T 
-t4))))) t3 (sym_eq T t3 (THead (Bind Abst) u t5) H22))) u1 (sym_eq T u1 v1 
-H21))) k (sym_eq K k (Flat Appl) H20))) H19)) H18)))]) in (H17 (refl_equal T 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t5)))))))) t2 H13)) t H11 H12 H9 
-H10))) | (pr0_upsilon b H9 v1 v2 H10 u0 u3 H11 t5 t6 H12) \Rightarrow 
-(\lambda (H13: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) 
-t)).(\lambda (H14: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O 
-v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) 
-(\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O 
-v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to 
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda 
-(t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b) u3 (THead 
-(Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u3 (THead 
-(Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b Abst)) \to 
+t4))))) k H21))))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon 
+b H9 v1 v2 H10 u0 u3 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead 
+(Flat Appl) v1 (THead (Bind b) u0 t5)) t)).(\lambda (H14: (eq T (THead (Bind 
+b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat 
+Appl) v1 (THead (Bind b) u0 t5)) (\lambda (_: T).((eq T (THead (Bind b) u3 
+(THead (Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to 
 ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda 
-(H16: (not (eq B b Abst))).(\lambda (H17: (pr0 v1 v2)).(\lambda (H18: (pr0 u0 
-u3)).(\lambda (H19: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: 
-T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b) u0 
-t5)) H13) in (let H21 \def (match H20 in eq return (\lambda (t7: T).(\lambda 
-(_: (eq ? ? t7)).((eq T t7 (THead (Flat Appl) v1 (THead (Bind b) u0 t5))) \to 
-(ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t8)))))) with 
-[refl_equal \Rightarrow (\lambda (H21: (eq T (THead k u1 t3) (THead (Flat 
-Appl) v1 (THead (Bind b) u0 t5)))).(let H22 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | 
-(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) 
-(THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H21) in ((let H23 \def (f_equal 
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) 
-(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H21) in ((let 
-H24 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) 
-with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
-\Rightarrow k0])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 
-t5)) H21) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T u1 v1) \to ((eq T 
-t3 (THead (Bind b) u0 t5)) \to (ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) 
-t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) 
-O v2) t6)) t7)))))) (\lambda (H25: (eq T u1 v1)).(eq_ind T v1 (\lambda (_: 
-T).((eq T t3 (THead (Bind b) u0 t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
-(Flat Appl) u2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Bind b) u3 (THead (Flat 
-Appl) (lift (S O) O v2) t6)) t8))))) (\lambda (H26: (eq T t3 (THead (Bind b) 
-u0 t5))).(eq_ind T (THead (Bind b) u0 t5) (\lambda (_: T).(ex2 T (\lambda 
-(t8: T).(pr0 (THead (Flat Appl) u2 t4) t8)) (\lambda (t8: T).(pr0 (THead 
-(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t8)))) (let H27 \def 
-(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: 
-T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: 
-T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) 
-v1 (THead (Bind b) u0 t5)) H13) in (let H28 \def (eq_ind T t3 (\lambda (t7: 
-T).(pr0 t7 t4)) H8 (THead (Bind b) u0 t5) H26) in (let H29 \def (match H28 in 
+T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda 
+(H15: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) 
+t2)).(eq_ind T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) 
+(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to 
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda 
+(t8: T).(pr0 t7 t8)))))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17: 
+(pr0 v1 v2)).(\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 t6)).(let H20 
+\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead 
+(Flat Appl) v1 (THead (Bind b) u0 t5)) H13) in (let H21 \def (f_equal T K 
+(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) 
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
+(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let 
+H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
+with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) 
+\Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 
+t5)) H20) in ((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return 
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 
+| (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 
+(THead (Bind b) u0 t5)) H20) in (\lambda (H24: (eq T u1 v1)).(\lambda (H25: 
+(eq K k (Flat Appl))).(let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall 
+(v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: 
+T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: 
+T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H13) 
+in (eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 
+(THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u3 (THead (Flat 
+Appl) (lift (S O) O v2) t6)) t7)))) (let H27 \def (eq_ind T u1 (\lambda (t7: 
+T).(pr0 t7 u2)) H7 v1 H24) in (let H28 \def (eq_ind T t3 (\lambda (t7: 
+T).(pr0 t7 t4)) H8 (THead (Bind b) u0 t5) H23) in (let H29 \def (match H28 in 
 pr0 return (\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T 
 t7 (THead (Bind b) u0 t5)) \to ((eq T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 
 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 
@@ -736,66 +702,64 @@ T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))
 (\lambda (H31: (eq T (THead (Bind b) u0 t5) t4)).(eq_ind T (THead (Bind b) u0 
 t5) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8) 
 t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) 
-O v2) t6)) t9)))) (let H32 \def (eq_ind T u1 (\lambda (t8: T).(pr0 t8 u2)) H7 
-v1 H25) in (ex2_ind T (\lambda (t8: T).(pr0 u2 t8)) (\lambda (t8: T).(pr0 v2 
-t8)) (ex2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u0 
-t5)) t8)) (\lambda (t8: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift 
-(S O) O v2) t6)) t8))) (\lambda (x: T).(\lambda (H33: (pr0 u2 x)).(\lambda 
-(H34: (pr0 v2 x)).(pr0_confluence__pr0_cong_upsilon_refl b H16 u0 u3 H18 t5 
-t6 H19 u2 v2 x H33 H34)))) (H27 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind 
-b) u0 t5)) u2 H32 v2 H17))) t4 H31)) t7 (sym_eq T t7 (THead (Bind b) u0 t5) 
-H29) H30))) | (pr0_comp u4 u5 H29 t7 t8 H30 k0) \Rightarrow (\lambda (H31: 
-(eq T (THead k0 u4 t7) (THead (Bind b) u0 t5))).(\lambda (H32: (eq T (THead 
-k0 u5 t8) t4)).((let H33 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | (TLRef _) 
-\Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7) (THead 
-(Bind b) u0 t5) H31) in ((let H34 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 | (TLRef _) 
-\Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) (THead 
-(Bind b) u0 t5) H31) in ((let H35 \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 u4 t7) (THead 
-(Bind b) u0 t5) H31) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4 u0) \to 
-((eq T t7 t5) \to ((eq T (THead k1 u5 t8) t4) \to ((pr0 u4 u5) \to ((pr0 t7 
-t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda 
-(t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) 
-t9))))))))) (\lambda (H36: (eq T u4 u0)).(eq_ind T u0 (\lambda (t9: T).((eq T 
-t7 t5) \to ((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 t9 u5) \to ((pr0 t7 
-t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) 
-(\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O 
-v2) t6)) t10)))))))) (\lambda (H37: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: 
-T).((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8) \to 
-(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: 
-T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) 
+O v2) t6)) t9)))) (ex2_ind T (\lambda (t8: T).(pr0 u2 t8)) (\lambda (t8: 
+T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead 
+(Bind b) u0 t5)) t8)) (\lambda (t8: T).(pr0 (THead (Bind b) u3 (THead (Flat 
+Appl) (lift (S O) O v2) t6)) t8))) (\lambda (x: T).(\lambda (H32: (pr0 u2 
+x)).(\lambda (H33: (pr0 v2 x)).(pr0_confluence__pr0_cong_upsilon_refl b H16 
+u0 u3 H18 t5 t6 H19 u2 v2 x H32 H33)))) (H26 v1 (tlt_head_sx (Flat Appl) v1 
+(THead (Bind b) u0 t5)) u2 H27 v2 H17)) t4 H31)) t7 (sym_eq T t7 (THead (Bind 
+b) u0 t5) H29) H30))) | (pr0_comp u4 u5 H29 t7 t8 H30 k0) \Rightarrow 
+(\lambda (H31: (eq T (THead k0 u4 t7) (THead (Bind b) u0 t5))).(\lambda (H32: 
+(eq T (THead k0 u5 t8) t4)).((let H33 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | 
+(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7) 
+(THead (Bind b) u0 t5) H31) in ((let H34 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 | 
+(TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) 
+(THead (Bind b) u0 t5) H31) in ((let H35 \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 u4 t7) 
+(THead (Bind b) u0 t5) H31) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4 
+u0) \to ((eq T t7 t5) \to ((eq T (THead k1 u5 t8) t4) \to ((pr0 u4 u5) \to 
+((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) 
+(\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) 
+t6)) t9))))))))) (\lambda (H36: (eq T u4 u0)).(eq_ind T u0 (\lambda (t9: 
+T).((eq T t7 t5) \to ((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 t9 u5) \to 
+((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) 
+t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S 
+O) O v2) t6)) t10)))))))) (\lambda (H37: (eq T t7 t5)).(eq_ind T t5 (\lambda 
+(t9: T).((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8) 
+\to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda 
+(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) 
 t10))))))) (\lambda (H38: (eq T (THead (Bind b) u5 t8) t4)).(eq_ind T (THead 
 (Bind b) u5 t8) (\lambda (t9: T).((pr0 u0 u5) \to ((pr0 t5 t8) \to (ex2 T 
 (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0 
 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t10)))))) 
-(\lambda (H39: (pr0 u0 u5)).(\lambda (H40: (pr0 t5 t8)).(let H41 \def (eq_ind 
-T u1 (\lambda (t9: T).(pr0 t9 u2)) H7 v1 H25) in (ex2_ind T (\lambda (t9: 
-T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0 
-(THead (Flat Appl) u2 (THead (Bind b) u5 t8)) t9)) (\lambda (t9: T).(pr0 
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda 
-(x: T).(\lambda (H42: (pr0 u2 x)).(\lambda (H43: (pr0 v2 x)).(ex2_ind T 
-(\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda 
-(t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8)) t9)) (\lambda (t9: 
+(\lambda (H39: (pr0 u0 u5)).(\lambda (H40: (pr0 t5 t8)).(ex2_ind T (\lambda 
+(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8)) t9)) (\lambda (t9: 
 T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))) 
-(\lambda (x0: T).(\lambda (H44: (pr0 t8 x0)).(\lambda (H45: (pr0 t6 
-x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u3 t9)) 
+(\lambda (x: T).(\lambda (H41: (pr0 t8 x)).(\lambda (H42: (pr0 t6 
+x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u3 t9)) 
 (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8)) 
 t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) 
-O v2) t6)) t9))) (\lambda (x1: T).(\lambda (H46: (pr0 u5 x1)).(\lambda (H47: 
-(pr0 u3 x1)).(pr0_confluence__pr0_cong_upsilon_cong b H16 u2 v2 x H42 H43 t8 
-t6 x0 H44 H45 u5 u3 x1 H46 H47)))) (H27 u0 (tlt_trans (THead (Bind b) u0 t5) 
-u0 (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) (tlt_head_sx (Bind b) u0 t5) 
-(tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t5))) u5 H39 u3 H18))))) (H27 
-t5 (tlt_trans (THead (Bind b) u0 t5) t5 (THead (Flat Appl) v1 (THead (Bind b) 
-u0 t5)) (tlt_head_dx (Bind b) u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind 
-b) u0 t5))) t8 H40 t6 H19))))) (H27 v1 (tlt_head_sx (Flat Appl) v1 (THead 
-(Bind b) u0 t5)) u2 H41 v2 H17))))) t4 H38)) t7 (sym_eq T t7 t5 H37))) u4 
-(sym_eq T u4 u0 H36))) k0 (sym_eq K k0 (Bind b) H35))) H34)) H33)) H32 H29 
-H30))) | (pr0_beta u v0 v3 H29 t7 t8 H30) \Rightarrow (\lambda (H31: (eq T 
-(THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (THead (Bind b) u0 
+O v2) t6)) t9))) (\lambda (x0: T).(\lambda (H43: (pr0 u5 x0)).(\lambda (H44: 
+(pr0 u3 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 
+v2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 
+t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift 
+(S O) O v2) t6)) t9))) (\lambda (x1: T).(\lambda (H45: (pr0 u2 x1)).(\lambda 
+(H46: (pr0 v2 x1)).(pr0_confluence__pr0_cong_upsilon_cong b H16 u2 v2 x1 H45 
+H46 t8 t6 x H41 H42 u5 u3 x0 H43 H44)))) (H26 v1 (tlt_head_sx (Flat Appl) v1 
+(THead (Bind b) u0 t5)) u2 H27 v2 H17))))) (H26 u0 (tlt_trans (THead (Bind b) 
+u0 t5) u0 (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) (tlt_head_sx (Bind b) 
+u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t5))) u5 H39 u3 
+H18))))) (H26 t5 (tlt_trans (THead (Bind b) u0 t5) t5 (THead (Flat Appl) v1 
+(THead (Bind b) u0 t5)) (tlt_head_dx (Bind b) u0 t5) (tlt_head_dx (Flat Appl) 
+v1 (THead (Bind b) u0 t5))) t8 H40 t6 H19)))) t4 H38)) t7 (sym_eq T t7 t5 
+H37))) u4 (sym_eq T u4 u0 H36))) k0 (sym_eq K k0 (Bind b) H35))) H34)) H33)) 
+H32 H29 H30))) | (pr0_beta u v0 v3 H29 t7 t8 H30) \Rightarrow (\lambda (H31: 
+(eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (THead (Bind b) u0 
 t5))).(\lambda (H32: (eq T (THead (Bind Abbr) v3 t8) t4)).((let H33 \def 
 (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (\lambda (e: 
 T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
@@ -853,31 +817,30 @@ u5)).(\lambda (H41: (pr0 t5 t8)).(\lambda (H42: (subst0 O u5 t8 w)).(let H43
 \def (eq_ind_r B b (\lambda (b0: B).(\forall (v: T).((tlt v (THead (Flat 
 Appl) v1 (THead (Bind b0) u0 t5))) \to (\forall (t9: T).((pr0 v t9) \to 
 (\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) 
-(\lambda (t11: T).(pr0 t10 t11)))))))))) H27 Abbr H36) in (let H44 \def 
-(eq_ind_r B b (\lambda (b0: B).(eq T t3 (THead (Bind b0) u0 t5))) H26 Abbr 
+(\lambda (t11: T).(pr0 t10 t11)))))))))) H26 Abbr H36) in (let H44 \def 
+(eq_ind_r B b (\lambda (b0: B).(eq T t3 (THead (Bind b0) u0 t5))) H23 Abbr 
 H36) in (let H45 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) 
-H16 Abbr H36) in (let H46 \def (eq_ind T u1 (\lambda (t9: T).(pr0 t9 u2)) H7 
-v1 H25) in (ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 
-t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 
-w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift 
-(S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H47: (pr0 u2 x)).(\lambda 
-(H48: (pr0 v2 x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: 
+H16 Abbr H36) in (ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: 
 T).(pr0 t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead 
 (Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead 
-(Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x0: T).(\lambda (H49: (pr0 
-t8 x0)).(\lambda (H50: (pr0 t6 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) 
+(Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H46: (pr0 
+t8 x)).(\lambda (H47: (pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) 
 (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) 
 u2 (THead (Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 
-(THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x1: T).(\lambda 
-(H51: (pr0 u5 x1)).(\lambda (H52: (pr0 u3 
-x1)).(pr0_confluence__pr0_cong_upsilon_delta H45 u5 t8 w H42 u2 v2 x H47 H48 
-t6 x0 H49 H50 u3 x1 H51 H52)))) (H43 u0 (tlt_trans (THead (Bind Abbr) u0 t5) 
-u0 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_sx (Bind Abbr) 
-u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) u5 H40 u3 
-H18))))) (H43 t5 (tlt_trans (THead (Bind Abbr) u0 t5) t5 (THead (Flat Appl) 
-v1 (THead (Bind Abbr) u0 t5)) (tlt_head_dx (Bind Abbr) u0 t5) (tlt_head_dx 
-(Flat Appl) v1 (THead (Bind Abbr) u0 t5))) t8 H41 t6 H19))))) (H43 v1 
-(tlt_head_sx (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) u2 H46 v2 H17))))))))) 
+(THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x0: T).(\lambda 
+(H48: (pr0 u5 x0)).(\lambda (H49: (pr0 u3 x0)).(ex2_ind T (\lambda (t9: 
+T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0 
+(THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 
+(THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))) 
+(\lambda (x1: T).(\lambda (H50: (pr0 u2 x1)).(\lambda (H51: (pr0 v2 
+x1)).(pr0_confluence__pr0_cong_upsilon_delta H45 u5 t8 w H42 u2 v2 x1 H50 H51 
+t6 x H46 H47 u3 x0 H48 H49)))) (H43 v1 (tlt_head_sx (Flat Appl) v1 (THead 
+(Bind Abbr) u0 t5)) u2 H27 v2 H17))))) (H43 u0 (tlt_trans (THead (Bind Abbr) 
+u0 t5) u0 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_sx (Bind 
+Abbr) u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) u5 H40 
+u3 H18))))) (H43 t5 (tlt_trans (THead (Bind Abbr) u0 t5) t5 (THead (Flat 
+Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_dx (Bind Abbr) u0 t5) 
+(tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) t8 H41 t6 H19)))))))) 
 t4 H39)) t7 (sym_eq T t7 t5 H38))) u4 (sym_eq T u4 u0 H37))) b H36)) H35)) 
 H34)) H33 H29 H30 H31))) | (pr0_zeta b0 H29 t7 t8 H30 u) \Rightarrow (\lambda 
 (H31: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 
@@ -922,8 +885,8 @@ Abst))).(\lambda (H40: (pr0 t7 t4)).(let H41 \def (eq_ind_r T t5 (\lambda
 (t9: T).(\forall (v: T).((tlt v (THead (Flat Appl) v1 (THead (Bind b) u0 
 t9))) \to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) 
 \to (ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 
-t12)))))))))) H27 (lift (S O) O t7) H37) in (let H42 \def (eq_ind_r T t5 
-(\lambda (t9: T).(eq T t3 (THead (Bind b) u0 t9))) H26 (lift (S O) O t7) H37) 
+t12)))))))))) H26 (lift (S O) O t7) H37) in (let H42 \def (eq_ind_r T t5 
+(\lambda (t9: T).(eq T t3 (THead (Bind b) u0 t9))) H23 (lift (S O) O t7) H37) 
 in (let H43 \def (eq_ind_r T t5 (\lambda (t9: T).(pr0 t9 t6)) H19 (lift (S O) 
 O t7) H37) in (ex2_ind T (\lambda (t9: T).(eq T t6 (lift (S O) O t9))) 
 (\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) 
@@ -932,136 +895,112 @@ u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift
 x))).(\lambda (H45: (pr0 t7 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: 
 T).(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda 
 (t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t9)) 
-t10)))) (let H46 \def (eq_ind T u1 (\lambda (t9: T).(pr0 t9 u2)) H7 v1 H25) 
-in (ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) 
+t10)))) (ex2_ind T (\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t4 t9)) 
 (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: 
 T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O 
-x))) t9))) (\lambda (x0: T).(\lambda (H47: (pr0 u2 x0)).(\lambda (H48: (pr0 
-v2 x0)).(ex2_ind T (\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t4 t9)) 
+x))) t9))) (\lambda (x0: T).(\lambda (H46: (pr0 x x0)).(\lambda (H47: (pr0 t4 
+x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) 
 (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: 
 T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O 
-x))) t9))) (\lambda (x1: T).(\lambda (H49: (pr0 x x1)).(\lambda (H50: (pr0 t4 
-x1)).(pr0_confluence__pr0_cong_upsilon_zeta b H39 u0 u3 H18 u2 v2 x0 H47 H48 
-x t4 x1 H49 H50)))) (H41 t7 (tlt_trans (THead (Bind b) u0 (lift (S O) O t7)) 
-t7 (THead (Flat Appl) v1 (THead (Bind b) u0 (lift (S O) O t7))) (lift_tlt_dx 
-(Bind b) u0 t7 (S O) O) (tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 (lift 
-(S O) O t7)))) x H45 t4 H40))))) (H41 v1 (tlt_head_sx (Flat Appl) v1 (THead 
-(Bind b) u0 (lift (S O) O t7))) u2 H46 v2 H17))) t6 H44)))) (pr0_gen_lift t7 
-t6 (S O) O H43))))))) t8 (sym_eq T t8 t4 H38))) t5 H37)) u (sym_eq T u u0 
-H36))) b0 (sym_eq B b0 b H35))) H34)) H33)) H32 H29 H30))) | (pr0_epsilon t7 
-t8 H29 u) \Rightarrow (\lambda (H30: (eq T (THead (Flat Cast) u t7) (THead 
-(Bind b) u0 t5))).(\lambda (H31: (eq T t8 t4)).((let H32 \def (eq_ind T 
-(THead (Flat Cast) u t7) (\lambda (e: T).(match e in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k0 _ _) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with 
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
-b) u0 t5) H30) in (False_ind ((eq T t8 t4) \to ((pr0 t7 t8) \to (ex2 T 
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))) H32)) 
-H31 H29)))]) in (H29 (refl_equal T (THead (Bind b) u0 t5)) (refl_equal T 
-t4))))) t3 (sym_eq T t3 (THead (Bind b) u0 t5) H26))) u1 (sym_eq T u1 v1 
-H25))) k (sym_eq K k (Flat Appl) H24))) H23)) H22)))]) in (H21 (refl_equal T 
-(THead (Flat Appl) v1 (THead (Bind b) u0 t5)))))))))) t2 H15)) t H13 H14 H9 
-H10 H11 H12))) | (pr0_delta u0 u3 H9 t5 t6 H10 w H11) \Rightarrow (\lambda 
-(H12: (eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H13: (eq T (THead (Bind 
-Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T 
-(THead (Bind Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O 
-u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: 
+x))) t9))) (\lambda (x1: T).(\lambda (H48: (pr0 u2 x1)).(\lambda (H49: (pr0 
+v2 x1)).(pr0_confluence__pr0_cong_upsilon_zeta b H39 u0 u3 H18 u2 v2 x1 H48 
+H49 x t4 x0 H46 H47)))) (H41 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b) 
+u0 (lift (S O) O t7))) u2 H27 v2 H17))))) (H41 t7 (tlt_trans (THead (Bind b) 
+u0 (lift (S O) O t7)) t7 (THead (Flat Appl) v1 (THead (Bind b) u0 (lift (S O) 
+O t7))) (lift_tlt_dx (Bind b) u0 t7 (S O) O) (tlt_head_dx (Flat Appl) v1 
+(THead (Bind b) u0 (lift (S O) O t7)))) x H45 t4 H40)) t6 H44)))) 
+(pr0_gen_lift t7 t6 (S O) O H43))))))) t8 (sym_eq T t8 t4 H38))) t5 H37)) u 
+(sym_eq T u u0 H36))) b0 (sym_eq B b0 b H35))) H34)) H33)) H32 H29 H30))) | 
+(pr0_epsilon t7 t8 H29 u) \Rightarrow (\lambda (H30: (eq T (THead (Flat Cast) 
+u t7) (THead (Bind b) u0 t5))).(\lambda (H31: (eq T t8 t4)).((let H32 \def 
+(eq_ind T (THead (Flat Cast) u t7) (\lambda (e: T).(match e in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda 
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
+True])])) I (THead (Bind b) u0 t5) H30) in (False_ind ((eq T t8 t4) \to ((pr0 
+t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) 
+(\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) 
+t6)) t9))))) H32)) H31 H29)))]) in (H29 (refl_equal T (THead (Bind b) u0 t5)) 
+(refl_equal T t4))))) k H25))))) H22)) H21))))))) t2 H15)) t H13 H14 H9 H10 
+H11 H12))) | (pr0_delta u0 u3 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12: 
+(eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr) 
+u3 w) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T (THead 
+(Bind Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 
+w) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: 
 T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind Abbr) u3 w) 
 t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to 
 ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
 k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (H15: (pr0 u0 
 u3)).(\lambda (H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u3 t6 w)).(let H18 
 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead 
-(Bind Abbr) u0 t5) H12) in (let H19 \def (match H18 in eq return (\lambda 
-(t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind Abbr) u0 t5)) \to 
-(ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 
-(THead (Bind Abbr) u3 w) t8)))))) with [refl_equal \Rightarrow (\lambda (H19: 
-(eq T (THead k u1 t3) (THead (Bind Abbr) u0 t5))).(let H20 \def (f_equal T T 
+(Bind Abbr) u0 t5) H12) in (let H19 \def (f_equal T K (\lambda (e: T).(match 
+e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind 
+Abbr) u0 t5) H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in 
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
+\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead 
+(Bind Abbr) u0 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match 
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) 
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead 
+(Bind Abbr) u0 t5) H18) in (\lambda (H22: (eq T u1 u0)).(\lambda (H23: (eq K 
+k (Bind Abbr))).(let H24 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: 
+T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v 
+t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
+t10)))))))))) H (THead (Bind Abbr) u0 t5) H12) in (eq_ind_r K (Bind Abbr) 
+(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda 
+(t7: T).(pr0 (THead (Bind Abbr) u3 w) t7)))) (let H25 \def (eq_ind T u1 
+(\lambda (t7: T).(pr0 t7 u2)) H7 u0 H22) in (let H26 \def (eq_ind T t3 
+(\lambda (t7: T).(pr0 t7 t4)) H8 t5 H21) in (ex2_ind T (\lambda (t7: T).(pr0 
+t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead 
+(Bind Abbr) u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7))) 
+(\lambda (x: T).(\lambda (H27: (pr0 t4 x)).(\lambda (H28: (pr0 t6 
+x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) 
+(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 t4) t7)) (\lambda (t7: 
+T).(pr0 (THead (Bind Abbr) u3 w) t7))) (\lambda (x0: T).(\lambda (H29: (pr0 
+u2 x0)).(\lambda (H30: (pr0 u3 x0)).(pr0_confluence__pr0_cong_delta u3 t6 w 
+H17 u2 x0 H29 H30 t4 x H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 
+H25 u3 H15))))) (H24 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 H16)))) k 
+H23))))) H20)) H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 
+t6 H10 u) \Rightarrow (\lambda (H11: (eq T (THead (Bind b) u (lift (S O) O 
+t5)) t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O) 
+O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 
+t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: 
+T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: 
+T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 
+(THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (not 
+(eq B b Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t 
+(\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Bind b) u (lift (S O) 
+O t5)) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e in T 
+return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind 
+b) u (lift (S O) O t5)) H16) in ((let H18 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | 
+(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) 
+(THead (Bind b) u (lift (S O) O t5)) H16) in ((let H19 \def (f_equal T T 
 (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
 \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) 
-(THead k u1 t3) (THead (Bind Abbr) u0 t5) H19) in ((let H21 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) 
-(THead k u1 t3) (THead (Bind Abbr) u0 t5) H19) in ((let H22 \def (f_equal T K 
-(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) 
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
-(THead k u1 t3) (THead (Bind Abbr) u0 t5) H19) in (eq_ind K (Bind Abbr) 
-(\lambda (k0: K).((eq T u1 u0) \to ((eq T t3 t5) \to (ex2 T (\lambda (t7: 
-T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) 
-t7)))))) (\lambda (H23: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t3 
-t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 t4) t8)) (\lambda 
-(t8: T).(pr0 (THead (Bind Abbr) u3 w) t8))))) (\lambda (H24: (eq T t3 
-t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind 
-Abbr) u2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u3 w) t8)))) (let 
-H25 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to 
-(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T 
-(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H 
-(THead (Bind Abbr) u0 t5) H12) in (let H26 \def (eq_ind T t3 (\lambda (t7: 
-T).(pr0 t7 t4)) H8 t5 H24) in (let H27 \def (eq_ind T u1 (\lambda (t7: 
-T).(pr0 t7 u2)) H7 u0 H23) in (ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) 
-(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) 
-u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7))) (\lambda (x: 
-T).(\lambda (H28: (pr0 u2 x)).(\lambda (H29: (pr0 u3 x)).(ex2_ind T (\lambda 
-(t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: 
-T).(pr0 (THead (Bind Abbr) u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind 
-Abbr) u3 w) t7))) (\lambda (x0: T).(\lambda (H30: (pr0 t4 x0)).(\lambda (H31: 
-(pr0 t6 x0)).(pr0_confluence__pr0_cong_delta u3 t6 w H17 u2 x H28 H29 t4 x0 
-H30 H31)))) (H25 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 H16))))) (H25 
-u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 H27 u3 H15))))) t3 (sym_eq T t3 t5 
-H24))) u1 (sym_eq T u1 u0 H23))) k (sym_eq K k (Bind Abbr) H22))) H21)) 
-H20)))]) in (H19 (refl_equal T (THead (Bind Abbr) u0 t5)))))))) t2 H14)) t 
-H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 t6 H10 u) \Rightarrow (\lambda 
-(H11: (eq T (THead (Bind b) u (lift (S O) O t5)) t)).(\lambda (H12: (eq T t6 
-t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t5)) (\lambda (_: T).((eq T t6 
-t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda 
-(H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b Abst)) \to 
-((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda 
-(t8: T).(pr0 t2 t8)))))) (\lambda (H14: (not (eq B b Abst))).(\lambda (H15: 
-(pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 
-t3) t7)) H4 (THead (Bind b) u (lift (S O) O t5)) H11) in (let H17 \def (match 
-H16 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead 
-(Bind b) u (lift (S O) O t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 
-t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) with [refl_equal \Rightarrow 
-(\lambda (H17: (eq T (THead k u1 t3) (THead (Bind b) u (lift (S O) O 
-t5)))).(let H18 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 
-| (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead (Bind b) u (lift (S 
-O) O t5)) H17) in ((let H19 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
-\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead 
-(Bind b) u (lift (S O) O t5)) H17) in ((let H20 \def (f_equal T K (\lambda 
-(e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k 
-| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) 
-(THead (Bind b) u (lift (S O) O t5)) H17) in (eq_ind K (Bind b) (\lambda (k0: 
-K).((eq T u1 u) \to ((eq T t3 (lift (S O) O t5)) \to (ex2 T (\lambda (t7: 
-T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))))) (\lambda 
-(H21: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t3 (lift (S O) O t5)) 
-\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 t4) t8)) (\lambda (t8: 
-T).(pr0 t2 t8))))) (\lambda (H22: (eq T t3 (lift (S O) O t5))).(eq_ind T 
-(lift (S O) O t5) (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind 
-b) u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))) (let H23 \def (eq_ind_r T t 
+(THead k u1 t3) (THead (Bind b) u (lift (S O) O t5)) H16) in (\lambda (H20: 
+(eq T u1 u)).(\lambda (H21: (eq K k (Bind b))).(let H22 \def (eq_ind_r T t 
 (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) 
 \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
 (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind b) u (lift (S O) O 
-t5)) H11) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 
-(lift (S O) O t5) H22) in (ex2_ind T (\lambda (t7: T).(eq T t4 (lift (S O) O 
-t7))) (\lambda (t7: T).(pr0 t5 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind 
-b) u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H25: 
-(eq T t4 (lift (S O) O x))).(\lambda (H26: (pr0 t5 x)).(eq_ind_r T (lift (S 
-O) O x) (\lambda (t7: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 t7) 
-t8)) (\lambda (t8: T).(pr0 t2 t8)))) (let H27 \def (eq_ind T u1 (\lambda (t7: 
-T).(pr0 t7 u2)) H7 u H21) in (ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda 
-(t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift (S 
-O) O x)) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: T).(\lambda (H28: 
-(pr0 x x0)).(\lambda (H29: (pr0 t2 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 
-(THead (Bind b) u2 (lift (S O) O x)) t7)) (\lambda (t7: T).(pr0 t2 t7)) x0 
-(pr0_zeta b H14 x x0 H28 u2) H29)))) (H23 t5 (lift_tlt_dx (Bind b) u t5 (S O) 
-O) x H26 t2 H15))) t4 H25)))) (pr0_gen_lift t5 t4 (S O) O H24)))) t3 (sym_eq 
-T t3 (lift (S O) O t5) H22))) u1 (sym_eq T u1 u H21))) k (sym_eq K k (Bind b) 
-H20))) H19)) H18)))]) in (H17 (refl_equal T (THead (Bind b) u (lift (S O) O 
-t5)))))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_epsilon t5 t6 
+t5)) H11) in (eq_ind_r K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7: 
+T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H23 \def 
+(eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u H20) in (let H24 \def (eq_ind 
+T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (lift (S O) O t5) H19) in (ex2_ind T 
+(\lambda (t7: T).(eq T t4 (lift (S O) O t7))) (\lambda (t7: T).(pr0 t5 t7)) 
+(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 t4) t7)) (\lambda (t7: 
+T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H25: (eq T t4 (lift (S O) O 
+x))).(\lambda (H26: (pr0 t5 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t7: 
+T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 t7) t8)) (\lambda (t8: 
+T).(pr0 t2 t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: 
+T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O 
+x)) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: T).(\lambda (H27: (pr0 
+x x0)).(\lambda (H28: (pr0 t2 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead 
+(Bind b) u2 (lift (S O) O x)) t7)) (\lambda (t7: T).(pr0 t2 t7)) x0 (pr0_zeta 
+b H14 x x0 H27 u2) H28)))) (H22 t5 (lift_tlt_dx (Bind b) u t5 (S O) O) x H26 
+t2 H15)) t4 H25)))) (pr0_gen_lift t5 t4 (S O) O H24)))) k H21))))) H18)) 
+H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_epsilon t5 t6 
 H9 u) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u t5) t)).(\lambda 
 (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T 
 t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) 
@@ -1069,218 +1008,198 @@ t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8))
 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 
 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H13: (pr0 t5 t2)).(let 
 H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead 
-(Flat Cast) u t5) H10) in (let H15 \def (match H14 in eq return (\lambda (t7: 
-T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Cast) u t5)) \to (ex2 T 
-(\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) 
-with [refl_equal \Rightarrow (\lambda (H15: (eq T (THead k u1 t3) (THead 
-(Flat Cast) u t5))).(let H16 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) 
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead 
-(Flat Cast) u t5) H15) in ((let H17 \def (f_equal T T (\lambda (e: T).(match 
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
-\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead 
-(Flat Cast) u t5) H15) in ((let H18 \def (f_equal T K (\lambda (e: T).(match 
-e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
+(Flat Cast) u t5) H10) in (let H15 \def (f_equal T K (\lambda (e: T).(match e 
+in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
 \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat 
-Cast) u t5) H15) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq T u1 u) \to 
-((eq T t3 t5) \to (ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda 
-(t7: T).(pr0 t2 t7)))))) (\lambda (H19: (eq T u1 u)).(eq_ind T u (\lambda (_: 
-T).((eq T t3 t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Flat Cast) u2 t4) 
-t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H20: (eq T t3 t5)).(eq_ind T 
-t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Flat Cast) u2 t4) 
-t8)) (\lambda (t8: T).(pr0 t2 t8)))) (let H21 \def (eq_ind_r T t (\lambda 
-(t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to 
-(\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u t5) H10) in 
-(let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 t5 H20) in (let 
-H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u H19) in (ex2_ind T 
-(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda 
-(t7: T).(pr0 (THead (Flat Cast) u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) 
-(\lambda (x: T).(\lambda (H24: (pr0 t4 x)).(\lambda (H25: (pr0 t2 
-x)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t4) t7)) 
-(\lambda (t7: T).(pr0 t2 t7)) x (pr0_epsilon t4 x H24 u2) H25)))) (H21 t5 
-(tlt_head_dx (Flat Cast) u t5) t4 H22 t2 H13))))) t3 (sym_eq T t3 t5 H20))) 
-u1 (sym_eq T u1 u H19))) k (sym_eq K k (Flat Cast) H18))) H17)) H16)))]) in 
-(H15 (refl_equal T (THead (Flat Cast) u t5)))))) t6 (sym_eq T t6 t2 H12))) t 
-H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t1 H6)) t H4 H5 
-H2 H3))) | (pr0_beta u v1 v2 H2 t3 t4 H3) \Rightarrow (\lambda (H4: (eq T 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t)).(\lambda (H5: (eq T 
-(THead (Bind Abbr) v2 t4) t1)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t3)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t4) t1) \to ((pr0 
-v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda 
-(t6: T).(pr0 t2 t6))))))) (\lambda (H6: (eq T (THead (Bind Abbr) v2 t4) 
-t1)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda (t5: T).((pr0 v1 v2) \to 
-((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 t6)) (\lambda (t6: T).(pr0 
-t2 t6)))))) (\lambda (H7: (pr0 v1 v2)).(\lambda (H8: (pr0 t3 t4)).(let H9 
-\def (match H1 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda (_: 
-(pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 
-(THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with 
-[(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 t)).(\lambda (H10: (eq T t5 
-t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: 
-T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))))) 
-(\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: 
-T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let 
-H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H11 (THead (Flat Appl) 
-v1 (THead (Bind Abst) u t3)) H4) in (eq_ind T (THead (Flat Appl) v1 (THead 
-(Bind Abst) u t3)) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind 
-Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def (eq_ind_r T t 
-(\lambda (t6: T).(eq T t5 t6)) H9 (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t3)) H4) in (let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: 
-T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v 
-t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 
-t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H4) in 
-(ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2 t4) t6)) (\lambda 
-(t6: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t6)) (THead 
-(Bind Abbr) v2 t4) (pr0_refl (THead (Bind Abbr) v2 t4)) (pr0_beta u v1 v2 H7 
-t3 t4 H8)))) t2 H12)) t (sym_eq T t t2 H11))) t5 (sym_eq T t5 t H9) H10))) | 
-(pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow (\lambda (H11: (eq T (THead k u1 
-t5) t)).(\lambda (H12: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) 
-(\lambda (_: T).((eq T (THead k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) 
-\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: 
-T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead k u2 t6) t2)).(eq_ind T 
-(THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T 
-(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t7 
-t8)))))) (\lambda (H14: (pr0 u1 u2)).(\lambda (H15: (pr0 t5 t6)).(let H16 
-\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t3)) t7)) H4 (THead k u1 t5) H11) in (let H17 \def (match H16 in eq 
-return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead k u1 t5)) 
-\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: 
-T).(pr0 (THead k u2 t6) t8)))))) with [refl_equal \Rightarrow (\lambda (H17: 
-(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5))).(let 
-H18 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow (THead (Bind Abst) u t3) | (TLRef _) \Rightarrow 
-(THead (Bind Abst) u t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H17) in ((let H19 \def 
+Cast) u t5) H14) in ((let H16 \def (f_equal T T (\lambda (e: T).(match e in T 
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
+\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead 
+(Flat Cast) u t5) H14) in ((let H17 \def (f_equal T T (\lambda (e: T).(match 
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) 
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead 
+(Flat Cast) u t5) H14) in (\lambda (H18: (eq T u1 u)).(\lambda (H19: (eq K k 
+(Flat Cast))).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: 
+T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v 
+t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
+t10)))))))))) H (THead (Flat Cast) u t5) H10) in (eq_ind_r K (Flat Cast) 
+(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda 
+(t7: T).(pr0 t2 t7)))) (let H21 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 
+u2)) H7 u H18) in (let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 
+t5 H17) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t2 
+t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t4) t7)) (\lambda 
+(t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H23: (pr0 t4 x)).(\lambda 
+(H24: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 
+t4) t7)) (\lambda (t7: T).(pr0 t2 t7)) x (pr0_epsilon t4 x H23 u2) H24)))) 
+(H20 t5 (tlt_head_dx (Flat Cast) u t5) t4 H22 t2 H13)))) k H19))))) H16)) 
+H15)))) t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) 
+(refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | (pr0_beta u v1 v2 H2 t3 t4 
+H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) 
+u t3)) t)).(\lambda (H5: (eq T (THead (Bind Abbr) v2 t4) t1)).(eq_ind T 
+(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (_: T).((eq T (THead 
+(Bind Abbr) v2 t4) t1) \to ((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda 
+(t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H6: (eq T 
+(THead (Bind Abbr) v2 t4) t1)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda 
+(t5: T).((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 
+t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 v1 v2)).(\lambda 
+(H8: (pr0 t3 t4)).(let H9 \def (match H1 in pr0 return (\lambda (t5: 
+T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) 
+\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: 
+T).(pr0 t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 
+t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) 
+\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: 
+T).(pr0 t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: 
+T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: 
+T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) 
+H11 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H4) in (eq_ind T (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (t6: T).(ex2 T (\lambda 
+(t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t6 t7)))) 
+(let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t3)) H4) in (let H14 \def (eq_ind_r T t 
+(\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) 
+\to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) 
+(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind 
+Abst) u t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2 
+t4) t6)) (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u 
+t3)) t6)) (THead (Bind Abbr) v2 t4) (pr0_refl (THead (Bind Abbr) v2 t4)) 
+(pr0_beta u v1 v2 H7 t3 t4 H8)))) t2 H12)) t (sym_eq T t t2 H11))) t5 (sym_eq 
+T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow (\lambda 
+(H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T (THead k u2 t6) 
+t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead k u2 t6) t2) \to 
+((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind 
+Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T 
+(THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) 
+\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) 
+t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u1 u2)).(\lambda 
+(H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead k u1 t5) H11) in (let 
+H17 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) 
+with [(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | 
+(THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind Abst) u 
+t3)) (THead k u1 t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | 
+(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H16) in ((let H19 \def 
 (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) 
-\Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 
-t5) H17) in ((let H20 \def (f_equal T K (\lambda (e: T).(match e in T return 
-(\lambda (_: T).K) with [(TSort _) \Rightarrow (Flat Appl) | (TLRef _) 
-\Rightarrow (Flat Appl) | (THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) 
-v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H17) in (eq_ind K (Flat Appl) 
-(\lambda (k0: K).((eq T v1 u1) \to ((eq T (THead (Bind Abst) u t3) t5) \to 
-(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: 
-T).(pr0 (THead k0 u2 t6) t7)))))) (\lambda (H21: (eq T v1 u1)).(eq_ind T u1 
-(\lambda (_: T).((eq T (THead (Bind Abst) u t3) t5) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat 
-Appl) u2 t6) t8))))) (\lambda (H22: (eq T (THead (Bind Abst) u t3) 
-t5)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (_: T).(ex2 T (\lambda (t8: 
-T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat 
-Appl) u2 t6) t8)))) (let H23 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead 
-k0 u1 t5) t)) H11 (Flat Appl) H20) in (let H24 \def (eq_ind_r T t5 (\lambda 
-(t7: T).(pr0 t7 t6)) H15 (THead (Bind Abst) u t3) H22) in (let H25 \def 
-(match H24 in pr0 return (\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 
-t7 t8)).((eq T t7 (THead (Bind Abst) u t3)) \to ((eq T t8 t6) \to (ex2 T 
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 
-(THead (Flat Appl) u2 t6) t9)))))))) with [(pr0_refl t7) \Rightarrow (\lambda 
-(H25: (eq T t7 (THead (Bind Abst) u t3))).(\lambda (H26: (eq T t7 
-t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (t8: T).((eq T t8 t6) \to 
-(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: 
-T).(pr0 (THead (Flat Appl) u2 t6) t9))))) (\lambda (H27: (eq T (THead (Bind 
-Abst) u t3) t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (t8: T).(ex2 T 
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 
-(THead (Flat Appl) u2 t8) t9)))) (let H28 \def (eq_ind_r T t5 (\lambda (t8: 
-T).(eq T (THead (Flat Appl) u1 t8) t)) H23 (THead (Bind Abst) u t3) H22) in 
-(let H29 \def (eq_ind_r T t (\lambda (t8: T).(\forall (v: T).((tlt v t8) \to 
-(\forall (t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v t10) \to (ex2 T 
-(\lambda (t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 t11)))))))))) H 
-(THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H28) in (let H30 \def (eq_ind 
-T v1 (\lambda (t8: T).(pr0 t8 v2)) H7 u1 H21) in (ex2_ind T (\lambda (t8: 
-T).(pr0 v2 t8)) (\lambda (t8: T).(pr0 u2 t8)) (ex2 T (\lambda (t8: T).(pr0 
+[(TSort _) \Rightarrow (THead (Bind Abst) u t3) | (TLRef _) \Rightarrow 
+(THead (Bind Abst) u t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H16) in (\lambda (H20: (eq 
+T v1 u1)).(\lambda (H21: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda 
+(k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda 
+(t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r K k (\lambda 
+(k0: K).(eq T (THead k0 u1 t5) t)) H11 (Flat Appl) H21) in (let H23 \def 
+(eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (THead (Bind Abst) u t3) 
+H19) in (let H24 \def (match H23 in pr0 return (\lambda (t7: T).(\lambda (t8: 
+T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead (Bind Abst) u t3)) \to ((eq T 
+t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) 
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9)))))))) with [(pr0_refl 
+t7) \Rightarrow (\lambda (H24: (eq T t7 (THead (Bind Abst) u t3))).(\lambda 
+(H25: (eq T t7 t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (t8: T).((eq 
+T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) 
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) (\lambda (H26: (eq T 
+(THead (Bind Abst) u t3) t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda 
+(t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda 
+(t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)))) (let H27 \def (eq_ind_r T t5 
+(\lambda (t8: T).(eq T (THead (Flat Appl) u1 t8) t)) H22 (THead (Bind Abst) u 
+t3) H19) in (let H28 \def (eq_ind_r T t (\lambda (t8: T).(\forall (v: 
+T).((tlt v t8) \to (\forall (t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v 
+t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 
+t11)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H27) in (let 
+H29 \def (eq_ind T v1 (\lambda (t8: T).(pr0 t8 v2)) H7 u1 H20) in (ex2_ind T 
+(\lambda (t8: T).(pr0 v2 t8)) (\lambda (t8: T).(pr0 u2 t8)) (ex2 T (\lambda 
+(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead 
+(Flat Appl) u2 (THead (Bind Abst) u t3)) t8))) (\lambda (x: T).(\lambda (H30: 
+(pr0 v2 x)).(\lambda (H31: (pr0 u2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 
 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 
-(THead (Bind Abst) u t3)) t8))) (\lambda (x: T).(\lambda (H31: (pr0 v2 
-x)).(\lambda (H32: (pr0 u2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 (THead 
-(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead 
-(Bind Abst) u t3)) t8)) (THead (Bind Abbr) x t4) (pr0_comp v2 x H31 t4 t4 
-(pr0_refl t4) (Bind Abbr)) (pr0_beta u u2 x H32 t3 t4 H8))))) (H29 u1 
-(tlt_head_sx (Flat Appl) u1 (THead (Bind Abst) u t3)) v2 H30 u2 H14))))) t6 
-H27)) t7 (sym_eq T t7 (THead (Bind Abst) u t3) H25) H26))) | (pr0_comp u0 u3 
-H25 t7 t8 H26 k0) \Rightarrow (\lambda (H27: (eq T (THead k0 u0 t7) (THead 
-(Bind Abst) u t3))).(\lambda (H28: (eq T (THead k0 u3 t8) t6)).((let H29 \def 
+(THead (Bind Abst) u t3)) t8)) (THead (Bind Abbr) x t4) (pr0_comp v2 x H30 t4 
+t4 (pr0_refl t4) (Bind Abbr)) (pr0_beta u u2 x H31 t3 t4 H8))))) (H28 u1 
+(tlt_head_sx (Flat Appl) u1 (THead (Bind Abst) u t3)) v2 H29 u2 H14))))) t6 
+H26)) t7 (sym_eq T t7 (THead (Bind Abst) u t3) H24) H25))) | (pr0_comp u0 u3 
+H24 t7 t8 H25 k0) \Rightarrow (\lambda (H26: (eq T (THead k0 u0 t7) (THead 
+(Bind Abst) u t3))).(\lambda (H27: (eq T (THead k0 u3 t8) t6)).((let H28 \def 
 (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
 [(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) 
-\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H27) in ((let H30 
+\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H29 
 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
 with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) 
-\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H27) in ((let H31 
+\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H30 
 \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 t7) (THead (Bind Abst) u t3) H27) in (eq_ind K 
+\Rightarrow k1])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in (eq_ind K 
 (Bind Abst) (\lambda (k1: K).((eq T u0 u) \to ((eq T t7 t3) \to ((eq T (THead 
 k1 u3 t8) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
 T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat 
-Appl) u2 t6) t9))))))))) (\lambda (H32: (eq T u0 u)).(eq_ind T u (\lambda 
+Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0 u)).(eq_ind T u (\lambda 
 (t9: T).((eq T t7 t3) \to ((eq T (THead (Bind Abst) u3 t8) t6) \to ((pr0 t9 
 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 
 t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10)))))))) 
-(\lambda (H33: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq T (THead 
+(\lambda (H32: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq T (THead 
 (Bind Abst) u3 t8) t6) \to ((pr0 u u3) \to ((pr0 t9 t8) \to (ex2 T (\lambda 
 (t10: T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead 
-(Flat Appl) u2 t6) t10))))))) (\lambda (H34: (eq T (THead (Bind Abst) u3 t8) 
+(Flat Appl) u2 t6) t10))))))) (\lambda (H33: (eq T (THead (Bind Abst) u3 t8) 
 t6)).(eq_ind T (THead (Bind Abst) u3 t8) (\lambda (t9: T).((pr0 u u3) \to 
 ((pr0 t3 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) 
 t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) t10)))))) (\lambda (_: 
-(pr0 u u3)).(\lambda (H36: (pr0 t3 t8)).(let H37 \def (eq_ind_r T t5 (\lambda 
-(t9: T).(eq T (THead (Flat Appl) u1 t9) t)) H23 (THead (Bind Abst) u t3) H22) 
-in (let H38 \def (eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) 
+(pr0 u u3)).(\lambda (H35: (pr0 t3 t8)).(let H36 \def (eq_ind_r T t5 (\lambda 
+(t9: T).(eq T (THead (Flat Appl) u1 t9) t)) H22 (THead (Bind Abst) u t3) H19) 
+in (let H37 \def (eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) 
 \to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to 
 (ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 
-t12)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H37) in (let 
-H39 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H7 u1 H21) in (ex2_ind T 
+t12)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H36) in (let 
+H38 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H7 u1 H20) in (ex2_ind T 
 (\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u2 t9)) (ex2 T (\lambda 
 (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead 
 (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9))) (\lambda (x: T).(\lambda 
-(H40: (pr0 v2 x)).(\lambda (H41: (pr0 u2 x)).(ex2_ind T (\lambda (t9: T).(pr0 
+(H39: (pr0 v2 x)).(\lambda (H40: (pr0 u2 x)).(ex2_ind T (\lambda (t9: T).(pr0 
 t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead 
 (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead 
-(Bind Abst) u3 t8)) t9))) (\lambda (x0: T).(\lambda (H42: (pr0 t8 
-x0)).(\lambda (H43: (pr0 t4 x0)).(ex_intro2 T (\lambda (t9: T).(pr0 (THead 
+(Bind Abst) u3 t8)) t9))) (\lambda (x0: T).(\lambda (H41: (pr0 t8 
+x0)).(\lambda (H42: (pr0 t4 x0)).(ex_intro2 T (\lambda (t9: T).(pr0 (THead 
 (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead 
-(Bind Abst) u3 t8)) t9)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H40 t4 x0 
-H43 (Bind Abbr)) (pr0_beta u3 u2 x H41 t8 x0 H42))))) (H38 t3 (tlt_trans 
+(Bind Abst) u3 t8)) t9)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H39 t4 x0 
+H42 (Bind Abbr)) (pr0_beta u3 u2 x H40 t8 x0 H41))))) (H37 t3 (tlt_trans 
 (THead (Bind Abst) u t3) t3 (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) 
 (tlt_head_dx (Bind Abst) u t3) (tlt_head_dx (Flat Appl) u1 (THead (Bind Abst) 
-u t3))) t8 H36 t4 H8))))) (H38 u1 (tlt_head_sx (Flat Appl) u1 (THead (Bind 
-Abst) u t3)) v2 H39 u2 H14))))))) t6 H34)) t7 (sym_eq T t7 t3 H33))) u0 
-(sym_eq T u0 u H32))) k0 (sym_eq K k0 (Bind Abst) H31))) H30)) H29)) H28 H25 
-H26))) | (pr0_beta u0 v0 v3 H25 t7 t8 H26) \Rightarrow (\lambda (H27: (eq T 
+u t3))) t8 H35 t4 H8))))) (H37 u1 (tlt_head_sx (Flat Appl) u1 (THead (Bind 
+Abst) u t3)) v2 H38 u2 H14))))))) t6 H33)) t7 (sym_eq T t7 t3 H32))) u0 
+(sym_eq T u0 u H31))) k0 (sym_eq K k0 (Bind Abst) H30))) H29)) H28)) H27 H24 
+H25))) | (pr0_beta u0 v0 v3 H24 t7 t8 H25) \Rightarrow (\lambda (H26: (eq T 
 (THead (Flat Appl) v0 (THead (Bind Abst) u0 t7)) (THead (Bind Abst) u 
-t3))).(\lambda (H28: (eq T (THead (Bind Abbr) v3 t8) t6)).((let H29 \def 
+t3))).(\lambda (H27: (eq T (THead (Bind Abbr) v3 t8) t6)).((let H28 \def 
 (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t7)) (\lambda (e: 
 T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
 False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in 
 K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) 
-\Rightarrow True])])) I (THead (Bind Abst) u t3) H27) in (False_ind ((eq T 
+\Rightarrow True])])) I (THead (Bind Abst) u t3) H26) in (False_ind ((eq T 
 (THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T 
 (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 
-(THead (Flat Appl) u2 t6) t9)))))) H29)) H28 H25 H26))) | (pr0_upsilon b H25 
-v0 v3 H26 u0 u3 H27 t7 t8 H28) \Rightarrow (\lambda (H29: (eq T (THead (Flat 
-Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) u t3))).(\lambda (H30: 
+(THead (Flat Appl) u2 t6) t9)))))) H28)) H27 H24 H25))) | (pr0_upsilon b H24 
+v0 v3 H25 u0 u3 H26 t7 t8 H27) \Rightarrow (\lambda (H28: (eq T (THead (Flat 
+Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) u t3))).(\lambda (H29: 
 (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t6)).((let 
-H31 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (\lambda (e: 
+H30 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (\lambda (e: 
 T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
 False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in 
 K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) 
-\Rightarrow True])])) I (THead (Bind Abst) u t3) H29) in (False_ind ((eq T 
+\Rightarrow True])])) I (THead (Bind Abst) u t3) H28) in (False_ind ((eq T 
 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t6) \to ((not 
 (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T 
 (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 
-(THead (Flat Appl) u2 t6) t9)))))))) H31)) H30 H25 H26 H27 H28))) | 
-(pr0_delta u0 u3 H25 t7 t8 H26 w H27) \Rightarrow (\lambda (H28: (eq T (THead 
-(Bind Abbr) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H29: (eq T (THead 
-(Bind Abbr) u3 w) t6)).((let H30 \def (eq_ind T (THead (Bind Abbr) u0 t7) 
+(THead (Flat Appl) u2 t6) t9)))))))) H30)) H29 H24 H25 H26 H27))) | 
+(pr0_delta u0 u3 H24 t7 t8 H25 w H26) \Rightarrow (\lambda (H27: (eq T (THead 
+(Bind Abbr) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H28: (eq T (THead 
+(Bind Abbr) u3 w) t6)).((let H29 \def (eq_ind T (THead (Bind Abbr) u0 t7) 
 (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) 
 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow 
 (match k0 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) u t3) H28) in (False_ind ((eq T (THead (Bind 
+False])])) I (THead (Bind Abst) u t3) H27) in (False_ind ((eq T (THead (Bind 
 Abbr) u3 w) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O u3 t8 w) \to 
 (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: 
-T).(pr0 (THead (Flat Appl) u2 t6) t9))))))) H30)) H29 H25 H26 H27))) | 
-(pr0_zeta b H25 t7 t8 H26 u0) \Rightarrow (\lambda (H27: (eq T (THead (Bind 
-b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t3))).(\lambda (H28: (eq T t8 
-t6)).((let H29 \def (f_equal T T (\lambda (e: T).(match e in T return 
+T).(pr0 (THead (Flat Appl) u2 t6) t9))))))) H29)) H28 H24 H25 H26))) | 
+(pr0_zeta b H24 t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead (Bind 
+b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t3))).(\lambda (H27: (eq T t8 
+t6)).((let H28 \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) (t9: T) on t9: T \def (match t9 with [(TSort n) 
 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
@@ -1293,46 +1212,45 @@ t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
 (THead k0 u3 t10) \Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 
 d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ 
 t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind 
-Abst) u t3) H27) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T 
+Abst) u t3) H26) in ((let H29 \def (f_equal T T (\lambda (e: T).(match e in T 
 return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) 
 \Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) u0 (lift (S 
-O) O t7)) (THead (Bind Abst) u t3) H27) in ((let H31 \def (f_equal T B 
+O) O t7)) (THead (Bind Abst) u t3) H26) in ((let H30 \def (f_equal T B 
 (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) 
 \Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow (match 
 k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
 \Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u 
-t3) H27) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S 
+t3) H26) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S 
 O) O t7) t3) \to ((eq T t8 t6) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to 
 (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: 
-T).(pr0 (THead (Flat Appl) u2 t6) t9))))))))) (\lambda (H32: (eq T u0 
+T).(pr0 (THead (Flat Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0 
 u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t7) t3) \to ((eq T t8 
 t6) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: 
 T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat 
-Appl) u2 t6) t10)))))))) (\lambda (H33: (eq T (lift (S O) O t7) t3)).(eq_ind 
+Appl) u2 t6) t10)))))))) (\lambda (H32: (eq T (lift (S O) O t7) t3)).(eq_ind 
 T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t6) \to ((not (eq B Abst Abst)) 
 \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) 
 t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10))))))) (\lambda 
-(H34: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not (eq B Abst Abst)) \to 
+(H33: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not (eq B Abst Abst)) \to 
 ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) 
 t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10)))))) (\lambda 
-(H35: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 t6)).(let H37 \def (match 
-(H35 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda 
+(H34: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 t6)).(let H36 \def (match 
+(H34 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda 
 (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead 
-(Flat Appl) u2 t6) t9)))) with []) in H37))) t8 (sym_eq T t8 t6 H34))) t3 
-H33)) u0 (sym_eq T u0 u H32))) b (sym_eq B b Abst H31))) H30)) H29)) H28 H25 
-H26))) | (pr0_epsilon t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead 
-(Flat Cast) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H27: (eq T t8 
-t6)).((let H28 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: 
+(Flat Appl) u2 t6) t9)))) with []) in H36))) t8 (sym_eq T t8 t6 H33))) t3 
+H32)) u0 (sym_eq T u0 u H31))) b (sym_eq B b Abst H30))) H29)) H28)) H27 H24 
+H25))) | (pr0_epsilon t7 t8 H24 u0) \Rightarrow (\lambda (H25: (eq T (THead 
+(Flat Cast) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H26: (eq T t8 
+t6)).((let H27 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: 
 T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
 False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in 
 K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) 
-\Rightarrow True])])) I (THead (Bind Abst) u t3) H26) in (False_ind ((eq T t8 
+\Rightarrow True])])) I (THead (Bind Abst) u t3) H25) in (False_ind ((eq T t8 
 t6) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 
-t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) H28)) H27 
-H25)))]) in (H25 (refl_equal T (THead (Bind Abst) u t3)) (refl_equal T 
-t6))))) t5 H22)) v1 (sym_eq T v1 u1 H21))) k H20)) H19)) H18)))]) in (H17 
-(refl_equal T (THead k u1 t5))))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta 
-u0 v0 v3 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v0 
+t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) H27)) H26 
+H24)))]) in (H24 (refl_equal T (THead (Bind Abst) u t3)) (refl_equal T 
+t6))))) k H21)))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u0 
+v0 v3 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v0 
 (THead (Bind Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v3 t6) 
 t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) (\lambda (_: 
 T).((eq T (THead (Bind Abbr) v3 t6) t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to 
@@ -1343,226 +1261,179 @@ t2)).(eq_ind T (THead (Bind Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to
 (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 v0 v3)).(\lambda (H15: 
 (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat 
 Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind 
-Abst) u0 t5)) H11) in (let H17 \def (match H16 in eq return (\lambda (t7: 
-T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Appl) v0 (THead (Bind 
-Abst) u0 t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) 
-t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v3 t6) t8)))))) with 
-[refl_equal \Rightarrow (\lambda (H17: (eq T (THead (Flat Appl) v1 (THead 
-(Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)))).(let 
-H18 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) 
-\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow 
-t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 
-(THead (Bind Abst) u0 t5)) H17) in ((let H19 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | 
-(TLRef _) \Rightarrow u | (THead _ _ t7) \Rightarrow (match t7 in T return 
+Abst) u0 t5)) H11) in (let H17 \def (f_equal T T (\lambda (e: T).(match e in 
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) 
+\Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 
+(THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) 
+H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match e in T return 
 (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | 
-(THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u 
-t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H17) in ((let H20 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) 
-\Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead 
-(Flat Appl) v0 (THead (Bind Abst) u0 t5)) H17) in (eq_ind T v0 (\lambda (_: 
-T).((eq T u u0) \to ((eq T t3 t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
-(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v3 t6) 
-t8)))))) (\lambda (H21: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T t3 
-t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda 
-(t8: T).(pr0 (THead (Bind Abbr) v3 t6) t8))))) (\lambda (H22: (eq T t3 
-t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind 
-Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v3 t6) t8)))) (let 
-H23 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to 
-(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T 
-(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H 
-(THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H11) in (let H24 \def 
-(eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 t5 H22) in (let H25 \def 
-(eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) H7 v0 H20) in (ex2_ind T (\lambda 
-(t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda (t7: 
-T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind 
-Abbr) v3 t6) t7))) (\lambda (x: T).(\lambda (H26: (pr0 v2 x)).(\lambda (H27: 
-(pr0 v3 x)).(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 
-t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda 
-(t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) (\lambda (x0: T).(\lambda (H28: 
-(pr0 t4 x0)).(\lambda (H29: (pr0 t6 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 
-(THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 
-t6) t7)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H26 t4 x0 H28 (Bind Abbr)) 
-(pr0_comp v3 x H27 t6 x0 H29 (Bind Abbr)))))) (H23 t5 (tlt_trans (THead (Bind 
-Abst) u0 t5) t5 (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) (tlt_head_dx 
-(Bind Abst) u0 t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind Abst) u0 t5))) t4 
-H24 t6 H15))))) (H23 v0 (tlt_head_sx (Flat Appl) v0 (THead (Bind Abst) u0 
-t5)) v2 H25 v3 H14))))) t3 (sym_eq T t3 t5 H22))) u (sym_eq T u u0 H21))) v1 
-(sym_eq T v1 v0 H20))) H19)) H18)))]) in (H17 (refl_equal T (THead (Flat 
-Appl) v0 (THead (Bind Abst) u0 t5)))))))) t2 H13)) t H11 H12 H9 H10))) | 
-(pr0_upsilon b H9 v0 v3 H10 u1 u2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: 
-(eq T (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) t)).(\lambda (H14: (eq T 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T 
-(THead (Flat Appl) v0 (THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b 
-Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda 
-(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 
-t8))))))))) (\lambda (H15: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v3) t6)) t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v3) t6)) (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v0 v3) \to 
-((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind 
-Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (H16: (not (eq 
-B b Abst))).(\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: 
-(pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind 
-b) u1 t5)) H13) in (let H21 \def (match H20 in eq return (\lambda (t7: 
-T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Appl) v0 (THead (Bind b) 
-u1 t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) 
-(\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) 
-t6)) t8)))))) with [refl_equal \Rightarrow (\lambda (H21: (eq T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 
-t5)))).(let H22 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 
-| (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) 
+(THead _ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with 
+[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) 
 \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead 
-(Flat Appl) v0 (THead (Bind b) u1 t5)) H21) in ((let H23 \def (f_equal T T 
+(Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in ((let H19 \def (f_equal T T 
 (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7) \Rightarrow (match 
-t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) 
-\Rightarrow u | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 
-(THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H21) 
-in ((let H24 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda 
-(_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | 
-(THead _ _ t7) \Rightarrow (match t7 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 (Flat Appl) v1 
-(THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H21) 
-in ((let H25 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda 
-(_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead 
-_ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) 
-(THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H21) in (eq_ind T v0 (\lambda 
-(_: T).((eq B Abst b) \to ((eq T u u1) \to ((eq T t3 t5) \to (ex2 T (\lambda 
-(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) t8))))))) (\lambda 
-(H26: (eq B Abst b)).(eq_ind B Abst (\lambda (b0: B).((eq T u u1) \to ((eq T 
-t3 t5) \to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) 
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow (match 
+t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) 
+\Rightarrow t3 | (THead _ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 
+(THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) 
+H16) in (\lambda (_: (eq T u u0)).(\lambda (H21: (eq T v1 v0)).(let H22 \def 
+(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: 
+T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: 
+T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) 
+v0 (THead (Bind Abst) u0 t5)) H11) in (let H23 \def (eq_ind T v1 (\lambda 
+(t7: T).(pr0 t7 v2)) H7 v0 H21) in (let H24 \def (eq_ind T t3 (\lambda (t7: 
+T).(pr0 t7 t4)) H8 t5 H19) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) 
+(\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) 
+v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) (\lambda 
+(x: T).(\lambda (H25: (pr0 t4 x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T 
+(\lambda (t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda 
+(t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead 
+(Bind Abbr) v3 t6) t7))) (\lambda (x0: T).(\lambda (H27: (pr0 v2 
+x0)).(\lambda (H28: (pr0 v3 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead 
+(Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)) 
+(THead (Bind Abbr) x0 x) (pr0_comp v2 x0 H27 t4 x H25 (Bind Abbr)) (pr0_comp 
+v3 x0 H28 t6 x H26 (Bind Abbr)))))) (H22 v0 (tlt_head_sx (Flat Appl) v0 
+(THead (Bind Abst) u0 t5)) v2 H23 v3 H14))))) (H22 t5 (tlt_trans (THead (Bind 
+Abst) u0 t5) t5 (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) (tlt_head_dx 
+(Bind Abst) u0 t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind Abst) u0 t5))) t4 
+H24 t6 H15)))))))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon 
+b H9 v0 v3 H10 u1 u2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead 
+(Flat Appl) v0 (THead (Bind b) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind 
+b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Flat 
+Appl) v0 (THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b Abst)) \to 
+((pr0 v0 v3) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) 
+(\lambda (H15: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) 
+t6)) t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) 
+t6)) (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) 
+\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) 
+t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (H16: (not (eq B b 
+Abst))).(\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 
+t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) 
+v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind b) u1 
+t5)) H13) in (let H21 \def (f_equal T T (\lambda (e: T).(match e in T return 
+(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 
+| (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u 
+t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H22 \def 
+(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with 
+[(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead _ _ t7) 
+\Rightarrow (match t7 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 (Flat Appl) v1 (THead (Bind Abst) u 
+t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H23 \def 
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
+[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7) 
+\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) \Rightarrow t8])])) 
+(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead 
+(Bind b) u1 t5)) H20) in ((let H24 \def (f_equal T T (\lambda (e: T).(match e 
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) 
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda 
+(_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead 
+_ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) 
+(THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in (\lambda (_: (eq T u 
+u1)).(\lambda (H26: (eq B Abst b)).(\lambda (_: (eq T v1 v0)).(eq_ind B Abst 
+(\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) 
 (\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O 
-v3) t6)) t7)))))) (\lambda (H27: (eq T u u1)).(eq_ind T u1 (\lambda (_: 
-T).((eq T t3 t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) 
-t8)) (\lambda (t8: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S 
-O) O v3) t6)) t8))))) (\lambda (H28: (eq T t3 t5)).(eq_ind T t5 (\lambda (_: 
-T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: 
-T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) 
-t8)))) (let H29 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H16 
-Abst H26) in (let H30 \def (match (H29 (refl_equal B Abst)) in False return 
-(\lambda (_: False).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) 
-t7)) (\lambda (t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S 
-O) O v3) t6)) t7)))) with []) in H30)) t3 (sym_eq T t3 t5 H28))) u (sym_eq T 
-u u1 H27))) b H26)) v1 (sym_eq T v1 v0 H25))) H24)) H23)) H22)))]) in (H21 
-(refl_equal T (THead (Flat Appl) v0 (THead (Bind b) u1 t5)))))))))) t2 H15)) 
-t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6 H10 w H11) 
-\Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5) t)).(\lambda (H13: 
-(eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u1 t5) 
-(\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 u2) \to 
-((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
-(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H14: 
-(eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) 
+v3) t6)) t7)))) (let H28 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 
+Abst))) H16 Abst H26) in (let H29 \def (match (H28 (refl_equal B Abst)) in 
+False return (\lambda (_: False).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind 
+Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat 
+Appl) (lift (S O) O v3) t6)) t7)))) with []) in H29)) b H26))))) H23)) H22)) 
+H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6 
+H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5) 
+t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind 
+Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 
+u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 
+(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda 
+(H14: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) 
 (\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to 
 (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: 
 T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t5 
 t6)).(\lambda (_: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r T t (\lambda 
 (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead 
-(Bind Abbr) u1 t5) H12) in (let H19 \def (match H18 in eq return (\lambda 
-(t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind Abbr) u1 t5)) \to 
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: 
-T).(pr0 (THead (Bind Abbr) u2 w) t8)))))) with [refl_equal \Rightarrow 
-(\lambda (H19: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead 
-(Bind Abbr) u1 t5))).(let H20 \def (eq_ind T (THead (Flat Appl) v1 (THead 
-(Bind Abst) u t3)) (\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 
-t5) H19) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) 
-t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7))) H20)))]) in (H19 
-(refl_equal T (THead (Bind Abbr) u1 t5)))))))) t2 H14)) t H12 H13 H9 H10 
-H11))) | (pr0_zeta b H9 t5 t6 H10 u0) \Rightarrow (\lambda (H11: (eq T (THead 
-(Bind b) u0 (lift (S O) O t5)) t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T 
-(THead (Bind b) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not 
-(eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind 
-Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 
-t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to 
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: 
-T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 t5 
-t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 
-(THead (Bind Abst) u t3)) t7)) H4 (THead (Bind b) u0 (lift (S O) O t5)) H11) 
-in (let H17 \def (match H16 in eq return (\lambda (t7: T).(\lambda (_: (eq ? 
-? t7)).((eq T t7 (THead (Bind b) u0 (lift (S O) O t5))) \to (ex2 T (\lambda 
-(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) 
-with [refl_equal \Rightarrow (\lambda (H17: (eq T (THead (Flat Appl) v1 
-(THead (Bind Abst) u t3)) (THead (Bind b) u0 (lift (S O) O t5)))).(let H18 
-\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\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 b) u0 (lift (S O) O t5)) H17) in 
-(False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) 
-(\lambda (t7: T).(pr0 t2 t7))) H18)))]) in (H17 (refl_equal T (THead (Bind b) 
-u0 (lift (S O) O t5)))))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | 
-(pr0_epsilon t5 t6 H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) 
-u0 t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) 
-(\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) 
-(\lambda (H12: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to 
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: 
-T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t 
+(Bind Abbr) u1 t5) H12) in (let H19 \def (eq_ind T (THead (Flat Appl) v1 
+(THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
+Abbr) u1 t5) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind 
+Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7))) 
+H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 t6 H10 u0) 
+\Rightarrow (\lambda (H11: (eq T (THead (Bind b) u0 (lift (S O) O t5)) 
+t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind b) u0 (lift (S O) O 
+t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) 
+\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: 
+T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: 
+T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 
+(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: 
+(not (eq B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t 
 (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) 
-H4 (THead (Flat Cast) u0 t5) H10) in (let H15 \def (match H14 in eq return 
-(\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Cast) u0 
-t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda 
-(t8: T).(pr0 t2 t8)))))) with [refl_equal \Rightarrow (\lambda (H15: (eq T 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Cast) u0 
-t5))).(let H16 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) 
-(\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) u0 t5) 
-H15) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) 
-t7)) (\lambda (t7: T).(pr0 t2 t7))) H16)))]) in (H15 (refl_equal T (THead 
-(Flat Cast) u0 t5)))))) t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 
-(refl_equal T t) (refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | 
-(pr0_upsilon b H2 v1 v2 H3 u1 u2 H4 t3 t4 H5) \Rightarrow (\lambda (H6: (eq T 
-(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t)).(\lambda (H7: (eq T (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1)).(eq_ind T (THead 
-(Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (_: T).((eq T (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1) \to ((not (eq B b Abst)) \to 
-((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: 
-T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))))) (\lambda (H8: (eq T 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1)).(eq_ind T 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (\lambda (t5: 
-T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) 
-\to (ex2 T (\lambda (t6: T).(pr0 t5 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))) 
-(\lambda (H9: (not (eq B b Abst))).(\lambda (H10: (pr0 v1 v2)).(\lambda (H11: 
-(pr0 u1 u2)).(\lambda (H12: (pr0 t3 t4)).(let H13 \def (match H1 in pr0 
-return (\lambda (t5: T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 
-t) \to ((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 
-t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H13: (eq T t5 
-t)).(\lambda (H14: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) 
+H4 (THead (Bind b) u0 (lift (S O) O t5)) H11) in (let H17 \def (eq_ind T 
+(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee in 
+T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
+True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H16) in (False_ind (ex2 T 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 
+t7))) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_epsilon 
+t5 t6 H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0 t5) 
+t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda 
+(_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
+(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq 
+T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))) 
+(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T 
+(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Cast) u0 
+t5) H10) in (let H15 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) 
+u t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
+\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_: 
+F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead 
+(Flat Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead 
+(Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) H15)))) t6 (sym_eq T 
+t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) 
+t1 H6)) t H4 H5 H2 H3))) | (pr0_upsilon b H2 v1 v2 H3 u1 u2 H4 t3 t4 H5) 
+\Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) 
+t)).(\lambda (H7: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
+v2) t4)) t1)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) 
+(\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
+v2) t4)) t1) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to 
+((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 
+t2 t6))))))))) (\lambda (H8: (eq T (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) t1)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b Abst)) \to ((pr0 v1 v2) 
+\to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 t6)) 
+(\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H9: (not (eq B b 
+Abst))).(\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 u1 u2)).(\lambda 
+(H12: (pr0 t3 t4)).(let H13 \def (match H1 in pr0 return (\lambda (t5: 
+T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) 
 \to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H15: (eq T t 
-t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind 
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 
-t7)))) (let H16 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H15 (THead 
-(Flat Appl) v1 (THead (Bind b) u1 t3)) H6) in (eq_ind T (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 (THead 
+(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with [(pr0_refl t5) 
+\Rightarrow (\lambda (H13: (eq T t5 t)).(\lambda (H14: (eq T t5 t2)).(eq_ind 
+T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead 
 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: 
-T).(pr0 t6 t7)))) (let H17 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) 
-H13 (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H6) in (let H18 \def 
-(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: 
-T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: 
-T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) H6) in (ex2_sym T (pr0 (THead (Flat Appl) v1 (THead 
-(Bind b) u1 t3))) (pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t4))) (pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t3 t4 H12 v1 
-v2 v2 H10 (pr0_refl v2))))) t2 H16)) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 
-t H13) H14))) | (pr0_comp u0 u3 H13 t5 t6 H14 k) \Rightarrow (\lambda (H15: 
-(eq T (THead k u0 t5) t)).(\lambda (H16: (eq T (THead k u3 t6) t2)).(eq_ind T 
+T).(pr0 t2 t7))))) (\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (_: 
+T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H16 \def (eq_ind_r 
+T t (\lambda (t6: T).(eq T t6 t2)) H15 (THead (Flat Appl) v1 (THead (Bind b) 
+u1 t3)) H6) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) 
+(\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H17 
+\def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H13 (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) H6) in (let H18 \def (eq_ind_r T t (\lambda (t6: 
+T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall 
+(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: 
+T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H6) 
+in (ex2_sym T (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 t3))) (pr0 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
+(pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t3 t4 H12 v1 v2 v2 H10 
+(pr0_refl v2))))) t2 H16)) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t H13) 
+H14))) | (pr0_comp u0 u3 H13 t5 t6 H14 k) \Rightarrow (\lambda (H15: (eq T 
+(THead k u0 t5) t)).(\lambda (H16: (eq T (THead k u3 t6) t2)).(eq_ind T 
 (THead k u0 t5) (\lambda (_: T).((eq T (THead k u3 t6) t2) \to ((pr0 u0 u3) 
 \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead 
 (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
@@ -1571,211 +1442,200 @@ t H13) H14))) | (pr0_comp u0 u3 H13 t5 t6 H14 k) \Rightarrow (\lambda (H15:
 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: 
 T).(pr0 t7 t8)))))) (\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 
 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) t7)) H6 (THead k u0 t5) H15) in (let H21 \def (match 
-H20 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead 
-k u0 t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead k u3 t6) 
-t8)))))) with [refl_equal \Rightarrow (\lambda (H21: (eq T (THead (Flat Appl) 
-v1 (THead (Bind b) u1 t3)) (THead k u0 t5))).(let H22 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead (Bind b) u1 
-t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind b) 
-u1 t3)) (THead k u0 t5) H21) in ((let H23 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | 
-(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat 
-Appl) v1 (THead (Bind b) u1 t3)) (THead k u0 t5) H21) in ((let H24 \def 
+(THead (Bind b) u1 t3)) t7)) H6 (THead k u0 t5) H15) in (let H21 \def 
 (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with 
 [(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | 
 (THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind b) u1 
-t3)) (THead k u0 t5) H21) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v1 
-u0) \to ((eq T (THead (Bind b) u1 t3) t5) \to (ex2 T (\lambda (t7: T).(pr0 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda 
-(t7: T).(pr0 (THead k0 u3 t6) t7)))))) (\lambda (H25: (eq T v1 u0)).(eq_ind T 
-u0 (\lambda (_: T).((eq T (THead (Bind b) u1 t3) t5) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) 
-(\lambda (t8: T).(pr0 (THead (Flat Appl) u3 t6) t8))))) (\lambda (H26: (eq T 
-(THead (Bind b) u1 t3) t5)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (_: 
-T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u3 t6) t8)))) 
-(let H27 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H15 
-(Flat Appl) H24) in (let H28 \def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 
-t6)) H19 (THead (Bind b) u1 t3) H26) in (let H29 \def (match H28 in pr0 
-return (\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T t7 
-(THead (Bind b) u1 t3)) \to ((eq T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda 
-(t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) with [(pr0_refl t7) 
-\Rightarrow (\lambda (H29: (eq T t7 (THead (Bind b) u1 t3))).(\lambda (H30: 
-(eq T t7 t6)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (t8: T).((eq T t8 t6) 
-\to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))) 
-(\lambda (H31: (eq T (THead (Bind b) u1 t3) t6)).(eq_ind T (THead (Bind b) u1 
+t3)) (THead k u0 t5) H20) in ((let H22 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | 
+(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat 
+Appl) v1 (THead (Bind b) u1 t3)) (THead k u0 t5) H20) in ((let H23 \def 
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
+[(TSort _) \Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead 
+(Bind b) u1 t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (THead k u0 t5) H20) in (\lambda (H24: (eq T v1 
+u0)).(\lambda (H25: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda (k0: 
+K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) (let H26 
+\def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H15 (Flat 
+Appl) H25) in (let H27 \def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H19 
+(THead (Bind b) u1 t3) H23) in (let H28 \def (match H27 in pr0 return 
+(\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead 
+(Bind b) u1 t3)) \to ((eq T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) with [(pr0_refl t7) \Rightarrow 
+(\lambda (H28: (eq T t7 (THead (Bind b) u1 t3))).(\lambda (H29: (eq T t7 
+t6)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (t8: T).((eq T t8 t6) \to (ex2 
+T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
+v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))) 
+(\lambda (H30: (eq T (THead (Bind b) u1 t3) t6)).(eq_ind T (THead (Bind b) u1 
 t3) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead 
 (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat 
-Appl) u3 t8) t9)))) (let H32 \def (eq_ind_r T t5 (\lambda (t8: T).(eq T 
-(THead (Flat Appl) u0 t8) t)) H27 (THead (Bind b) u1 t3) H26) in (let H33 
+Appl) u3 t8) t9)))) (let H31 \def (eq_ind_r T t5 (\lambda (t8: T).(eq T 
+(THead (Flat Appl) u0 t8) t)) H26 (THead (Bind b) u1 t3) H23) in (let H32 
 \def (eq_ind_r T t (\lambda (t8: T).(\forall (v: T).((tlt v t8) \to (\forall 
 (t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda 
 (t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 t11)))))))))) H (THead 
-(Flat Appl) u0 (THead (Bind b) u1 t3)) H32) in (let H34 \def (eq_ind T v1 
-(\lambda (t8: T).(pr0 t8 v2)) H10 u0 H25) in (ex2_ind T (\lambda (t8: T).(pr0 
+(Flat Appl) u0 (THead (Bind b) u1 t3)) H31) in (let H33 \def (eq_ind T v1 
+(\lambda (t8: T).(pr0 t8 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t8: T).(pr0 
 v2 t8)) (\lambda (t8: T).(pr0 u3 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead 
 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: 
 T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)) t8))) (\lambda (x: 
-T).(\lambda (H35: (pr0 v2 x)).(\lambda (H36: (pr0 u3 x)).(ex2_sym T (pr0 
+T).(\lambda (H34: (pr0 v2 x)).(\lambda (H35: (pr0 u3 x)).(ex2_sym T (pr0 
 (THead (Flat Appl) u3 (THead (Bind b) u1 t3))) (pr0 (THead (Bind b) u2 (THead 
 (Flat Appl) (lift (S O) O v2) t4))) (pr0_confluence__pr0_cong_upsilon_refl b 
-H9 u1 u2 H11 t3 t4 H12 u3 v2 x H36 H35))))) (H33 u0 (tlt_head_sx (Flat Appl) 
-u0 (THead (Bind b) u1 t3)) v2 H34 u3 H18))))) t6 H31)) t7 (sym_eq T t7 (THead 
-(Bind b) u1 t3) H29) H30))) | (pr0_comp u4 u5 H29 t7 t8 H30 k0) \Rightarrow 
-(\lambda (H31: (eq T (THead k0 u4 t7) (THead (Bind b) u1 t3))).(\lambda (H32: 
-(eq T (THead k0 u5 t8) t6)).((let H33 \def (f_equal T T (\lambda (e: 
+H9 u1 u2 H11 t3 t4 H12 u3 v2 x H35 H34))))) (H32 u0 (tlt_head_sx (Flat Appl) 
+u0 (THead (Bind b) u1 t3)) v2 H33 u3 H18))))) t6 H30)) t7 (sym_eq T t7 (THead 
+(Bind b) u1 t3) H28) H29))) | (pr0_comp u4 u5 H28 t7 t8 H29 k0) \Rightarrow 
+(\lambda (H30: (eq T (THead k0 u4 t7) (THead (Bind b) u1 t3))).(\lambda (H31: 
+(eq T (THead k0 u5 t8) t6)).((let H32 \def (f_equal T T (\lambda (e: 
 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | 
 (TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7) 
-(THead (Bind b) u1 t3) H31) in ((let H34 \def (f_equal T T (\lambda (e: 
+(THead (Bind b) u1 t3) H30) in ((let H33 \def (f_equal T T (\lambda (e: 
 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 | 
 (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) 
-(THead (Bind b) u1 t3) H31) in ((let H35 \def (f_equal T K (\lambda (e: 
+(THead (Bind b) u1 t3) H30) in ((let H34 \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 u4 t7) 
-(THead (Bind b) u1 t3) H31) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4 
+(THead (Bind b) u1 t3) H30) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4 
 u1) \to ((eq T t7 t3) \to ((eq T (THead k1 u5 t8) t6) \to ((pr0 u4 u5) \to 
 ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat 
 Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 
-t6) t9))))))))) (\lambda (H36: (eq T u4 u1)).(eq_ind T u1 (\lambda (t9: 
+t6) t9))))))))) (\lambda (H35: (eq T u4 u1)).(eq_ind T u1 (\lambda (t9: 
 T).((eq T t7 t3) \to ((eq T (THead (Bind b) u5 t8) t6) \to ((pr0 t9 u5) \to 
 ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead 
 (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat 
-Appl) u3 t6) t10)))))))) (\lambda (H37: (eq T t7 t3)).(eq_ind T t3 (\lambda 
+Appl) u3 t6) t10)))))))) (\lambda (H36: (eq T t7 t3)).(eq_ind T t3 (\lambda 
 (t9: T).((eq T (THead (Bind b) u5 t8) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) 
 \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift 
 (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) 
-t10))))))) (\lambda (H38: (eq T (THead (Bind b) u5 t8) t6)).(eq_ind T (THead 
+t10))))))) (\lambda (H37: (eq T (THead (Bind b) u5 t8) t6)).(eq_ind T (THead 
 (Bind b) u5 t8) (\lambda (t9: T).((pr0 u1 u5) \to ((pr0 t3 t8) \to (ex2 T 
 (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
 v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t9) t10)))))) 
-(\lambda (H39: (pr0 u1 u5)).(\lambda (H40: (pr0 t3 t8)).(let H41 \def 
-(eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t)) H27 
-(THead (Bind b) u1 t3) H26) in (let H42 \def (eq_ind_r T t (\lambda (t9: 
+(\lambda (H38: (pr0 u1 u5)).(\lambda (H39: (pr0 t3 t8)).(let H40 \def 
+(eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t)) H26 
+(THead (Bind b) u1 t3) H23) in (let H41 \def (eq_ind_r T t (\lambda (t9: 
 T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v t10) \to 
 (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 t12)) 
 (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u0 (THead (Bind 
-b) u1 t3)) H41) in (let H43 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) 
-H10 u0 H25) in (ex2_ind T (\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 
+b) u1 t3)) H40) in (let H42 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) 
+H10 u0 H24) in (ex2_ind T (\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 
 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) 
 (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 
-(THead (Bind b) u5 t8)) t9))) (\lambda (x: T).(\lambda (H44: (pr0 v2 
-x)).(\lambda (H45: (pr0 u3 x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) 
+(THead (Bind b) u5 t8)) t9))) (\lambda (x: T).(\lambda (H43: (pr0 v2 
+x)).(\lambda (H44: (pr0 u3 x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) 
 (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 
 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead 
-(Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x0: T).(\lambda (H46: 
-(pr0 t8 x0)).(\lambda (H47: (pr0 t4 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 
+(Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x0: T).(\lambda (H45: 
+(pr0 t8 x0)).(\lambda (H46: (pr0 t4 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 
 t9)) (\lambda (t9: T).(pr0 u2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind 
 b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 
 (THead (Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x1: T).(\lambda 
-(H48: (pr0 u5 x1)).(\lambda (H49: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat 
+(H47: (pr0 u5 x1)).(\lambda (H48: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat 
 Appl) u3 (THead (Bind b) u5 t8))) (pr0 (THead (Bind b) u2 (THead (Flat Appl) 
 (lift (S O) O v2) t4))) (pr0_confluence__pr0_cong_upsilon_cong b H9 u3 v2 x 
-H45 H44 t8 t4 x0 H46 H47 u5 u2 x1 H48 H49))))) (H42 u1 (tlt_trans (THead 
+H44 H43 t8 t4 x0 H45 H46 u5 u2 x1 H47 H48))))) (H41 u1 (tlt_trans (THead 
 (Bind b) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind b) u1 t3)) (tlt_head_sx 
-(Bind b) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t3))) u5 H39 
-u2 H11))))) (H42 t3 (tlt_trans (THead (Bind b) u1 t3) t3 (THead (Flat Appl) 
+(Bind b) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t3))) u5 H38 
+u2 H11))))) (H41 t3 (tlt_trans (THead (Bind b) u1 t3) t3 (THead (Flat Appl) 
 u0 (THead (Bind b) u1 t3)) (tlt_head_dx (Bind b) u1 t3) (tlt_head_dx (Flat 
-Appl) u0 (THead (Bind b) u1 t3))) t8 H40 t4 H12))))) (H42 u0 (tlt_head_sx 
-(Flat Appl) u0 (THead (Bind b) u1 t3)) v2 H43 u3 H18))))))) t6 H38)) t7 
-(sym_eq T t7 t3 H37))) u4 (sym_eq T u4 u1 H36))) k0 (sym_eq K k0 (Bind b) 
-H35))) H34)) H33)) H32 H29 H30))) | (pr0_beta u v0 v3 H29 t7 t8 H30) 
-\Rightarrow (\lambda (H31: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u 
-t7)) (THead (Bind b) u1 t3))).(\lambda (H32: (eq T (THead (Bind Abbr) v3 t8) 
-t6)).((let H33 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) 
+Appl) u0 (THead (Bind b) u1 t3))) t8 H39 t4 H12))))) (H41 u0 (tlt_head_sx 
+(Flat Appl) u0 (THead (Bind b) u1 t3)) v2 H42 u3 H18))))))) t6 H37)) t7 
+(sym_eq T t7 t3 H36))) u4 (sym_eq T u4 u1 H35))) k0 (sym_eq K k0 (Bind b) 
+H34))) H33)) H32)) H31 H28 H29))) | (pr0_beta u v0 v3 H28 t7 t8 H29) 
+\Rightarrow (\lambda (H30: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u 
+t7)) (THead (Bind b) u1 t3))).(\lambda (H31: (eq T (THead (Bind Abbr) v3 t8) 
+t6)).((let H32 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) 
 (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) 
 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow 
 (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False 
-| (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t3) H31) in (False_ind 
+| (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t3) H30) in (False_ind 
 ((eq T (THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to 
 (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
 O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))) 
-H33)) H32 H29 H30))) | (pr0_upsilon b0 H29 v0 v3 H30 u4 u5 H31 t7 t8 H32) 
-\Rightarrow (\lambda (H33: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 
-t7)) (THead (Bind b) u1 t3))).(\lambda (H34: (eq T (THead (Bind b0) u5 (THead 
-(Flat Appl) (lift (S O) O v3) t8)) t6)).((let H35 \def (eq_ind T (THead (Flat 
+H32)) H31 H28 H29))) | (pr0_upsilon b0 H28 v0 v3 H29 u4 u5 H30 t7 t8 H31) 
+\Rightarrow (\lambda (H32: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 
+t7)) (THead (Bind b) u1 t3))).(\lambda (H33: (eq T (THead (Bind b0) u5 (THead 
+(Flat Appl) (lift (S O) O v3) t8)) t6)).((let H34 \def (eq_ind T (THead (Flat 
 Appl) v0 (THead (Bind b0) u4 t7)) (\lambda (e: T).(match e in T return 
 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
 \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda 
 (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind b) u1 t3) H33) in (False_ind ((eq T (THead (Bind b0) 
+True])])) I (THead (Bind b) u1 t3) H32) in (False_ind ((eq T (THead (Bind b0) 
 u5 (THead (Flat Appl) (lift (S O) O v3) t8)) t6) \to ((not (eq B b0 Abst)) 
 \to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
 T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) 
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) H35)) H34 H29 H30 
-H31 H32))) | (pr0_delta u4 u5 H29 t7 t8 H30 w H31) \Rightarrow (\lambda (H32: 
-(eq T (THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3))).(\lambda (H33: (eq T 
-(THead (Bind Abbr) u5 w) t6)).((let H34 \def (f_equal T T (\lambda (e: 
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) H34)) H33 H28 H29 
+H30 H31))) | (pr0_delta u4 u5 H28 t7 t8 H29 w H30) \Rightarrow (\lambda (H31: 
+(eq T (THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3))).(\lambda (H32: (eq T 
+(THead (Bind Abbr) u5 w) t6)).((let H33 \def (f_equal T T (\lambda (e: 
 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | 
 (TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead (Bind 
-Abbr) u4 t7) (THead (Bind b) u1 t3) H32) in ((let H35 \def (f_equal T T 
+Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H34 \def (f_equal T T 
 (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
 \Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) 
-(THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3) H32) in ((let H36 \def 
+(THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H35 \def 
 (f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with 
 [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k0 _ _) 
 \Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) 
 \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7) 
-(THead (Bind b) u1 t3) H32) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u1) 
+(THead (Bind b) u1 t3) H31) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u1) 
 \to ((eq T t7 t3) \to ((eq T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u4 u5) 
 \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 
 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda 
-(t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))))) (\lambda (H37: (eq T u4 
+(t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))))) (\lambda (H36: (eq T u4 
 u1)).(eq_ind T u1 (\lambda (t9: T).((eq T t7 t3) \to ((eq T (THead (Bind 
 Abbr) u5 w) t6) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to 
 (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift 
 (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) 
-t10))))))))) (\lambda (H38: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq 
+t10))))))))) (\lambda (H37: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq 
 T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) \to ((subst0 
 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead 
 (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat 
-Appl) u3 t6) t10)))))))) (\lambda (H39: (eq T (THead (Bind Abbr) u5 w) 
+Appl) u3 t6) t10)))))))) (\lambda (H38: (eq T (THead (Bind Abbr) u5 w) 
 t6)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u1 u5) \to 
 ((pr0 t3 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead 
 (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: 
-T).(pr0 (THead (Flat Appl) u3 t9) t10))))))) (\lambda (H40: (pr0 u1 
-u5)).(\lambda (H41: (pr0 t3 t8)).(\lambda (H42: (subst0 O u5 t8 w)).(let H43 
-\def (eq_ind_r B b (\lambda (b0: B).(eq T (THead (Bind b0) u1 t3) t5)) H26 
-Abbr H36) in (let H44 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 
-Abst))) H9 Abbr H36) in (let H45 \def (eq_ind_r B b (\lambda (b0: B).(eq T 
+T).(pr0 (THead (Flat Appl) u3 t9) t10))))))) (\lambda (H39: (pr0 u1 
+u5)).(\lambda (H40: (pr0 t3 t8)).(\lambda (H41: (subst0 O u5 t8 w)).(let H42 
+\def (eq_ind_r B b (\lambda (b0: B).(eq T (THead (Bind b0) u1 t3) t5)) H23 
+Abbr H35) in (let H43 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 
+Abst))) H9 Abbr H35) in (let H44 \def (eq_ind_r B b (\lambda (b0: B).(eq T 
 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1)) H8 Abbr 
-H36) in (let H46 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat 
-Appl) u0 t9) t)) H27 (THead (Bind Abbr) u1 t3) H43) in (let H47 \def 
+H35) in (let H45 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat 
+Appl) u0 t9) t)) H26 (THead (Bind Abbr) u1 t3) H42) in (let H46 \def 
 (eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: 
 T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: 
 T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat 
-Appl) u0 (THead (Bind Abbr) u1 t3)) H46) in (let H48 \def (eq_ind T v1 
-(\lambda (t9: T).(pr0 t9 v2)) H10 u0 H25) in (ex2_ind T (\lambda (t9: T).(pr0 
+Appl) u0 (THead (Bind Abbr) u1 t3)) H45) in (let H47 \def (eq_ind T v1 
+(\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t9: T).(pr0 
 v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead 
 (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: 
 T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9))) (\lambda (x: 
-T).(\lambda (H49: (pr0 v2 x)).(\lambda (H50: (pr0 u3 x)).(ex2_ind T (\lambda 
+T).(\lambda (H48: (pr0 v2 x)).(\lambda (H49: (pr0 u3 x)).(ex2_ind T (\lambda 
 (t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: 
 T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) 
 (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9))) 
-(\lambda (x0: T).(\lambda (H51: (pr0 t8 x0)).(\lambda (H52: (pr0 t4 
+(\lambda (x0: T).(\lambda (H50: (pr0 t8 x0)).(\lambda (H51: (pr0 t4 
 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u2 t9)) 
 (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift 
 (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead 
-(Bind Abbr) u5 w)) t9))) (\lambda (x1: T).(\lambda (H53: (pr0 u5 
-x1)).(\lambda (H54: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead 
+(Bind Abbr) u5 w)) t9))) (\lambda (x1: T).(\lambda (H52: (pr0 u5 
+x1)).(\lambda (H53: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead 
 (Bind Abbr) u5 w))) (pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) 
-O v2) t4))) (pr0_confluence__pr0_cong_upsilon_delta H44 u5 t8 w H42 u3 v2 x 
-H50 H49 t4 x0 H51 H52 u2 x1 H53 H54))))) (H47 u1 (tlt_trans (THead (Bind 
+O v2) t4))) (pr0_confluence__pr0_cong_upsilon_delta H43 u5 t8 w H41 u3 v2 x 
+H49 H48 t4 x0 H50 H51 u2 x1 H52 H53))))) (H46 u1 (tlt_trans (THead (Bind 
 Abbr) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_sx 
 (Bind Abbr) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) u5 
-H40 u2 H11))))) (H47 t3 (tlt_trans (THead (Bind Abbr) u1 t3) t3 (THead (Flat 
+H39 u2 H11))))) (H46 t3 (tlt_trans (THead (Bind Abbr) u1 t3) t3 (THead (Flat 
 Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_dx (Bind Abbr) u1 t3) 
-(tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) t8 H41 t4 H12))))) 
-(H47 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) v2 H48 u3 
-H18))))))))))) t6 H39)) t7 (sym_eq T t7 t3 H38))) u4 (sym_eq T u4 u1 H37))) b 
-H36)) H35)) H34)) H33 H29 H30 H31))) | (pr0_zeta b0 H29 t7 t8 H30 u) 
-\Rightarrow (\lambda (H31: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead 
-(Bind b) u1 t3))).(\lambda (H32: (eq T t8 t6)).((let H33 \def (f_equal T T 
+(tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) t8 H40 t4 H12))))) 
+(H46 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) v2 H47 u3 
+H18))))))))))) t6 H38)) t7 (sym_eq T t7 t3 H37))) u4 (sym_eq T u4 u1 H36))) b 
+H35)) H34)) H33)) H32 H28 H29 H30))) | (pr0_zeta b0 H28 t7 t8 H29 u) 
+\Rightarrow (\lambda (H30: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead 
+(Bind b) u1 t3))).(\lambda (H31: (eq T t8 t6)).((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) (t9: T) on t9: T 
 \def (match t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow 
@@ -1788,76 +1648,75 @@ d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _)
 | (THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 
 d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ 
 t9) \Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) 
-u1 t3) H31) in ((let H34 \def (f_equal T T (\lambda (e: T).(match e in T 
+u1 t3) H30) in ((let H33 \def (f_equal T T (\lambda (e: T).(match e in T 
 return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) 
 \Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift (S 
-O) O t7)) (THead (Bind b) u1 t3) H31) in ((let H35 \def (f_equal T B (\lambda 
+O) O t7)) (THead (Bind b) u1 t3) H30) in ((let H34 \def (f_equal T B (\lambda 
 (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 
 | (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 in K return 
 (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow 
-b0])])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u1 t3) H31) in 
+b0])])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u1 t3) H30) in 
 (eq_ind B b (\lambda (b1: B).((eq T u u1) \to ((eq T (lift (S O) O t7) t3) 
 \to ((eq T t8 t6) \to ((not (eq B b1 Abst)) \to ((pr0 t7 t8) \to (ex2 T 
 (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
 t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))))))) 
-(\lambda (H36: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O 
+(\lambda (H35: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O 
 t7) t3) \to ((eq T t8 t6) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to (ex2 
 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
 v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))))))) 
-(\lambda (H37: (eq T (lift (S O) O t7) t3)).(eq_ind T (lift (S O) O t7) 
+(\lambda (H36: (eq T (lift (S O) O t7) t3)).(eq_ind T (lift (S O) O t7) 
 (\lambda (_: T).((eq T t8 t6) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to 
 (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
 O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) 
-t10))))))) (\lambda (H38: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not 
+t10))))))) (\lambda (H37: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not 
 (eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead 
 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: 
-T).(pr0 (THead (Flat Appl) u3 t6) t10)))))) (\lambda (H39: (not (eq B b 
-Abst))).(\lambda (H40: (pr0 t7 t6)).(let H41 \def (eq_ind_r T t3 (\lambda 
-(t9: T).(eq T (THead (Bind b) u1 t9) t5)) H26 (lift (S O) O t7) H37) in (let 
-H42 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t)) 
-H27 (THead (Bind b) u1 (lift (S O) O t7)) H41) in (let H43 \def (eq_ind_r T t 
+T).(pr0 (THead (Flat Appl) u3 t6) t10)))))) (\lambda (H38: (not (eq B b 
+Abst))).(\lambda (H39: (pr0 t7 t6)).(let H40 \def (eq_ind_r T t3 (\lambda 
+(t9: T).(eq T (THead (Bind b) u1 t9) t5)) H23 (lift (S O) O t7) H36) in (let 
+H41 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t)) 
+H26 (THead (Bind b) u1 (lift (S O) O t7)) H40) in (let H42 \def (eq_ind_r T t 
 (\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v 
 t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 
 t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u0 (THead 
-(Bind b) u1 (lift (S O) O t7))) H42) in (let H44 \def (eq_ind_r T t3 (\lambda 
-(t9: T).(pr0 t9 t4)) H12 (lift (S O) O t7) H37) in (ex2_ind T (\lambda (t9: 
+(Bind b) u1 (lift (S O) O t7))) H41) in (let H43 \def (eq_ind_r T t3 (\lambda 
+(t9: T).(pr0 t9 t4)) H12 (lift (S O) O t7) H36) in (ex2_ind T (\lambda (t9: 
 T).(eq T t4 (lift (S O) O t9))) (\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda 
 (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
 t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda (x: 
-T).(\lambda (H45: (eq T t4 (lift (S O) O x))).(\lambda (H46: (pr0 t7 
+T).(\lambda (H44: (eq T t4 (lift (S O) O x))).(\lambda (H45: (pr0 t7 
 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: T).(ex2 T (\lambda (t10: 
 T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t9)) t10)) 
-(\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))) (let H47 \def 
-(eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H10 u0 H25) in (ex2_ind T (\lambda 
+(\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))) (let H46 \def 
+(eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda 
 (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: 
 T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O 
 x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda (x0: 
-T).(\lambda (H48: (pr0 v2 x0)).(\lambda (H49: (pr0 u3 x0)).(ex2_ind T 
+T).(\lambda (H47: (pr0 v2 x0)).(\lambda (H48: (pr0 u3 x0)).(ex2_ind T 
 (\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda 
 (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S 
 O) O x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda 
-(x1: T).(\lambda (H50: (pr0 x x1)).(\lambda (H51: (pr0 t6 x1)).(ex2_sym T 
+(x1: T).(\lambda (H49: (pr0 x x1)).(\lambda (H50: (pr0 t6 x1)).(ex2_sym T 
 (pr0 (THead (Flat Appl) u3 t6)) (pr0 (THead (Bind b) u2 (THead (Flat Appl) 
 (lift (S O) O v2) (lift (S O) O x)))) (pr0_confluence__pr0_cong_upsilon_zeta 
-b H39 u1 u2 H11 u3 v2 x0 H49 H48 x t6 x1 H50 H51))))) (H43 t7 (tlt_trans 
+b H38 u1 u2 H11 u3 v2 x0 H48 H47 x t6 x1 H49 H50))))) (H42 t7 (tlt_trans 
 (THead (Bind b) u1 (lift (S O) O t7)) t7 (THead (Flat Appl) u0 (THead (Bind 
 b) u1 (lift (S O) O t7))) (lift_tlt_dx (Bind b) u1 t7 (S O) O) (tlt_head_dx 
-(Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7)))) x H46 t6 H40))))) (H43 
-u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7))) v2 H47 
-u3 H18))) t4 H45)))) (pr0_gen_lift t7 t4 (S O) O H44)))))))) t8 (sym_eq T t8 
-t6 H38))) t3 H37)) u (sym_eq T u u1 H36))) b0 (sym_eq B b0 b H35))) H34)) 
-H33)) H32 H29 H30))) | (pr0_epsilon t7 t8 H29 u) \Rightarrow (\lambda (H30: 
-(eq T (THead (Flat Cast) u t7) (THead (Bind b) u1 t3))).(\lambda (H31: (eq T 
-t8 t6)).((let H32 \def (eq_ind T (THead (Flat Cast) u t7) (\lambda (e: 
+(Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7)))) x H45 t6 H39))))) (H42 
+u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7))) v2 H46 
+u3 H18))) t4 H44)))) (pr0_gen_lift t7 t4 (S O) O H43)))))))) t8 (sym_eq T t8 
+t6 H37))) t3 H36)) u (sym_eq T u u1 H35))) b0 (sym_eq B b0 b H34))) H33)) 
+H32)) H31 H28 H29))) | (pr0_epsilon t7 t8 H28 u) \Rightarrow (\lambda (H29: 
+(eq T (THead (Flat Cast) u t7) (THead (Bind b) u1 t3))).(\lambda (H30: (eq T 
+t8 t6)).((let H31 \def (eq_ind T (THead (Flat Cast) u t7) (\lambda (e: 
 T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
 False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in 
 K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) 
-\Rightarrow True])])) I (THead (Bind b) u1 t3) H30) in (False_ind ((eq T t8 
+\Rightarrow True])])) I (THead (Bind b) u1 t3) H29) in (False_ind ((eq T t8 
 t6) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 
 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead 
-(Flat Appl) u3 t6) t9))))) H32)) H31 H29)))]) in (H29 (refl_equal T (THead 
-(Bind b) u1 t3)) (refl_equal T t6))))) t5 H26)) v1 (sym_eq T v1 u0 H25))) k 
-H24)) H23)) H22)))]) in (H21 (refl_equal T (THead k u0 t5))))))) t2 H17)) t 
+(Flat Appl) u3 t6) t9))))) H31)) H30 H28)))]) in (H28 (refl_equal T (THead 
+(Bind b) u1 t3)) (refl_equal T t6))))) k H25)))) H22)) H21))))) t2 H17)) t 
 H15 H16 H13 H14))) | (pr0_beta u v0 v3 H13 t5 t6 H14) \Rightarrow (\lambda 
 (H15: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) t)).(\lambda 
 (H16: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T (THead (Flat Appl) v0 
@@ -1870,562 +1729,448 @@ t2)).(eq_ind T (THead (Bind Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to
 Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda 
 (_: (pr0 v0 v3)).(\lambda (_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t 
 (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 
-(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H15) in (let H21 \def (match 
-H20 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead 
-(Flat Appl) v0 (THead (Bind Abst) u t5))) \to (ex2 T (\lambda (t8: T).(pr0 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda 
-(t8: T).(pr0 (THead (Bind Abbr) v3 t6) t8)))))) with [refl_equal \Rightarrow 
-(\lambda (H21: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead 
-(Flat Appl) v0 (THead (Bind Abst) u t5)))).(let H22 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow (match 
-t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) 
-\Rightarrow t3 | (THead _ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H21) 
-in ((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda 
+(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H15) in (let H21 \def 
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
+[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) 
+\Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat 
+Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H22 \def (f_equal T B 
+(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) 
+\Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t7) \Rightarrow (match 
+t7 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 
+(Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind 
+Abst) u t5)) H20) in ((let H23 \def (f_equal T T (\lambda (e: T).(match e in 
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
+\Rightarrow u1 | (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda 
 (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead 
-_ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t8 _) \Rightarrow 
+_ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) 
+(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H24 \def 
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
+[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) 
+\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow 
 t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 
-(THead (Bind Abst) u t5)) H21) in ((let H24 \def (f_equal T B (\lambda (e: 
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | 
-(TLRef _) \Rightarrow b | (THead _ _ t7) \Rightarrow (match t7 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 (Flat Appl) v1 
-(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H21) 
-in ((let H25 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda 
-(_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead 
-_ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) 
-(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H21) in (eq_ind T v0 (\lambda 
-(_: T).((eq B b Abst) \to ((eq T u1 u) \to ((eq T t3 t5) \to (ex2 T (\lambda 
-(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v3 t6) t8))))))) (\lambda (H26: 
-(eq B b Abst)).(eq_ind B Abst (\lambda (b0: B).((eq T u1 u) \to ((eq T t3 t5) 
-\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) 
-t7)))))) (\lambda (H27: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t3 
-t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v3 t6) 
-t8))))) (\lambda (H28: (eq T t3 t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T 
-(\lambda (t8: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v3 t6) t8)))) (let H29 
-\def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall 
-(t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda 
-(t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat 
-Appl) v0 (THead (Bind Abst) u t5)) H15) in (let H30 \def (eq_ind T t3 
-(\lambda (t7: T).(pr0 t7 t4)) H12 t5 H28) in (let H31 \def (eq_ind T u1 
-(\lambda (t7: T).(pr0 t7 u2)) H11 u H27) in (let H32 \def (eq_ind B b 
-(\lambda (b0: B).(not (eq B b0 Abst))) H9 Abst H26) in (let H33 \def (match 
-(H32 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda 
+(THead (Bind Abst) u t5)) H20) in (\lambda (_: (eq T u1 u)).(\lambda (H26: 
+(eq B b Abst)).(\lambda (H27: (eq T v1 v0)).(let H28 \def (eq_ind_r T t 
+(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) 
+\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind 
+Abst) u t5)) H15) in (let H29 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) 
+H10 v0 H27) in (eq_ind_r B Abst (\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0 
+(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda 
+(t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) (let H30 \def (eq_ind B b 
+(\lambda (b0: B).(not (eq B b0 Abst))) H9 Abst H26) in (let H31 \def (match 
+(H30 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda 
 (t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) with []) in 
-H33))))) t3 (sym_eq T t3 t5 H28))) u1 (sym_eq T u1 u H27))) b (sym_eq B b 
-Abst H26))) v1 (sym_eq T v1 v0 H25))) H24)) H23)) H22)))]) in (H21 
-(refl_equal T (THead (Flat Appl) v0 (THead (Bind Abst) u t5)))))))) t2 H17)) 
-t H15 H16 H13 H14))) | (pr0_upsilon b0 H13 v0 v3 H14 u0 u3 H15 t5 t6 H16) 
-\Rightarrow (\lambda (H17: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u0 
-t5)) t)).(\lambda (H18: (eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S 
-O) O v3) t6)) t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) 
-(\lambda (_: T).((eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O 
-v3) t6)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to 
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) 
-(\lambda (H19: (eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) 
-t6)) t2)).(eq_ind T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) 
-t6)) (\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) 
-\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) 
-(\lambda (_: (not (eq B b0 Abst))).(\lambda (H21: (pr0 v0 v3)).(\lambda (H22: 
-(pr0 u0 u3)).(\lambda (H23: (pr0 t5 t6)).(let H24 \def (eq_ind_r T t (\lambda 
-(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead 
-(Flat Appl) v0 (THead (Bind b0) u0 t5)) H17) in (let H25 \def (match H24 in 
-eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat 
-Appl) v0 (THead (Bind b0) u0 t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: 
-T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) 
-t8)))))) with [refl_equal \Rightarrow (\lambda (H25: (eq T (THead (Flat Appl) 
-v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 
-t5)))).(let H26 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 
-| (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) 
-\Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead 
-(Flat Appl) v0 (THead (Bind b0) u0 t5)) H25) in ((let H27 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ _ t7) \Rightarrow (match 
-t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
-\Rightarrow u1 | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H25) 
-in ((let H28 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda 
+t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) with []) in H31)) 
+b H26))))))) H23)) H22)) H21))))) t2 H17)) t H15 H16 H13 H14))) | 
+(pr0_upsilon b0 H13 v0 v3 H14 u0 u3 H15 t5 t6 H16) \Rightarrow (\lambda (H17: 
+(eq T (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) t)).(\lambda (H18: (eq T 
+(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T 
+(THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (\lambda (_: T).((eq T (THead 
+(Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b0 
+Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda 
+(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H19: (eq T (THead (Bind 
+b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Bind 
+b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) (\lambda (t7: T).((not (eq B 
+b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T 
+(\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not (eq B b0 
+Abst))).(\lambda (H21: (pr0 v0 v3)).(\lambda (H22: (pr0 u0 u3)).(\lambda 
+(H23: (pr0 t5 t6)).(let H24 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead 
+(Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Flat Appl) v0 (THead 
+(Bind b0) u0 t5)) H17) in (let H25 \def (f_equal T T (\lambda (e: T).(match e 
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) 
+\Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) 
+in ((let H26 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda 
 (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead _ 
 _ t7) \Rightarrow (match t7 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 b1) \Rightarrow b1 | (Flat _) 
 \Rightarrow b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead 
-(Flat Appl) v0 (THead (Bind b0) u0 t5)) H25) in ((let H29 \def (f_equal T T 
+(Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) in ((let H27 \def (f_equal T T 
 (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) 
-(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead 
-(Bind b0) u0 t5)) H25) in (eq_ind T v0 (\lambda (_: T).((eq B b b0) \to ((eq 
-T u1 u0) \to ((eq T t3 t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 
-(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t8))))))) 
-(\lambda (H30: (eq B b b0)).(eq_ind B b0 (\lambda (b1: B).((eq T u1 u0) \to 
-((eq T t3 t5) \to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b1) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind 
-b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t7)))))) (\lambda (H31: (eq 
-T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t3 t5) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) 
-(\lambda (t8: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O 
-v3) t6)) t8))))) (\lambda (H32: (eq T t3 t5)).(eq_ind T t5 (\lambda (_: 
-T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead (Bind b0) u3 (THead (Flat 
-Appl) (lift (S O) O v3) t6)) t8)))) (let H33 \def (eq_ind_r T t (\lambda (t7: 
-T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall 
-(t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: 
-T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) 
-H17) in (let H34 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H12 t5 H32) 
-in (let H35 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H11 u0 H31) in 
-(let H36 \def (eq_ind B b (\lambda (b1: B).(not (eq B b1 Abst))) H9 b0 H30) 
-in (let H37 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) H10 v0 H29) in 
-(ex2_ind T (\lambda (t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T 
+\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ _ t7) \Rightarrow (match 
+t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
+\Rightarrow u1 | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) 
+in ((let H28 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda 
+(_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead 
+_ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow 
+t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 
+(THead (Bind b0) u0 t5)) H24) in (\lambda (H29: (eq T u1 u0)).(\lambda (H30: 
+(eq B b b0)).(\lambda (H31: (eq T v1 v0)).(let H32 \def (eq_ind_r T t 
+(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) 
+\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind 
+b0) u0 t5)) H17) in (let H33 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) 
+H10 v0 H31) in (eq_ind_r B b0 (\lambda (b1: B).(ex2 T (\lambda (t7: T).(pr0 
+(THead (Bind b1) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda 
+(t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) 
+t7)))) (let H34 \def (eq_ind B b (\lambda (b1: B).(not (eq B b1 Abst))) H9 b0 
+H30) in (let H35 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H11 u0 H29) 
+in (let H36 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H12 t5 H28) in 
+(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T 
 (\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O 
 v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) 
-(lift (S O) O v3) t6)) t7))) (\lambda (x: T).(\lambda (H38: (pr0 v2 
-x)).(\lambda (H39: (pr0 v3 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) 
+(lift (S O) O v3) t6)) t7))) (\lambda (x: T).(\lambda (H37: (pr0 t4 
+x)).(\lambda (H38: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) 
 (\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0) 
 u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 
 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t7))) (\lambda 
-(x0: T).(\lambda (H40: (pr0 u2 x0)).(\lambda (H41: (pr0 u3 x0)).(ex2_ind T 
-(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda 
+(x0: T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: (pr0 u3 x0)).(ex2_ind T 
+(\lambda (t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda 
 (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
 t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) 
-O v3) t6)) t7))) (\lambda (x1: T).(\lambda (H42: (pr0 t4 x1)).(\lambda (H43: 
-(pr0 t6 x1)).(pr0_confluence__pr0_upsilon_upsilon b0 H36 v2 v3 x H38 H39 u2 
-u3 x0 H40 H41 t4 t6 x1 H42 H43)))) (H33 t5 (tlt_trans (THead (Bind b0) u0 t5) 
-t5 (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (tlt_head_dx (Bind b0) u0 
-t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t5))) t4 H34 t6 H23))))) 
-(H33 u0 (tlt_trans (THead (Bind b0) u0 t5) u0 (THead (Flat Appl) v0 (THead 
-(Bind b0) u0 t5)) (tlt_head_sx (Bind b0) u0 t5) (tlt_head_dx (Flat Appl) v0 
-(THead (Bind b0) u0 t5))) u2 H35 u3 H22))))) (H33 v0 (tlt_head_sx (Flat Appl) 
-v0 (THead (Bind b0) u0 t5)) v2 H37 v3 H21))))))) t3 (sym_eq T t3 t5 H32))) u1 
-(sym_eq T u1 u0 H31))) b (sym_eq B b b0 H30))) v1 (sym_eq T v1 v0 H29))) 
-H28)) H27)) H26)))]) in (H25 (refl_equal T (THead (Flat Appl) v0 (THead (Bind 
-b0) u0 t5)))))))))) t2 H19)) t H17 H18 H13 H14 H15 H16))) | (pr0_delta u0 u3 
-H13 t5 t6 H14 w H15) \Rightarrow (\lambda (H16: (eq T (THead (Bind Abbr) u0 
-t5) t)).(\lambda (H17: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead 
-(Bind Abbr) u0 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u3 w) t2) \to 
-((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda 
-(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H18: (eq T (THead (Bind 
-Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 
+O v3) t6)) t7))) (\lambda (x1: T).(\lambda (H41: (pr0 v2 x1)).(\lambda (H42: 
+(pr0 v3 x1)).(pr0_confluence__pr0_upsilon_upsilon b0 H34 v2 v3 x1 H41 H42 u2 
+u3 x0 H39 H40 t4 t6 x H37 H38)))) (H32 v0 (tlt_head_sx (Flat Appl) v0 (THead 
+(Bind b0) u0 t5)) v2 H33 v3 H21))))) (H32 u0 (tlt_trans (THead (Bind b0) u0 
+t5) u0 (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (tlt_head_sx (Bind b0) 
+u0 t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t5))) u2 H35 u3 
+H22))))) (H32 t5 (tlt_trans (THead (Bind b0) u0 t5) t5 (THead (Flat Appl) v0 
+(THead (Bind b0) u0 t5)) (tlt_head_dx (Bind b0) u0 t5) (tlt_head_dx (Flat 
+Appl) v0 (THead (Bind b0) u0 t5))) t4 H36 t6 H23))))) b H30))))))) H27)) 
+H26)) H25))))))) t2 H19)) t H17 H18 H13 H14 H15 H16))) | (pr0_delta u0 u3 H13 
+t5 t6 H14 w H15) \Rightarrow (\lambda (H16: (eq T (THead (Bind Abbr) u0 t5) 
+t)).(\lambda (H17: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind 
+Abbr) u0 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u3 w) t2) \to ((pr0 u0 
 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 
 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda 
-(t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5 
+(t8: T).(pr0 t2 t8)))))))) (\lambda (H18: (eq T (THead (Bind Abbr) u3 w) 
+t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to 
+((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: 
+T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5 
 t6)).(\lambda (_: (subst0 O u3 t6 w)).(let H22 \def (eq_ind_r T t (\lambda 
 (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead 
-(Bind Abbr) u0 t5) H16) in (let H23 \def (match H22 in eq return (\lambda 
-(t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind Abbr) u0 t5)) \to 
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u3 w) t8)))))) 
-with [refl_equal \Rightarrow (\lambda (H23: (eq T (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) (THead (Bind Abbr) u0 t5))).(let H24 \def (eq_ind T 
-(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (\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) u0 t5) H23) in (False_ind (ex2 T (\lambda (t7: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) 
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7))) H24)))]) in (H23 
-(refl_equal T (THead (Bind Abbr) u0 t5)))))))) t2 H18)) t H16 H17 H13 H14 
-H15))) | (pr0_zeta b0 H13 t5 t6 H14 u) \Rightarrow (\lambda (H15: (eq T 
-(THead (Bind b0) u (lift (S O) O t5)) t)).(\lambda (H16: (eq T t6 
-t2)).(eq_ind T (THead (Bind b0) u (lift (S O) O t5)) (\lambda (_: T).((eq T 
-t6 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) 
-(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 
-(\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda 
-(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b0 
-Abst))).(\lambda (_: (pr0 t5 t2)).(let H20 \def (eq_ind_r T t (\lambda (t7: 
-T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Bind 
-b0) u (lift (S O) O t5)) H15) in (let H21 \def (match H20 in eq return 
-(\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind b0) u (lift 
-(S O) O t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) 
-with [refl_equal \Rightarrow (\lambda (H21: (eq T (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) (THead (Bind b0) u (lift (S O) O t5)))).(let H22 \def 
-(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (\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 b0) u (lift (S O) O t5)) H21) in 
-(False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) H22)))]) in (H21 
-(refl_equal T (THead (Bind b0) u (lift (S O) O t5)))))))) t6 (sym_eq T t6 t2 
-H17))) t H15 H16 H13 H14))) | (pr0_epsilon t5 t6 H13 u) \Rightarrow (\lambda 
-(H14: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H15: (eq T t6 t2)).(eq_ind 
-T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to 
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H16: (eq T t6 
-t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) 
-(\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 t2)).(let H18 \def 
-(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 
-t3)) t7)) H6 (THead (Flat Cast) u t5) H14) in (let H19 \def (match H18 in eq 
-return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat 
-Cast) u t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) with 
-[refl_equal \Rightarrow (\lambda (H19: (eq T (THead (Flat Appl) v1 (THead 
-(Bind b) u1 t3)) (THead (Flat Cast) u t5))).(let H20 \def (eq_ind T (THead 
-(Flat Appl) v1 (THead (Bind b) u1 t3)) (\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) u t5) H19) in (False_ind (ex2 T 
-(\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) H20)))]) in (H19 (refl_equal T 
-(THead (Flat Cast) u t5)))))) t6 (sym_eq T t6 t2 H16))) t H14 H15 H13)))]) in 
-(H13 (refl_equal T t) (refl_equal T t2))))))) t1 H8)) t H6 H7 H2 H3 H4 H5))) 
-| (pr0_delta u1 u2 H2 t3 t4 H3 w H4) \Rightarrow (\lambda (H5: (eq T (THead 
-(Bind Abbr) u1 t3) t)).(\lambda (H6: (eq T (THead (Bind Abbr) u2 w) 
-t1)).(eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (_: T).((eq T (THead (Bind 
-Abbr) u2 w) t1) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to 
-(ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))) 
-(\lambda (H7: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead (Bind 
-Abbr) u2 w) (\lambda (t5: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 
-t4 w) \to (ex2 T (\lambda (t6: T).(pr0 t5 t6)) (\lambda (t6: T).(pr0 t2 
-t6))))))) (\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (pr0 t3 t4)).(\lambda 
-(H10: (subst0 O u2 t4 w)).(let H11 \def (match H1 in pr0 return (\lambda (t5: 
-T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) 
-\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: 
-T).(pr0 t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H11: (eq T t5 
-t)).(\lambda (H12: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) 
-\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: 
-T).(pr0 t2 t7))))) (\lambda (H13: (eq T t t2)).(eq_ind T t2 (\lambda (_: 
-T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: 
-T).(pr0 t2 t7)))) (let H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) 
-H13 (THead (Bind Abbr) u1 t3) H5) in (eq_ind T (THead (Bind Abbr) u1 t3) 
-(\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) 
-(\lambda (t7: T).(pr0 t6 t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6: 
-T).(eq T t5 t6)) H11 (THead (Bind Abbr) u1 t3) H5) in (let H16 \def (eq_ind_r 
-T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v 
-t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) 
-(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t3) H5) in 
-(ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind Abbr) u2 w) t6)) (\lambda 
-(t6: T).(pr0 (THead (Bind Abbr) u1 t3) t6)) (THead (Bind Abbr) u2 w) 
-(pr0_refl (THead (Bind Abbr) u2 w)) (pr0_delta u1 u2 H8 t3 t4 H9 w H10)))) t2 
-H14)) t (sym_eq T t t2 H13))) t5 (sym_eq T t5 t H11) H12))) | (pr0_comp u0 u3 
-H11 t5 t6 H12 k) \Rightarrow (\lambda (H13: (eq T (THead k u0 t5) 
-t)).(\lambda (H14: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u0 t5) 
-(\lambda (_: T).((eq T (THead k u3 t6) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) 
-\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: 
-T).(pr0 t2 t8))))))) (\lambda (H15: (eq T (THead k u3 t6) t2)).(eq_ind T 
-(THead k u3 t6) (\lambda (t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T 
-(\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 
-t8)))))) (\lambda (H16: (pr0 u0 u3)).(\lambda (H17: (pr0 t5 t6)).(let H18 
-\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 
-(THead k u0 t5) H13) in (let H19 \def (match H18 in eq return (\lambda (t7: 
-T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead k u0 t5)) \to (ex2 T (\lambda 
-(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 (THead k u3 
-t6) t8)))))) with [refl_equal \Rightarrow (\lambda (H19: (eq T (THead (Bind 
-Abbr) u1 t3) (THead k u0 t5))).(let H20 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | 
-(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind 
-Abbr) u1 t3) (THead k u0 t5) H19) in ((let H21 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | 
-(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind 
-Abbr) u1 t3) (THead k u0 t5) H19) in ((let H22 \def (f_equal T K (\lambda (e: 
-T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind 
-Abbr) | (TLRef _) \Rightarrow (Bind Abbr) | (THead k0 _ _) \Rightarrow k0])) 
-(THead (Bind Abbr) u1 t3) (THead k u0 t5) H19) in (eq_ind K (Bind Abbr) 
-(\lambda (k0: K).((eq T u1 u0) \to ((eq T t3 t5) \to (ex2 T (\lambda (t7: 
-T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) 
-t7)))))) (\lambda (H23: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t3 
-t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda 
-(t8: T).(pr0 (THead (Bind Abbr) u3 t6) t8))))) (\lambda (H24: (eq T t3 
-t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind 
-Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u3 t6) t8)))) (let 
-H25 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H13 (Bind 
-Abbr) H22) in (let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: 
+(Bind Abbr) u0 t5) H16) in (let H23 \def (eq_ind T (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
+Abbr) u0 t5) H22) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 
+(THead (Bind Abbr) u3 w) t7))) H23)))))) t2 H18)) t H16 H17 H13 H14 H15))) | 
+(pr0_zeta b0 H13 t5 t6 H14 u) \Rightarrow (\lambda (H15: (eq T (THead (Bind 
+b0) u (lift (S O) O t5)) t)).(\lambda (H16: (eq T t6 t2)).(eq_ind T (THead 
+(Bind b0) u (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B 
+b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 
+t8))))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not 
+(eq B b0 Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: 
+T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (_: (pr0 t5 
+t2)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) t7)) H6 (THead (Bind b0) u (lift (S O) O t5)) H15) in 
+(let H21 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) 
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | 
+(Flat _) \Rightarrow True])])) I (THead (Bind b0) u (lift (S O) O t5)) H20) 
+in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) H21))))) t6 
+(sym_eq T t6 t2 H17))) t H15 H16 H13 H14))) | (pr0_epsilon t5 t6 H13 u) 
+\Rightarrow (\lambda (H14: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H15: 
+(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 
+t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 
+t8)))))) (\lambda (H16: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 
+t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: 
+(pr0 t5 t2)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat 
+Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Flat Cast) u t5) H14) in 
+(let H19 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) 
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | 
+(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl 
+\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t5) 
+H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) 
+H19)))) t6 (sym_eq T t6 t2 H16))) t H14 H15 H13)))]) in (H13 (refl_equal T t) 
+(refl_equal T t2))))))) t1 H8)) t H6 H7 H2 H3 H4 H5))) | (pr0_delta u1 u2 H2 
+t3 t4 H3 w H4) \Rightarrow (\lambda (H5: (eq T (THead (Bind Abbr) u1 t3) 
+t)).(\lambda (H6: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead (Bind 
+Abbr) u1 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t1) \to ((pr0 u1 
+u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t6: T).(pr0 
+t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H7: (eq T (THead (Bind 
+Abbr) u2 w) t1)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t5: T).((pr0 u1 
+u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t6: T).(pr0 
+t5 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H8: (pr0 u1 
+u2)).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (subst0 O u2 t4 w)).(let H11 
+\def (match H1 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda (_: 
+(pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 
+(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with 
+[(pr0_refl t5) \Rightarrow (\lambda (H11: (eq T t5 t)).(\lambda (H12: (eq T 
+t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: 
+T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))))) 
+(\lambda (H13: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: 
+T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let 
+H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H13 (THead (Bind Abbr) 
+u1 t3) H5) in (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (t6: T).(ex2 T 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t6 
+t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H11 (THead 
+(Bind Abbr) u1 t3) H5) in (let H16 \def (eq_ind_r T t (\lambda (t6: 
+T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall 
+(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: 
+T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t3) H5) in (ex_intro2 T 
+(\lambda (t6: T).(pr0 (THead (Bind Abbr) u2 w) t6)) (\lambda (t6: T).(pr0 
+(THead (Bind Abbr) u1 t3) t6)) (THead (Bind Abbr) u2 w) (pr0_refl (THead 
+(Bind Abbr) u2 w)) (pr0_delta u1 u2 H8 t3 t4 H9 w H10)))) t2 H14)) t (sym_eq 
+T t t2 H13))) t5 (sym_eq T t5 t H11) H12))) | (pr0_comp u0 u3 H11 t5 t6 H12 
+k) \Rightarrow (\lambda (H13: (eq T (THead k u0 t5) t)).(\lambda (H14: (eq T 
+(THead k u3 t6) t2)).(eq_ind T (THead k u0 t5) (\lambda (_: T).((eq T (THead 
+k u3 t6) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
+(\lambda (H15: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u3 t6) (\lambda 
+(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
+(Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H16: (pr0 
+u0 u3)).(\lambda (H17: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: 
+T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead k u0 t5) H13) in (let H19 
+\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) 
+with [(TSort _) \Rightarrow (Bind Abbr) | (TLRef _) \Rightarrow (Bind Abbr) | 
+(THead k0 _ _) \Rightarrow k0])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) 
+H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return 
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 
+| (THead _ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) 
+H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return 
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 
+| (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) 
+H18) in (\lambda (H22: (eq T u1 u0)).(\lambda (H23: (eq K (Bind Abbr) 
+k)).(eq_ind K (Bind Abbr) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 
+(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) 
+(let H24 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H13 
+(Bind Abbr) H23) in (let H25 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: 
 T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v 
 t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
-t10)))))))))) H (THead (Bind Abbr) u0 t5) H25) in (let H27 \def (eq_ind T t3 
-(\lambda (t7: T).(pr0 t7 t4)) H9 t5 H24) in (let H28 \def (eq_ind T u1 
-(\lambda (t7: T).(pr0 t7 u2)) H8 u0 H23) in (ex2_ind T (\lambda (t7: T).(pr0 
-u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead 
+t10)))))))))) H (THead (Bind Abbr) u0 t5) H24) in (let H26 \def (eq_ind T u1 
+(\lambda (t7: T).(pr0 t7 u2)) H8 u0 H22) in (let H27 \def (eq_ind T t3 
+(\lambda (t7: T).(pr0 t7 t4)) H9 t5 H21) in (ex2_ind T (\lambda (t7: T).(pr0 
+t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead 
 (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 t6) t7))) 
-(\lambda (x: T).(\lambda (H29: (pr0 u2 x)).(\lambda (H30: (pr0 u3 
-x)).(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) 
+(\lambda (x: T).(\lambda (H28: (pr0 t4 x)).(\lambda (H29: (pr0 t6 
+x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) 
 (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: 
-T).(pr0 (THead (Bind Abbr) u3 t6) t7))) (\lambda (x0: T).(\lambda (H31: (pr0 
-t4 x0)).(\lambda (H32: (pr0 t6 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u3 
+T).(pr0 (THead (Bind Abbr) u3 t6) t7))) (\lambda (x0: T).(\lambda (H30: (pr0 
+u2 x0)).(\lambda (H31: (pr0 u3 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u3 
 t6)) (pr0 (THead (Bind Abbr) u2 w)) (pr0_confluence__pr0_cong_delta u2 t4 w 
-H10 u3 x H30 H29 t6 x0 H32 H31))))) (H26 t5 (tlt_head_dx (Bind Abbr) u0 t5) 
-t4 H27 t6 H17))))) (H26 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 H28 u3 
-H16)))))) t3 (sym_eq T t3 t5 H24))) u1 (sym_eq T u1 u0 H23))) k H22)) H21)) 
-H20)))]) in (H19 (refl_equal T (THead k u0 t5))))))) t2 H15)) t H13 H14 H11 
-H12))) | (pr0_beta u v1 v2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t5)) t)).(\lambda (H14: (eq T 
-(THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 
-v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) 
-u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H15: (eq T (THead 
-(Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: 
-T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
-(Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v1 
-v2)).(\lambda (_: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: 
-T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t5)) H13) in (let H19 \def (match H18 in eq return (\lambda (t7: 
-T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) 
-(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t6) t8)))))) with [refl_equal 
-\Rightarrow (\lambda (H19: (eq T (THead (Bind Abbr) u1 t3) (THead (Flat Appl) 
-v1 (THead (Bind Abst) u t5)))).(let H20 \def (eq_ind T (THead (Bind Abbr) u1 
-t3) (\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) v1 
-(THead (Bind Abst) u t5)) H19) in (False_ind (ex2 T (\lambda (t7: T).(pr0 
-(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) 
-t7))) H20)))]) in (H19 (refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) 
-u t5)))))))) t2 H15)) t H13 H14 H11 H12))) | (pr0_upsilon b H11 v1 v2 H12 u0 
-u3 H13 t5 t6 H14) \Rightarrow (\lambda (H15: (eq T (THead (Flat Appl) v1 
-(THead (Bind b) u0 t5)) t)).(\lambda (H16: (eq T (THead (Bind b) u3 (THead 
-(Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 
-(THead (Bind b) u0 t5)) (\lambda (_: T).((eq T (THead (Bind b) u3 (THead 
-(Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 
-v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
-(Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H17: 
-(eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) 
-t2)).(eq_ind T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) 
-(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to 
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) 
-(\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not (eq B b 
-Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 
-t5 t6)).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) 
-u1 t3) t7)) H5 (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H15) in (let H23 
-\def (match H22 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq 
-T t7 (THead (Flat Appl) v1 (THead (Bind b) u0 t5))) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 (THead (Bind b) 
-u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t8)))))) with [refl_equal 
-\Rightarrow (\lambda (H23: (eq T (THead (Bind Abbr) u1 t3) (THead (Flat Appl) 
-v1 (THead (Bind b) u0 t5)))).(let H24 \def (eq_ind T (THead (Bind Abbr) u1 
-t3) (\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) v1 
-(THead (Bind b) u0 t5)) H23) in (False_ind (ex2 T (\lambda (t7: T).(pr0 
-(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u3 (THead 
-(Flat Appl) (lift (S O) O v2) t6)) t7))) H24)))]) in (H23 (refl_equal T 
-(THead (Flat Appl) v1 (THead (Bind b) u0 t5)))))))))) t2 H17)) t H15 H16 H11 
-H12 H13 H14))) | (pr0_delta u0 u3 H11 t5 t6 H12 w0 H13) \Rightarrow (\lambda 
-(H14: (eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H15: (eq T (THead (Bind 
-Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T 
-(THead (Bind Abbr) u3 w0) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 
-O u3 t6 w0) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) 
-(\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H16: (eq T (THead (Bind Abbr) 
-u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u3 w0) (\lambda (t7: T).((pr0 u0 u3) 
-\to ((pr0 t5 t6) \to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda (t8: T).(pr0 
-(THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda 
-(H17: (pr0 u0 u3)).(\lambda (H18: (pr0 t5 t6)).(\lambda (H19: (subst0 O u3 t6 
-w0)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 
-t3) t7)) H5 (THead (Bind Abbr) u0 t5) H14) in (let H21 \def (match H20 in eq 
-return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind 
-Abbr) u0 t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) 
-(\lambda (t8: T).(pr0 (THead (Bind Abbr) u3 w0) t8)))))) with [refl_equal 
-\Rightarrow (\lambda (H21: (eq T (THead (Bind Abbr) u1 t3) (THead (Bind Abbr) 
-u0 t5))).(let H22 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 
-| (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind 
-Abbr) u0 t5) H21) in ((let H23 \def (f_equal T T (\lambda (e: T).(match e in 
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
-\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) 
-(THead (Bind Abbr) u0 t5) H21) in (eq_ind T u0 (\lambda (_: T).((eq T t3 t5) 
-\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: 
-T).(pr0 (THead (Bind Abbr) u3 w0) t8))))) (\lambda (H24: (eq T t3 
-t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind 
-Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u3 w0) t8)))) (let 
-H25 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to 
-(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T 
-(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H 
-(THead (Bind Abbr) u0 t5) H14) in (let H26 \def (eq_ind T t3 (\lambda (t7: 
-T).(pr0 t7 t4)) H9 t5 H24) in (let H27 \def (eq_ind T u1 (\lambda (t7: 
-T).(pr0 t7 u2)) H8 u0 H23) in (ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) 
-(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) 
-u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w0) t7))) (\lambda (x: 
-T).(\lambda (H28: (pr0 u2 x)).(\lambda (H29: (pr0 u3 x)).(ex2_ind T (\lambda 
-(t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: 
-T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind 
-Abbr) u3 w0) t7))) (\lambda (x0: T).(\lambda (H30: (pr0 t4 x0)).(\lambda 
-(H31: (pr0 t6 x0)).(pr0_confluence__pr0_delta_delta u2 t4 w H10 u3 t6 w0 H19 
-x H28 H29 x0 H30 H31)))) (H25 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 
-H18))))) (H25 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 H27 u3 H17))))) t3 
-(sym_eq T t3 t5 H24))) u1 (sym_eq T u1 u0 H23))) H22)))]) in (H21 (refl_equal 
-T (THead (Bind Abbr) u0 t5)))))))) t2 H16)) t H14 H15 H11 H12 H13))) | 
-(pr0_zeta b H11 t5 t6 H12 u) \Rightarrow (\lambda (H13: (eq T (THead (Bind b) 
-u (lift (S O) O t5)) t)).(\lambda (H14: (eq T t6 t2)).(eq_ind T (THead (Bind 
-b) u (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b 
-Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) 
-u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H15: (eq T t6 
-t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to 
+H10 u3 x0 H31 H30 t6 x H29 H28))))) (H25 u0 (tlt_head_sx (Bind Abbr) u0 t5) 
+u2 H26 u3 H16))))) (H25 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H27 t6 
+H17)))))) k H23)))) H20)) H19))))) t2 H15)) t H13 H14 H11 H12))) | (pr0_beta 
+u v1 v2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 
+(THead (Bind Abst) u t5)) t)).(\lambda (H14: (eq T (THead (Bind Abbr) v2 t6) 
+t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_: 
+T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to 
 (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: 
-T).(pr0 t2 t8)))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 
-t5 t2)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) 
-u1 t3) t7)) H5 (THead (Bind b) u (lift (S O) O t5)) H13) in (let H19 \def 
-(match H18 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 
-(THead (Bind b) u (lift (S O) O t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
-(Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) with [refl_equal 
-\Rightarrow (\lambda (H19: (eq T (THead (Bind Abbr) u1 t3) (THead (Bind b) u 
-(lift (S O) O t5)))).(let H20 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) 
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) 
-(THead (Bind b) u (lift (S O) O t5)) H19) in ((let H21 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) 
-(THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift (S O) O t5)) H19) in ((let 
-H22 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) 
-with [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) 
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) 
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t3) 
-(THead (Bind b) u (lift (S O) O t5)) H19) in (eq_ind B Abbr (\lambda (_: 
-B).((eq T u1 u) \to ((eq T t3 (lift (S O) O t5)) \to (ex2 T (\lambda (t7: 
-T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))))) 
-(\lambda (H23: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t3 (lift (S O) 
-O t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) 
-(\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H24: (eq T t3 (lift (S O) O 
-t5))).(eq_ind T (lift (S O) O t5) (\lambda (_: T).(ex2 T (\lambda (t8: 
-T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))) (let 
-H25 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H16 Abbr H22) 
-in (let H26 \def (eq_ind_r B b (\lambda (b0: B).(eq T (THead (Bind b0) u 
-(lift (S O) O t5)) t)) H13 Abbr H22) in (let H27 \def (eq_ind_r T t (\lambda 
+T).(pr0 t2 t8))))))) (\lambda (H15: (eq T (THead (Bind Abbr) v2 t6) 
+t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to 
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) 
+(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 
+t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) 
+u1 t3) t7)) H5 (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H13) in (let 
+H19 \def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T 
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
+(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
+False])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H18) in 
+(False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H19))))) t2 H15)) t H13 
+H14 H11 H12))) | (pr0_upsilon b H11 v1 v2 H12 u0 u3 H13 t5 t6 H14) 
+\Rightarrow (\lambda (H15: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 
+t5)) t)).(\lambda (H16: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S 
+O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) 
+(\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O 
+v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to 
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) 
+(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H17: (eq T (THead (Bind b) u3 
+(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u3 
+(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b 
+Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda 
+(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 
+t8)))))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 
+v2)).(\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5 t6)).(let H22 \def 
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead 
+(Flat Appl) v1 (THead (Bind b) u0 t5)) H15) in (let H23 \def (eq_ind T (THead 
+(Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat 
+Appl) v1 (THead (Bind b) u0 t5)) H22) in (False_ind (ex2 T (\lambda (t7: 
+T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) 
+u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t7))) H23))))))) t2 H17)) t H15 
+H16 H11 H12 H13 H14))) | (pr0_delta u0 u3 H11 t5 t6 H12 w0 H13) \Rightarrow 
+(\lambda (H14: (eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H15: (eq T 
+(THead (Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda 
+(_: T).((eq T (THead (Bind Abbr) u3 w0) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) 
+\to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) 
+u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H16: (eq T (THead 
+(Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u3 w0) (\lambda (t7: 
+T).((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda 
+(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) 
+(\lambda (H17: (pr0 u0 u3)).(\lambda (H18: (pr0 t5 t6)).(\lambda (H19: 
+(subst0 O u3 t6 w0)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T 
+(THead (Bind Abbr) u1 t3) t7)) H5 (THead (Bind Abbr) u0 t5) H14) in (let H21 
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
+with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) 
+\Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind Abbr) u0 t5) H20) in 
+((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
+T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ 
+t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind Abbr) u0 t5) 
+H20) in (\lambda (H23: (eq T u1 u0)).(let H24 \def (eq_ind_r T t (\lambda 
 (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to 
 (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) u (lift (S O) O 
-t5)) H26) in (let H28 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 
-(lift (S O) O t5) H24) in (ex2_ind T (\lambda (t7: T).(eq T t4 (lift (S O) O 
-t7))) (\lambda (t7: T).(pr0 t5 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind 
-Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda 
-(H29: (eq T t4 (lift (S O) O x))).(\lambda (H30: (pr0 t5 x)).(let H31 \def 
-(eq_ind T t4 (\lambda (t7: T).(subst0 O u2 t7 w)) H10 (lift (S O) O x) H29) 
-in (let H32 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u H23) in 
-(ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T 
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 
-t7))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t2 
-x0)).(pr0_confluence__pr0_delta_epsilon u2 (lift (S O) O x) w H31 x (pr0_refl 
-(lift (S O) O x)) t2)))) (H27 t5 (lift_tlt_dx (Bind Abbr) u t5 (S O) O) x H30 
-t2 H17))))))) (pr0_gen_lift t5 t4 (S O) O H28)))))) t3 (sym_eq T t3 (lift (S 
-O) O t5) H24))) u1 (sym_eq T u1 u H23))) b H22)) H21)) H20)))]) in (H19 
-(refl_equal T (THead (Bind b) u (lift (S O) O t5)))))))) t6 (sym_eq T t6 t2 
-H15))) t H13 H14 H11 H12))) | (pr0_epsilon t5 t6 H11 u) \Rightarrow (\lambda 
-(H12: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H13: (eq T t6 t2)).(eq_ind 
-T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to 
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: 
-T).(pr0 t2 t8)))))) (\lambda (H14: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: 
-T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) 
-t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 t2)).(let H16 \def 
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) u0 t5) H14) in 
+(let H25 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u0 H23) in (let 
+H26 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 t5 H22) in (ex2_ind T 
+(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda 
+(t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind 
+Abbr) u3 w0) t7))) (\lambda (x: T).(\lambda (H27: (pr0 t4 x)).(\lambda (H28: 
+(pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 
+t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: 
+T).(pr0 (THead (Bind Abbr) u3 w0) t7))) (\lambda (x0: T).(\lambda (H29: (pr0 
+u2 x0)).(\lambda (H30: (pr0 u3 x0)).(pr0_confluence__pr0_delta_delta u2 t4 w 
+H10 u3 t6 w0 H19 x0 H29 H30 x H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 
+t5) u2 H25 u3 H17))))) (H24 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 
+H18))))))) H21)))))) t2 H16)) t H14 H15 H11 H12 H13))) | (pr0_zeta b H11 t5 
+t6 H12 u) \Rightarrow (\lambda (H13: (eq T (THead (Bind b) u (lift (S O) O 
+t5)) t)).(\lambda (H14: (eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O) 
+O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 
+t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda 
+(t8: T).(pr0 t2 t8))))))) (\lambda (H15: (eq T t6 t2)).(eq_ind T t2 (\lambda 
+(t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) 
+(\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5 t2)).(let H18 \def 
 (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead 
-(Flat Cast) u t5) H12) in (let H17 \def (match H16 in eq return (\lambda (t7: 
-T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Cast) u t5)) \to (ex2 T 
-(\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 
-t8)))))) with [refl_equal \Rightarrow (\lambda (H17: (eq T (THead (Bind Abbr) 
-u1 t3) (THead (Flat Cast) u t5))).(let H18 \def (eq_ind T (THead (Bind Abbr) 
-u1 t3) (\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) u t5) 
-H17) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) 
-(\lambda (t7: T).(pr0 t2 t7))) H18)))]) in (H17 (refl_equal T (THead (Flat 
-Cast) u t5)))))) t6 (sym_eq T t6 t2 H14))) t H12 H13 H11)))]) in (H11 
-(refl_equal T t) (refl_equal T t2)))))) t1 H7)) t H5 H6 H2 H3 H4))) | 
-(pr0_zeta b H2 t3 t4 H3 u) \Rightarrow (\lambda (H4: (eq T (THead (Bind b) u 
-(lift (S O) O t3)) t)).(\lambda (H5: (eq T t4 t1)).(eq_ind T (THead (Bind b) 
-u (lift (S O) O t3)) (\lambda (_: T).((eq T t4 t1) \to ((not (eq B b Abst)) 
-\to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: 
-T).(pr0 t2 t6))))))) (\lambda (H6: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: 
-T).((not (eq B b Abst)) \to ((pr0 t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 
-t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (not (eq B b 
-Abst))).(\lambda (H8: (pr0 t3 t1)).(let H9 \def (match H1 in pr0 return 
-(\lambda (t5: T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to 
-((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 
-t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 
-t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) 
-\to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) 
-(\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: 
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t 
-(\lambda (t6: T).(eq T t6 t2)) H11 (THead (Bind b) u (lift (S O) O t3)) H4) 
-in (eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\lambda (t6: T).(ex2 T 
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def 
-(eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead (Bind b) u (lift (S O) 
-O t3)) H4) in (let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: 
-T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v 
-t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 
-t9)))))))))) H (THead (Bind b) u (lift (S O) O t3)) H4) in (ex_intro2 T 
-(\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 (THead (Bind b) u (lift 
-(S O) O t3)) t6)) t1 (pr0_refl t1) (pr0_zeta b H7 t3 t1 H8 u)))) t2 H12)) t 
-(sym_eq T t t2 H11))) t5 (sym_eq T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 
-H10 k) \Rightarrow (\lambda (H11: (eq T (THead k u1 t5) t)).(\lambda (H12: 
-(eq T (THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T 
-(THead k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda 
-(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T 
-(THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) 
-\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: 
-T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H15: (pr0 t5 
-t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u 
-(lift (S O) O t3)) t7)) H4 (THead k u1 t5) H11) in (let H17 \def (match H16 
-in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead k u1 
-t5)) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead k 
-u2 t6) t8)))))) with [refl_equal \Rightarrow (\lambda (H17: (eq T (THead 
-(Bind b) u (lift (S O) O t3)) (THead k u1 t5))).(let H18 \def (f_equal T T 
+(Bind b) u (lift (S O) O t5)) H13) in (let H19 \def (f_equal T B (\lambda (e: 
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abbr | 
+(TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow (match k in K return 
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
+Abbr])])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift (S O) O t5)) H18) 
+in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda 
+(_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead 
+_ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift 
+(S O) O t5)) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in 
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) 
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) 
+(THead (Bind b) u (lift (S O) O t5)) H18) in (\lambda (H22: (eq T u1 
+u)).(\lambda (H23: (eq B Abbr b)).(let H24 \def (eq_ind_r B b (\lambda (b0: 
+B).(not (eq B b0 Abst))) H16 Abbr H23) in (let H25 \def (eq_ind_r B b 
+(\lambda (b0: B).(eq T (THead (Bind b0) u (lift (S O) O t5)) t)) H13 Abbr 
+H23) in (let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v 
+t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to 
+(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
+t10)))))))))) H (THead (Bind Abbr) u (lift (S O) O t5)) H25) in (let H27 \def 
+(eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u H22) in (let H28 \def (eq_ind 
+T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 (lift (S O) O t5) H21) in (ex2_ind T 
+(\lambda (t7: T).(eq T t4 (lift (S O) O t7))) (\lambda (t7: T).(pr0 t5 t7)) 
+(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: 
+T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H29: (eq T t4 (lift (S O) O 
+x))).(\lambda (H30: (pr0 t5 x)).(let H31 \def (eq_ind T t4 (\lambda (t7: 
+T).(subst0 O u2 t7 w)) H10 (lift (S O) O x) H29) in (ex2_ind T (\lambda (t7: 
+T).(pr0 x t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 
+(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: 
+T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t2 
+x0)).(pr0_confluence__pr0_delta_epsilon u2 (lift (S O) O x) w H31 x (pr0_refl 
+(lift (S O) O x)) t2)))) (H26 t5 (lift_tlt_dx (Bind Abbr) u t5 (S O) O) x H30 
+t2 H17)))))) (pr0_gen_lift t5 t4 (S O) O H28)))))))))) H20)) H19))))) t6 
+(sym_eq T t6 t2 H15))) t H13 H14 H11 H12))) | (pr0_epsilon t5 t6 H11 u) 
+\Rightarrow (\lambda (H12: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H13: 
+(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 
+t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 
+w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (eq T t6 
+t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))) 
+(\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T 
+(THead (Bind Abbr) u1 t3) t7)) H5 (THead (Flat Cast) u t5) H12) in (let H17 
+\def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T 
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
+(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
+False])])) I (THead (Flat Cast) u t5) H16) in (False_ind (ex2 T (\lambda (t7: 
+T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) H17)))) 
+t6 (sym_eq T t6 t2 H14))) t H12 H13 H11)))]) in (H11 (refl_equal T t) 
+(refl_equal T t2)))))) t1 H7)) t H5 H6 H2 H3 H4))) | (pr0_zeta b H2 t3 t4 H3 
+u) \Rightarrow (\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t3)) 
+t)).(\lambda (H5: (eq T t4 t1)).(eq_ind T (THead (Bind b) u (lift (S O) O 
+t3)) (\lambda (_: T).((eq T t4 t1) \to ((not (eq B b Abst)) \to ((pr0 t3 t4) 
+\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) 
+(\lambda (H6: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((not (eq B b 
+Abst)) \to ((pr0 t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda 
+(t6: T).(pr0 t2 t6)))))) (\lambda (H7: (not (eq B b Abst))).(\lambda (H8: 
+(pr0 t3 t1)).(let H9 \def (match H1 in pr0 return (\lambda (t5: T).(\lambda 
+(t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T 
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with 
+[(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 t)).(\lambda (H10: (eq T t5 
+t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: 
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H11: (eq T t 
+t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
+(\lambda (t7: T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6: 
+T).(eq T t6 t2)) H11 (THead (Bind b) u (lift (S O) O t3)) H4) in (eq_ind T 
+(THead (Bind b) u (lift (S O) O t3)) (\lambda (t6: T).(ex2 T (\lambda (t7: 
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def (eq_ind_r T t 
+(\lambda (t6: T).(eq T t5 t6)) H9 (THead (Bind b) u (lift (S O) O t3)) H4) in 
+(let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to 
+(\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T 
+(\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead 
+(Bind b) u (lift (S O) O t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 t1 
+t6)) (\lambda (t6: T).(pr0 (THead (Bind b) u (lift (S O) O t3)) t6)) t1 
+(pr0_refl t1) (pr0_zeta b H7 t3 t1 H8 u)))) t2 H12)) t (sym_eq T t t2 H11))) 
+t5 (sym_eq T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow 
+(\lambda (H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T (THead k u2 t6) 
+t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead k u2 t6) t2) \to 
+((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) 
+(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead k u2 t6) 
+t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) 
+\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))) 
+(\lambda (_: (pr0 u1 u2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r 
+T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 
+(THead k u1 t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e 
+in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef 
+_) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u 
+(lift (S O) O t3)) (THead k u1 t5) H16) in ((let H18 \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) (t7: T) on t7: T 
-\def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow 
-(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) 
-| (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map f d u0) (lref_map f (s k0 
-d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) 
-\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T 
-\def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow 
-(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) 
-| (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map f d u0) (lref_map f (s k0 
-d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ 
-t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead k u1 t5) 
-H17) in ((let H19 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | 
-(THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead 
-k u1 t5) H17) in ((let H20 \def (f_equal T K (\lambda (e: T).(match e in T 
-return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef _) 
-\Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u 
-(lift (S O) O t3)) (THead k u1 t5) H17) in (eq_ind K (Bind b) (\lambda (k0: 
-K).((eq T u u1) \to ((eq T (lift (S O) O t3) t5) \to (ex2 T (\lambda (t7: 
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead k0 u2 t6) t7)))))) (\lambda 
-(H21: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O t3) t5) 
-\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Bind 
-b) u2 t6) t8))))) (\lambda (H22: (eq T (lift (S O) O t3) t5)).(eq_ind T (lift 
-(S O) O t3) (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda 
-(t8: T).(pr0 (THead (Bind b) u2 t6) t8)))) (let H23 \def (eq_ind_r K k 
-(\lambda (k0: K).(eq T (THead k0 u1 t5) t)) H11 (Bind b) H20) in (let H24 
-\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (lift (S O) O t3) H22) 
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) \Rightarrow t7])) 
+(THead (Bind b) u (lift (S O) O t3)) (THead k u1 t5) H16) in ((let H19 \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) (t7: 
+T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) 
+\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false 
+\Rightarrow (f i)])) | (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map f d 
+u0) (lref_map f (s k0 d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S 
+O))) O t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) 
+(d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort 
+n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | 
+false \Rightarrow (f i)])) | (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map 
+f d u0) (lref_map f (s k0 d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S 
+O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O 
+t3)) (THead k u1 t5) H16) in (\lambda (_: (eq T u u1)).(\lambda (H21: (eq K 
+(Bind b) k)).(eq_ind K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 
+t1 t7)) (\lambda (t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r 
+K k (\lambda (k0: K).(eq T (THead k0 u1 t5) t)) H11 (Bind b) H21) in (let H23 
+\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (lift (S O) O t3) H19) 
 in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda (t7: 
 T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 
-(THead (Bind b) u2 t6) t7))) (\lambda (x: T).(\lambda (H25: (eq T t6 (lift (S 
-O) O x))).(\lambda (H26: (pr0 t3 x)).(let H27 \def (eq_ind_r T t5 (\lambda 
-(t7: T).(eq T (THead (Bind b) u1 t7) t)) H23 (lift (S O) O t3) H22) in (let 
-H28 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to 
+(THead (Bind b) u2 t6) t7))) (\lambda (x: T).(\lambda (H24: (eq T t6 (lift (S 
+O) O x))).(\lambda (H25: (pr0 t3 x)).(let H26 \def (eq_ind_r T t5 (\lambda 
+(t7: T).(eq T (THead (Bind b) u1 t7) t)) H22 (lift (S O) O t3) H19) in (let 
+H27 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to 
 (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T 
 (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H 
-(THead (Bind b) u1 (lift (S O) O t3)) H27) in (eq_ind_r T (lift (S O) O x) 
+(THead (Bind b) u1 (lift (S O) O t3)) H26) in (eq_ind_r T (lift (S O) O x) 
 (\lambda (t7: T).(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 
 (THead (Bind b) u2 t7) t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7)) 
 (\lambda (t7: T).(pr0 t1 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda 
 (t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t7))) (\lambda (x0: 
-T).(\lambda (H29: (pr0 x x0)).(\lambda (H30: (pr0 t1 x0)).(ex_intro2 T 
+T).(\lambda (H28: (pr0 x x0)).(\lambda (H29: (pr0 t1 x0)).(ex_intro2 T 
 (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift 
-(S O) O x)) t7)) x0 H30 (pr0_zeta b H7 x x0 H29 u2))))) (H28 t3 (lift_tlt_dx 
-(Bind b) u1 t3 (S O) O) x H26 t1 H8)) t6 H25)))))) (pr0_gen_lift t3 t6 (S O) 
-O H24)))) t5 H22)) u (sym_eq T u u1 H21))) k H20)) H19)) H18)))]) in (H17 
-(refl_equal T (THead k u1 t5))))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta 
+(S O) O x)) t7)) x0 H29 (pr0_zeta b H7 x x0 H28 u2))))) (H27 t3 (lift_tlt_dx 
+(Bind b) u1 t3 (S O) O) x H25 t1 H8)) t6 H24)))))) (pr0_gen_lift t3 t6 (S O) 
+O H23)))) k H21)))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta 
 u0 v1 v2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 
 (THead (Bind Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6) 
 t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) (\lambda (_: 
@@ -2436,69 +2181,60 @@ Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T
 (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: 
 (pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda 
 (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) 
-v1 (THead (Bind Abst) u0 t5)) H11) in (let H17 \def (match H16 in eq return 
-(\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Appl) v1 
-(THead (Bind Abst) u0 t5))) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda 
-(t8: T).(pr0 (THead (Bind Abbr) v2 t6) t8)))))) with [refl_equal \Rightarrow 
-(\lambda (H17: (eq T (THead (Bind b) u (lift (S O) O t3)) (THead (Flat Appl) 
-v1 (THead (Bind Abst) u0 t5)))).(let H18 \def (eq_ind T (THead (Bind b) u 
-(lift (S O) O t3)) (\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) v1 
-(THead (Bind Abst) u0 t5)) H17) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 
-t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H18)))]) in (H17 
-(refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)))))))) t2 H13)) 
-t H11 H12 H9 H10))) | (pr0_upsilon b0 H9 v1 v2 H10 u1 u2 H11 t5 t6 H12) 
-\Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1 
-t5)) t)).(\lambda (H14: (eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b0) u1 t5)) 
-(\lambda (_: T).((eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t6)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to 
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 
-t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b0) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b0) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 v1 
-v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 
-t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not (eq B b0 
-Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 
-t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u 
-(lift (S O) O t3)) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b0) u1 t5)) 
-H13) in (let H21 \def (match H20 in eq return (\lambda (t7: T).(\lambda (_: 
-(eq ? ? t7)).((eq T t7 (THead (Flat Appl) v1 (THead (Bind b0) u1 t5))) \to 
-(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Bind b0) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t8)))))) with [refl_equal 
-\Rightarrow (\lambda (H21: (eq T (THead (Bind b) u (lift (S O) O t3)) (THead 
-(Flat Appl) v1 (THead (Bind b0) u1 t5)))).(let H22 \def (eq_ind T (THead 
-(Bind b) u (lift (S O) O t3)) (\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 
+v1 (THead (Bind Abst) u0 t5)) H11) in (let H17 \def (eq_ind T (THead (Bind b) 
+u (lift (S O) O t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
 [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat 
-Appl) v1 (THead (Bind b0) u1 t5)) H21) in (False_ind (ex2 T (\lambda (t7: 
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t6)) t7))) H22)))]) in (H21 (refl_equal T (THead (Flat 
-Appl) v1 (THead (Bind b0) u1 t5)))))))))) t2 H15)) t H13 H14 H9 H10 H11 
-H12))) | (pr0_delta u1 u2 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12: (eq 
-T (THead (Bind Abbr) u1 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) 
-t2)).(eq_ind T (THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind 
-Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to 
-(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) 
-(\lambda (H14: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind 
-Abbr) u2 w) (\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 
-t6 w) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 
-t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H16: (pr0 t5 t6)).(\lambda 
-(H17: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T 
-(THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Bind Abbr) u1 t5) H12) 
-in (let H19 \def (match H18 in eq return (\lambda (t7: T).(\lambda (_: (eq ? 
-? t7)).((eq T t7 (THead (Bind Abbr) u1 t5)) \to (ex2 T (\lambda (t8: T).(pr0 
-t1 t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)))))) with 
-[refl_equal \Rightarrow (\lambda (H19: (eq T (THead (Bind b) u (lift (S O) O 
-t3)) (THead (Bind Abbr) u1 t5))).(let H20 \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) (t7: T) on t7: T \def (match t7 
-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 
-t8) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t8))]) in 
+Appl) v1 (THead (Bind Abst) u0 t5)) H16) in (False_ind (ex2 T (\lambda (t7: 
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) 
+H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b0 H9 v1 v2 H10 u1 u2 
+H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 (THead 
+(Bind b0) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind b0) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead 
+(Bind b0) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b0) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) 
+\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) 
+(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b0) u2 
+(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b0) u2 
+(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b0 
+Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda 
+(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not 
+(eq B b0 Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead 
+(Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b0) 
+u1 t5)) H13) in (let H21 \def (eq_ind T (THead (Bind b) u (lift (S O) O t3)) 
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | 
+(Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 (THead (Bind b0) u1 
+t5)) H20) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: 
+T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t7))) 
+H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6 
+H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5) 
+t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind 
+Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 
+u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 
+t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind 
+Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u1 
+u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 
+t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda 
+(H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r 
+T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 
+(THead (Bind Abbr) u1 t5) H12) in (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) u (lift (S O) O t3)) (THead (Bind Abbr) u1 t5) H18) in 
+((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
+T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 
+_) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) 
+u1 t5) H18) in ((let H21 \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) (t7: T) on t7: T \def (match t7 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 t8) 
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t8))]) in 
 lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow 
 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match 
 t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
@@ -2506,128 +2242,96 @@ t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
 (THead k u0 t8) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
 t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t7) 
 \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) u1 
-t5) H19) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | 
-(THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead 
-(Bind Abbr) u1 t5) H19) in ((let H22 \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 t3)) (THead (Bind Abbr) u1 t5) H19) in (eq_ind B 
-Abbr (\lambda (_: B).((eq T u u1) \to ((eq T (lift (S O) O t3) t5) \to (ex2 T 
+t5) H18) in (\lambda (_: (eq T u u1)).(\lambda (H23: (eq B b Abbr)).(let H24 
+\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H16 (lift (S O) O t3) H21) 
+in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda (t7: 
+T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 
+(THead (Bind Abbr) u2 w) t7))) (\lambda (x: T).(\lambda (H25: (eq T t6 (lift 
+(S O) O x))).(\lambda (H26: (pr0 t3 x)).(let H27 \def (eq_ind_r T t5 (\lambda 
+(t7: T).(eq T (THead (Bind Abbr) u1 t7) t)) H12 (lift (S O) O t3) H21) in 
+(let H28 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to 
+(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T 
+(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H 
+(THead (Bind Abbr) u1 (lift (S O) O t3)) H27) in (let H29 \def (eq_ind T t6 
+(\lambda (t7: T).(subst0 O u2 t7 w)) H17 (lift (S O) O x) H25) in (let H30 
+\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 Abbr H23) in 
+(ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: T).(pr0 t1 t7)) (ex2 T 
 (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) 
-t7)))))) (\lambda (H23: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T 
-(lift (S O) O t3) t5) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: 
-T).(pr0 (THead (Bind Abbr) u2 w) t8))))) (\lambda (H24: (eq T (lift (S O) O 
-t3) t5)).(eq_ind T (lift (S O) O t3) (\lambda (_: T).(ex2 T (\lambda (t8: 
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)))) (let 
-H25 \def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H16 (lift (S O) O t3) 
-H24) in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda 
-(t7: T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: 
-T).(pr0 (THead (Bind Abbr) u2 w) t7))) (\lambda (x: T).(\lambda (H26: (eq T 
-t6 (lift (S O) O x))).(\lambda (H27: (pr0 t3 x)).(let H28 \def (eq_ind_r T t5 
-(\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t7) t)) H12 (lift (S O) O t3) 
-H24) in (let H29 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v 
-t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to 
-(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
-t10)))))))))) H (THead (Bind Abbr) u1 (lift (S O) O t3)) H28) in (let H30 
-\def (eq_ind T t6 (\lambda (t7: T).(subst0 O u2 t7 w)) H17 (lift (S O) O x) 
-H26) in (let H31 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 
-Abbr H22) in (ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: T).(pr0 t1 
-t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind 
-Abbr) u2 w) t7))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 
-t1 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u2 w)) (pr0 t1) 
-(pr0_confluence__pr0_delta_epsilon u2 (lift (S O) O x) w H30 x (pr0_refl 
-(lift (S O) O x)) t1))))) (H29 t3 (lift_tlt_dx (Bind Abbr) u1 t3 (S O) O) x 
-H27 t1 H8))))))))) (pr0_gen_lift t3 t6 (S O) O H25))) t5 H24)) u (sym_eq T u 
-u1 H23))) b (sym_eq B b Abbr H22))) H21)) H20)))]) in (H19 (refl_equal T 
-(THead (Bind Abbr) u1 t5)))))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta 
-b0 H9 t5 t6 H10 u0) \Rightarrow (\lambda (H11: (eq T (THead (Bind b0) u0 
-(lift (S O) O t5)) t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind 
-b0) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b0 
-Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda 
-(t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda 
-(t7: T).((not (eq B b0 Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: 
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b0 
-Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: 
-T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Bind b0) u0 
-(lift (S O) O t5)) H11) in (let H17 \def (match H16 in eq return (\lambda 
-(t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind b0) u0 (lift (S O) O 
-t5))) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 
-t8)))))) with [refl_equal \Rightarrow (\lambda (H17: (eq T (THead (Bind b) u 
-(lift (S O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)))).(let H18 \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) (t7: 
-T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) 
-\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false 
-\Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f d u1) 
-(lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O 
-t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) 
-(t7: T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | 
-(TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | 
-false \Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f 
-d u1) (lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S 
-O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O 
-t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H17) in ((let H19 \def (f_equal T 
+t7))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t1 
+x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u2 w)) (pr0 t1) 
+(pr0_confluence__pr0_delta_epsilon u2 (lift (S O) O x) w H29 x (pr0_refl 
+(lift (S O) O x)) t1))))) (H28 t3 (lift_tlt_dx (Bind Abbr) u1 t3 (S O) O) x 
+H26 t1 H8))))))))) (pr0_gen_lift t3 t6 (S O) O H24)))))) H20)) H19)))))) t2 
+H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b0 H9 t5 t6 H10 u0) \Rightarrow 
+(\lambda (H11: (eq T (THead (Bind b0) u0 (lift (S O) O t5)) t)).(\lambda 
+(H12: (eq T t6 t2)).(eq_ind T (THead (Bind b0) u0 (lift (S O) O t5)) (\lambda 
+(_: T).((eq T t6 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T 
+(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda 
+(H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b0 Abst)) \to 
+((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 
+t2 t8)))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (H15: (pr0 t5 
+t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u 
+(lift (S O) O t3)) t7)) H4 (THead (Bind b0) u0 (lift (S O) O t5)) H11) in 
+(let H17 \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 b1) 
+\Rightarrow b1 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O 
+t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in ((let H18 \def (f_equal T 
 T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
 \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) \Rightarrow t7])) 
 (THead (Bind b) u (lift (S O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)) 
-H17) in ((let H20 \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 
-b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S 
-O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H17) in (eq_ind B b0 
-(\lambda (_: B).((eq T u u0) \to ((eq T (lift (S O) O t3) (lift (S O) O t5)) 
-\to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))) 
-(\lambda (H21: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T (lift (S O) O 
-t3) (lift (S O) O t5)) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: 
-T).(pr0 t2 t8))))) (\lambda (H22: (eq T (lift (S O) O t3) (lift (S O) O 
-t5))).(eq_ind T (lift (S O) O t3) (\lambda (_: T).(ex2 T (\lambda (t8: 
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))) (let H23 \def (eq_ind_r T t 
-(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) 
-\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind b0) u0 (lift (S O) O 
-t5)) H11) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H8 t5 
-(lift_inj t3 t5 (S O) O H22)) in (let H25 \def (eq_ind B b (\lambda (b1: 
-B).(not (eq B b1 Abst))) H7 b0 H20) in (ex2_ind T (\lambda (t7: T).(pr0 t1 
-t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
-(\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H26: (pr0 t1 
-x)).(\lambda (H27: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) 
-(\lambda (t7: T).(pr0 t2 t7)) x H26 H27)))) (H23 t5 (lift_tlt_dx (Bind b0) u0 
-t5 (S O) O) t1 H24 t2 H15))))) (lift (S O) O t5) H22)) u (sym_eq T u u0 
-H21))) b (sym_eq B b b0 H20))) H19)) H18)))]) in (H17 (refl_equal T (THead 
-(Bind b0) u0 (lift (S O) O t5)))))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 
-H10))) | (pr0_epsilon t5 t6 H9 u0) \Rightarrow (\lambda (H10: (eq T (THead 
-(Flat Cast) u0 t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat 
-Cast) u0 t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T 
-(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda 
-(H12: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T 
-(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: 
-(pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind 
-b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Cast) u0 t5) H10) in (let H15 
-\def (match H14 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq 
-T t7 (THead (Flat Cast) u0 t5)) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) 
-(\lambda (t8: T).(pr0 t2 t8)))))) with [refl_equal \Rightarrow (\lambda (H15: 
-(eq T (THead (Bind b) u (lift (S O) O t3)) (THead (Flat Cast) u0 t5))).(let 
-H16 \def (eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\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) u0 t5) H15) in (False_ind (ex2 T 
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) H16)))]) in (H15 
-(refl_equal T (THead (Flat Cast) u0 t5)))))) t6 (sym_eq T t6 t2 H12))) t H10 
-H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t4 (sym_eq T t4 t1 
-H6))) t H4 H5 H2 H3))) | (pr0_epsilon t3 t4 H2 u) \Rightarrow (\lambda (H3: 
-(eq T (THead (Flat Cast) u t3) t)).(\lambda (H4: (eq T t4 t1)).(eq_ind T 
-(THead (Flat Cast) u t3) (\lambda (_: T).((eq T t4 t1) \to ((pr0 t3 t4) \to 
-(ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))) 
-(\lambda (H5: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((pr0 t3 t5) \to 
-(ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))) 
-(\lambda (H6: (pr0 t3 t1)).(let H7 \def (match H1 in pr0 return (\lambda (t5: 
-T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) 
-\to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) 
-with [(pr0_refl t5) \Rightarrow (\lambda (H7: (eq T t5 t)).(\lambda (H8: (eq 
-T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: 
+H16) in ((let H19 \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) (t7: T) on t7: T \def (match t7 with [(TSort n) 
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u1 t8) 
+\Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t8))]) in 
+lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow 
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match 
+t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
+(THead k u1 t8) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) 
+t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t7) 
+\Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind b0) u0 
+(lift (S O) O t5)) H16) in (\lambda (_: (eq T u u0)).(\lambda (H21: (eq B b 
+b0)).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) 
+\to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T 
+(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H 
+(THead (Bind b0) u0 (lift (S O) O t5)) H11) in (let H23 \def (eq_ind T t3 
+(\lambda (t7: T).(pr0 t7 t1)) H8 t5 (lift_inj t3 t5 (S O) O H19)) in (let H24 
+\def (eq_ind B b (\lambda (b1: B).(not (eq B b1 Abst))) H7 b0 H21) in 
+(ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T 
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: 
+T).(\lambda (H25: (pr0 t1 x)).(\lambda (H26: (pr0 t2 x)).(ex_intro2 T 
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) x H25 H26)))) 
+(H22 t5 (lift_tlt_dx (Bind b0) u0 t5 (S O) O) t1 H23 t2 H15)))))))) H18)) 
+H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_epsilon t5 t6 
+H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0 t5) 
+t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda 
+(_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 
+t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6 t2)).(eq_ind T 
+t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) 
+(\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 t2)).(let H14 \def 
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) 
+t7)) H4 (THead (Flat Cast) u0 t5) H10) in (let H15 \def (eq_ind T (THead 
+(Bind b) u (lift (S O) O t3)) (\lambda (ee: T).(match ee in T return (\lambda 
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat 
+Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda 
+(t7: T).(pr0 t2 t7))) H15)))) t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in 
+(H9 (refl_equal T t) (refl_equal T t2))))) t4 (sym_eq T t4 t1 H6))) t H4 H5 
+H2 H3))) | (pr0_epsilon t3 t4 H2 u) \Rightarrow (\lambda (H3: (eq T (THead 
+(Flat Cast) u t3) t)).(\lambda (H4: (eq T t4 t1)).(eq_ind T (THead (Flat 
+Cast) u t3) (\lambda (_: T).((eq T t4 t1) \to ((pr0 t3 t4) \to (ex2 T 
+(\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H5: 
+(eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((pr0 t3 t5) \to (ex2 T (\lambda 
+(t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))) (\lambda (H6: (pr0 t3 
+t1)).(let H7 \def (match H1 in pr0 return (\lambda (t5: T).(\lambda (t6: 
+T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T 
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with 
+[(pr0_refl t5) \Rightarrow (\lambda (H7: (eq T t5 t)).(\lambda (H8: (eq T t5 
+t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: 
 T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H9: (eq T t 
 t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
 (\lambda (t7: T).(pr0 t2 t7)))) (let H10 \def (eq_ind_r T t (\lambda (t6: 
@@ -2649,168 +2353,129 @@ T (THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T
 \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: 
 T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13: (pr0 t5 
 t6)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u 
-t3) t7)) H3 (THead k u1 t5) H9) in (let H15 \def (match H14 in eq return 
-(\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead k u1 t5)) \to 
-(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead k u2 t6) 
-t8)))))) with [refl_equal \Rightarrow (\lambda (H15: (eq T (THead (Flat Cast) 
-u t3) (THead k u1 t5))).(let H16 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) 
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Cast) u t3) 
-(THead k u1 t5) H15) in ((let H17 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) 
-\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t3) 
-(THead k u1 t5) H15) in ((let H18 \def (f_equal T K (\lambda (e: T).(match e 
-in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Flat Cast) | 
-(TLRef _) \Rightarrow (Flat Cast) | (THead k0 _ _) \Rightarrow k0])) (THead 
-(Flat Cast) u t3) (THead k u1 t5) H15) in (eq_ind K (Flat Cast) (\lambda (k0: 
-K).((eq T u u1) \to ((eq T t3 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
-(\lambda (t7: T).(pr0 (THead k0 u2 t6) t7)))))) (\lambda (H19: (eq T u 
-u1)).(eq_ind T u1 (\lambda (_: T).((eq T t3 t5) \to (ex2 T (\lambda (t8: 
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Flat Cast) u2 t6) t8))))) 
-(\lambda (H20: (eq T t3 t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda 
-(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Flat Cast) u2 t6) t8)))) 
-(let H21 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u1 t5) t)) H9 
-(Flat Cast) H18) in (let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: 
-T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v 
-t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
-t10)))))))))) H (THead (Flat Cast) u1 t5) H21) in (let H23 \def (eq_ind T t3 
-(\lambda (t7: T).(pr0 t7 t1)) H6 t5 H20) in (ex2_ind T (\lambda (t7: T).(pr0 
-t1 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
-(\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6) t7))) (\lambda (x: 
-T).(\lambda (H24: (pr0 t1 x)).(\lambda (H25: (pr0 t6 x)).(ex_intro2 T 
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6) 
-t7)) x H24 (pr0_epsilon t6 x H25 u2))))) (H22 t5 (tlt_head_dx (Flat Cast) u1 
-t5) t1 H23 t6 H13))))) t3 (sym_eq T t3 t5 H20))) u (sym_eq T u u1 H19))) k 
-H18)) H17)) H16)))]) in (H15 (refl_equal T (THead k u1 t5))))))) t2 H11)) t 
-H9 H10 H7 H8))) | (pr0_beta u0 v1 v2 H7 t5 t6 H8) \Rightarrow (\lambda (H9: 
-(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) t)).(\lambda (H10: (eq 
-T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind 
-Abst) u0 t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 
-v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda 
-(t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v2 t6) 
-t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to 
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 
-t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H14 \def 
-(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead 
-(Flat Appl) v1 (THead (Bind Abst) u0 t5)) H9) in (let H15 \def (match H14 in 
-eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat 
-Appl) v1 (THead (Bind Abst) u0 t5))) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) 
-(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t6) t8)))))) with [refl_equal 
-\Rightarrow (\lambda (H15: (eq T (THead (Flat Cast) u t3) (THead (Flat Appl) 
-v1 (THead (Bind Abst) u0 t5)))).(let H16 \def (eq_ind T (THead (Flat Cast) u 
-t3) (\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) v1 (THead (Bind Abst) u0 t5)) H15) in (False_ind (ex2 T (\lambda 
-(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) 
-H16)))]) in (H15 (refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u0 
-t5)))))))) t2 H11)) t H9 H10 H7 H8))) | (pr0_upsilon b H7 v1 v2 H8 u1 u2 H9 
-t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead 
-(Bind b) u1 t5)) t)).(\lambda (H12: (eq T (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead 
-(Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to 
-((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) 
-(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H13: (eq T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b 
+t3) t7)) H3 (THead k u1 t5) H9) in (let H15 \def (f_equal T K (\lambda (e: 
+T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Flat 
+Cast) | (TLRef _) \Rightarrow (Flat Cast) | (THead k0 _ _) \Rightarrow k0])) 
+(THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H16 \def (f_equal T T 
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) \Rightarrow t7])) 
+(THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H17 \def (f_equal T T 
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) 
+(THead (Flat Cast) u t3) (THead k u1 t5) H14) in (\lambda (_: (eq T u 
+u1)).(\lambda (H19: (eq K (Flat Cast) k)).(eq_ind K (Flat Cast) (\lambda (k0: 
+K).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead k0 u2 
+t6) t7)))) (let H20 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u1 
+t5) t)) H9 (Flat Cast) H19) in (let H21 \def (eq_ind_r T t (\lambda (t7: 
+T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall 
+(t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: 
+T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u1 t5) H20) in (let H22 \def 
+(eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 H17) in (ex2_ind T (\lambda 
+(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: 
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6) t7))) 
+(\lambda (x: T).(\lambda (H23: (pr0 t1 x)).(\lambda (H24: (pr0 t6 
+x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead 
+(Flat Cast) u2 t6) t7)) x H23 (pr0_epsilon t6 x H24 u2))))) (H21 t5 
+(tlt_head_dx (Flat Cast) u1 t5) t1 H22 t6 H13))))) k H19)))) H16)) H15))))) 
+t2 H11)) t H9 H10 H7 H8))) | (pr0_beta u0 v1 v2 H7 t5 t6 H8) \Rightarrow 
+(\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) 
+t)).(\lambda (H10: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead 
+(Flat Appl) v1 (THead (Bind Abst) u0 t5)) (\lambda (_: T).((eq T (THead (Bind 
+Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T 
+(THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda 
+(t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 
+t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: 
+(pr0 t5 t6)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat 
+Cast) u t3) t7)) H3 (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) H9) in 
+(let H15 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda (ee: T).(match ee 
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef 
+_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return 
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow 
+(match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow False | 
+Cast \Rightarrow True])])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u0 
+t5)) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: 
+T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H15))))) t2 H11)) t H9 H10 H7 H8))) | 
+(pr0_upsilon b H7 v1 v2 H8 u1 u2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq 
+T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) t)).(\lambda (H12: (eq T 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T 
+(THead (Flat Appl) v1 (THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b 
 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda 
-(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not 
-(eq B b Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda 
-(_: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead 
-(Flat Cast) u t3) t7)) H3 (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) H11) 
-in (let H19 \def (match H18 in eq return (\lambda (t7: T).(\lambda (_: (eq ? 
-? t7)).((eq T t7 (THead (Flat Appl) v1 (THead (Bind b) u1 t5))) \to (ex2 T 
-(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t6)) t8)))))) with [refl_equal \Rightarrow 
-(\lambda (H19: (eq T (THead (Flat Cast) u t3) (THead (Flat Appl) v1 (THead 
-(Bind b) u1 t5)))).(let H20 \def (eq_ind T (THead (Flat Cast) u t3) (\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) v1 (THead (Bind b) 
-u1 t5)) H19) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t7))) 
-H20)))]) in (H19 (refl_equal T (THead (Flat Appl) v1 (THead (Bind b) u1 
-t5)))))))))) t2 H13)) t H11 H12 H7 H8 H9 H10))) | (pr0_delta u1 u2 H7 t5 t6 
-H8 w H9) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr) u1 t5) 
-t)).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind 
-Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 
-u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 
-t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H12: (eq T (THead (Bind 
-Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u1 
-u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 
-t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda 
-(_: (pr0 t5 t6)).(\lambda (_: (subst0 O u2 t6 w)).(let H16 \def (eq_ind_r T t 
-(\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Bind Abbr) u1 
-t5) H10) in (let H17 \def (match H16 in eq return (\lambda (t7: T).(\lambda 
-(_: (eq ? ? t7)).((eq T t7 (THead (Bind Abbr) u1 t5)) \to (ex2 T (\lambda 
-(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)))))) 
-with [refl_equal \Rightarrow (\lambda (H17: (eq T (THead (Flat Cast) u t3) 
-(THead (Bind Abbr) u1 t5))).(let H18 \def (eq_ind T (THead (Flat Cast) u t3) 
-(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) 
+(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H13: (eq 
+T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: 
+T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) 
+\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) 
+(\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 
+u1 u2)).(\lambda (_: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: 
+T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Appl) v1 (THead (Bind 
+b) u1 t5)) H11) in (let H19 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda 
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
 (match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | 
-(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t5) H17) in (False_ind 
-(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) 
-u2 w) t7))) H18)))]) in (H17 (refl_equal T (THead (Bind Abbr) u1 t5)))))))) 
-t2 H12)) t H10 H11 H7 H8 H9))) | (pr0_zeta b H7 t5 t6 H8 u0) \Rightarrow 
-(\lambda (H9: (eq T (THead (Bind b) u0 (lift (S O) O t5)) t)).(\lambda (H10: 
-(eq T t6 t2)).(eq_ind T (THead (Bind b) u0 (lift (S O) O t5)) (\lambda (_: 
-T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T 
-(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda 
-(H11: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b Abst)) \to 
-((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 
-t2 t8)))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 t5 t2)).(let 
-H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) 
-H3 (THead (Bind b) u0 (lift (S O) O t5)) H9) in (let H15 \def (match H14 in 
-eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind 
-b) u0 (lift (S O) O t5))) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda 
-(t8: T).(pr0 t2 t8)))))) with [refl_equal \Rightarrow (\lambda (H15: (eq T 
-(THead (Flat Cast) u t3) (THead (Bind b) u0 (lift (S O) O t5)))).(let H16 
-\def (eq_ind T (THead (Flat Cast) u t3) (\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 b) u0 (lift (S O) O t5)) H15) in (False_ind (ex2 T 
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) H16)))]) in (H15 
-(refl_equal T (THead (Bind b) u0 (lift (S O) O t5)))))))) t6 (sym_eq T t6 t2 
-H11))) t H9 H10 H7 H8))) | (pr0_epsilon t5 t6 H7 u0) \Rightarrow (\lambda 
-(H8: (eq T (THead (Flat Cast) u0 t5) t)).(\lambda (H9: (eq T t6 t2)).(eq_ind 
-T (THead (Flat Cast) u0 t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) 
-\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) 
-(\lambda (H10: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to 
-(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) 
-(\lambda (H11: (pr0 t5 t2)).(let H12 \def (eq_ind_r T t (\lambda (t7: T).(eq 
-T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Cast) u0 t5) H8) in (let H13 
-\def (match H12 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq 
-T t7 (THead (Flat Cast) u0 t5)) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) 
-(\lambda (t8: T).(pr0 t2 t8)))))) with [refl_equal \Rightarrow (\lambda (H13: 
-(eq T (THead (Flat Cast) u t3) (THead (Flat Cast) u0 t5))).(let H14 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) 
-\Rightarrow t7])) (THead (Flat Cast) u t3) (THead (Flat Cast) u0 t5) H13) in 
-((let H15 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 
-_) \Rightarrow t7])) (THead (Flat Cast) u t3) (THead (Flat Cast) u0 t5) H13) 
-in (eq_ind T u0 (\lambda (_: T).((eq T t3 t5) \to (ex2 T (\lambda (t8: 
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H16: (eq T t3 
-t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 t1 t8)) 
-(\lambda (t8: T).(pr0 t2 t8)))) (let H17 \def (eq_ind_r T t (\lambda (t7: 
-T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall 
-(t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: 
-T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u0 t5) H8) in (let H18 \def 
-(eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 H16) in (ex2_ind T (\lambda 
-(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: 
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H19: 
-(pr0 t1 x)).(\lambda (H20: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 
-t7)) (\lambda (t7: T).(pr0 t2 t7)) x H19 H20)))) (H17 t5 (tlt_head_dx (Flat 
-Cast) u0 t5) t1 H18 t2 H11)))) t3 (sym_eq T t3 t5 H16))) u (sym_eq T u u0 
-H15))) H14)))]) in (H13 (refl_equal T (THead (Flat Cast) u0 t5)))))) t6 
-(sym_eq T t6 t2 H10))) t H8 H9 H7)))]) in (H7 (refl_equal T t) (refl_equal T 
-t2)))) t4 (sym_eq T t4 t1 H5))) t H3 H4 H2)))]) in (H2 (refl_equal T t) 
-(refl_equal T t1))))))))) t0).
+(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl 
+\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) v1 
+(THead (Bind b) u1 t5)) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 
+t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) 
+O v2) t6)) t7))) H19))))))) t2 H13)) t H11 H12 H7 H8 H9 H10))) | (pr0_delta 
+u1 u2 H7 t5 t6 H8 w H9) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr) 
+u1 t5) t)).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T 
+(THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) 
+\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda 
+(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H12: (eq T 
+(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda 
+(t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T 
+(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: 
+(pr0 u1 u2)).(\lambda (_: (pr0 t5 t6)).(\lambda (_: (subst0 O u2 t6 w)).(let 
+H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) 
+H3 (THead (Bind Abbr) u1 t5) H10) in (let H17 \def (eq_ind T (THead (Flat 
+Cast) u t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 
+t5) H16) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: 
+T).(pr0 (THead (Bind Abbr) u2 w) t7))) H17)))))) t2 H12)) t H10 H11 H7 H8 
+H9))) | (pr0_zeta b H7 t5 t6 H8 u0) \Rightarrow (\lambda (H9: (eq T (THead 
+(Bind b) u0 (lift (S O) O t5)) t)).(\lambda (H10: (eq T t6 t2)).(eq_ind T 
+(THead (Bind b) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not 
+(eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) 
+(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T t6 t2)).(eq_ind T t2 
+(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda 
+(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: (not (eq 
+B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda 
+(t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Bind b) u0 (lift (S O) 
+O t5)) H9) in (let H15 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda (ee: 
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K 
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) 
+\Rightarrow True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H14) in 
+(False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 
+t7))) H15))))) t6 (sym_eq T t6 t2 H11))) t H9 H10 H7 H8))) | (pr0_epsilon t5 
+t6 H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Flat Cast) u0 t5) 
+t)).(\lambda (H9: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda 
+(_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 
+t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H10: (eq T t6 t2)).(eq_ind T 
+t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) 
+(\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H11: (pr0 t5 t2)).(let H12 \def 
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead 
+(Flat Cast) u0 t5) H8) in (let H13 \def (f_equal T T (\lambda (e: T).(match e 
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) 
+\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t3) 
+(THead (Flat Cast) u0 t5) H12) in ((let H14 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | 
+(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Flat 
+Cast) u t3) (THead (Flat Cast) u0 t5) H12) in (\lambda (_: (eq T u u0)).(let 
+H16 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to 
+(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T 
+(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H 
+(THead (Flat Cast) u0 t5) H8) in (let H17 \def (eq_ind T t3 (\lambda (t7: 
+T).(pr0 t7 t1)) H6 t5 H14) in (ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) 
+(\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda 
+(t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H18: (pr0 t1 x)).(\lambda 
+(H19: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: 
+T).(pr0 t2 t7)) x H18 H19)))) (H16 t5 (tlt_head_dx (Flat Cast) u0 t5) t1 H17 
+t2 H11)))))) H13)))) t6 (sym_eq T t6 t2 H10))) t H8 H9 H7)))]) in (H7 
+(refl_equal T t) (refl_equal T t2)))) t4 (sym_eq T t4 t1 H5))) t H3 H4 
+H2)))]) in (H2 (refl_equal T t) (refl_equal T t1))))))))) t0).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/props.ma
index c10df0103..a55260e65 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/props.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr0/defs.ma".
+include "LambdaDelta-1/pr0/defs.ma".
 
-include "subst0/subst0.ma".
+include "LambdaDelta-1/subst0/subst0.ma".
 
 theorem pr0_lift:
  \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (h: nat).(\forall 
@@ -27,105 +27,105 @@ theorem pr0_lift:
 (lift h d t0)))))) (\lambda (t: T).(\lambda (h: nat).(\lambda (d: 
 nat).(pr0_refl (lift h d t))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda 
 (_: (pr0 u1 u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 
-(lift h d u1) (lift h d u2)))))).(\lambda (t0: T).(\lambda (t3: T).(\lambda 
-(_: (pr0 t0 t3)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 
-(lift h d t0) (lift h d t3)))))).(\lambda (k: K).(\lambda (h: nat).(\lambda 
-(d: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) t0)) (\lambda (t: 
-T).(pr0 t (lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) 
-(lift h (s k d) t3)) (\lambda (t: T).(pr0 (THead k (lift h d u1) (lift h (s k 
-d) t0)) t)) (pr0_comp (lift h d u1) (lift h d u2) (H1 h d) (lift h (s k d) 
-t0) (lift h (s k d) t3) (H3 h (s k d)) k) (lift h d (THead k u2 t3)) 
-(lift_head k u2 t3 h d)) (lift h d (THead k u1 t0)) (lift_head k u1 t0 h 
+(lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda 
+(_: (pr0 t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 
+(lift h d t3) (lift h d t4)))))).(\lambda (k: K).(\lambda (h: nat).(\lambda 
+(d: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) t3)) (\lambda (t: 
+T).(pr0 t (lift h d (THead k u2 t4)))) (eq_ind_r T (THead k (lift h d u2) 
+(lift h (s k d) t4)) (\lambda (t: T).(pr0 (THead k (lift h d u1) (lift h (s k 
+d) t3)) t)) (pr0_comp (lift h d u1) (lift h d u2) (H1 h d) (lift h (s k d) 
+t3) (lift h (s k d) t4) (H3 h (s k d)) k) (lift h d (THead k u2 t4)) 
+(lift_head k u2 t4 h d)) (lift h d (THead k u1 t3)) (lift_head k u1 t3 h 
 d))))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: 
 (pr0 v1 v2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h 
-d v1) (lift h d v2)))))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (pr0 
-t0 t3)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t0) 
-(lift h d t3)))))).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead 
+d v1) (lift h d v2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 
+t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) 
+(lift h d t4)))))).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead 
 (Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) (THead (Bind Abst) u 
-t0))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) v2 t3)))) (eq_ind_r 
+t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r 
 T (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s 
-(Flat Appl) d)) t0)) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) 
-(lift h d (THead (Bind Abbr) v2 t3)))) (eq_ind_r T (THead (Bind Abbr) (lift h 
-d v2) (lift h (s (Bind Abbr) d) t3)) (\lambda (t: T).(pr0 (THead (Flat Appl) 
+(Flat Appl) d)) t3)) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) 
+(lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r T (THead (Bind Abbr) (lift h 
+d v2) (lift h (s (Bind Abbr) d) t4)) (\lambda (t: T).(pr0 (THead (Flat Appl) 
 (lift h d v1) (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s 
-(Bind Abst) (s (Flat Appl) d)) t0))) t)) (pr0_beta (lift h (s (Flat Appl) d) 
+(Bind Abst) (s (Flat Appl) d)) t3))) t)) (pr0_beta (lift h (s (Flat Appl) d) 
 u) (lift h d v1) (lift h d v2) (H1 h d) (lift h (s (Bind Abst) (s (Flat Appl) 
-d)) t0) (lift h (s (Bind Abbr) d) t3) (H3 h (s (Bind Abbr) d))) (lift h d 
-(THead (Bind Abbr) v2 t3)) (lift_head (Bind Abbr) v2 t3 h d)) (lift h (s 
-(Flat Appl) d) (THead (Bind Abst) u t0)) (lift_head (Bind Abst) u t0 h (s 
-(Flat Appl) d))) (lift h d (THead (Flat Appl) v1 (THead (Bind Abst) u t0))) 
-(lift_head (Flat Appl) v1 (THead (Bind Abst) u t0) h d))))))))))))) (\lambda 
+d)) t3) (lift h (s (Bind Abbr) d) t4) (H3 h (s (Bind Abbr) d))) (lift h d 
+(THead (Bind Abbr) v2 t4)) (lift_head (Bind Abbr) v2 t4 h d)) (lift h (s 
+(Flat Appl) d) (THead (Bind Abst) u t3)) (lift_head (Bind Abst) u t3 h (s 
+(Flat Appl) d))) (lift h d (THead (Flat Appl) v1 (THead (Bind Abst) u t3))) 
+(lift_head (Flat Appl) v1 (THead (Bind Abst) u t3) h d))))))))))))) (\lambda 
 (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
 T).(\lambda (_: (pr0 v1 v2)).(\lambda (H2: ((\forall (h: nat).(\forall (d: 
 nat).(pr0 (lift h d v1) (lift h d v2)))))).(\lambda (u1: T).(\lambda (u2: 
 T).(\lambda (_: (pr0 u1 u2)).(\lambda (H4: ((\forall (h: nat).(\forall (d: 
-nat).(pr0 (lift h d u1) (lift h d u2)))))).(\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (_: (pr0 t0 t3)).(\lambda (H6: ((\forall (h: nat).(\forall (d: 
-nat).(pr0 (lift h d t0) (lift h d t3)))))).(\lambda (h: nat).(\lambda (d: 
+nat).(pr0 (lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (_: (pr0 t3 t4)).(\lambda (H6: ((\forall (h: nat).(\forall (d: 
+nat).(pr0 (lift h d t3) (lift h d t4)))))).(\lambda (h: nat).(\lambda (d: 
 nat).(eq_ind_r T (THead (Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) 
-(THead (Bind b) u1 t0))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t3))))) (eq_ind_r T (THead (Bind b) 
-(lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) t0)) 
+(THead (Bind b) u1 t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead (Bind b) 
+(lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) t3)) 
 (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) (lift h d (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3))))) (eq_ind_r T (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead 
 (Bind b) (lift h d u2) (lift h (s (Bind b) d) (THead (Flat Appl) (lift (S O) 
-O v2) t3))) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) (THead 
+O v2) t4))) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) (THead 
 (Bind b) (lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) 
-t0))) t)) (eq_ind_r T (THead (Flat Appl) (lift h (s (Bind b) d) (lift (S O) O 
-v2)) (lift h (s (Flat Appl) (s (Bind b) d)) t3)) (\lambda (t: T).(pr0 (THead 
+t3))) t)) (eq_ind_r T (THead (Flat Appl) (lift h (s (Bind b) d) (lift (S O) O 
+v2)) (lift h (s (Flat Appl) (s (Bind b) d)) t4)) (\lambda (t: T).(pr0 (THead 
 (Flat Appl) (lift h d v1) (THead (Bind b) (lift h (s (Flat Appl) d) u1) (lift 
-h (s (Bind b) (s (Flat Appl) d)) t0))) (THead (Bind b) (lift h d u2) t))) 
+h (s (Bind b) (s (Flat Appl) d)) t3))) (THead (Bind b) (lift h d u2) t))) 
 (eq_ind nat (plus (S O) d) (\lambda (n: nat).(pr0 (THead (Flat Appl) (lift h 
-d v1) (THead (Bind b) (lift h d u1) (lift h n t0))) (THead (Bind b) (lift h d 
-u2) (THead (Flat Appl) (lift h n (lift (S O) O v2)) (lift h n t3))))) 
+d v1) (THead (Bind b) (lift h d u1) (lift h n t3))) (THead (Bind b) (lift h d 
+u2) (THead (Flat Appl) (lift h n (lift (S O) O v2)) (lift h n t4))))) 
 (eq_ind_r T (lift (S O) O (lift h d v2)) (\lambda (t: T).(pr0 (THead (Flat 
 Appl) (lift h d v1) (THead (Bind b) (lift h d u1) (lift h (plus (S O) d) 
-t0))) (THead (Bind b) (lift h d u2) (THead (Flat Appl) t (lift h (plus (S O) 
-d) t3))))) (pr0_upsilon b H0 (lift h d v1) (lift h d v2) (H2 h d) (lift h d 
-u1) (lift h d u2) (H4 h d) (lift h (plus (S O) d) t0) (lift h (plus (S O) d) 
-t3) (H6 h (plus (S O) d))) (lift h (plus (S O) d) (lift (S O) O v2)) (lift_d 
+t3))) (THead (Bind b) (lift h d u2) (THead (Flat Appl) t (lift h (plus (S O) 
+d) t4))))) (pr0_upsilon b H0 (lift h d v1) (lift h d v2) (H2 h d) (lift h d 
+u1) (lift h d u2) (H4 h d) (lift h (plus (S O) d) t3) (lift h (plus (S O) d) 
+t4) (H6 h (plus (S O) d))) (lift h (plus (S O) d) (lift (S O) O v2)) (lift_d 
 v2 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) (lift h (s (Bind b) 
-d) (THead (Flat Appl) (lift (S O) O v2) t3)) (lift_head (Flat Appl) (lift (S 
-O) O v2) t3 h (s (Bind b) d))) (lift h d (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t3))) (lift_head (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t3) h d)) (lift h (s (Flat Appl) d) (THead (Bind b) u1 t0)) 
-(lift_head (Bind b) u1 t0 h (s (Flat Appl) d))) (lift h d (THead (Flat Appl) 
-v1 (THead (Bind b) u1 t0))) (lift_head (Flat Appl) v1 (THead (Bind b) u1 t0) 
+d) (THead (Flat Appl) (lift (S O) O v2) t4)) (lift_head (Flat Appl) (lift (S 
+O) O v2) t4 h (s (Bind b) d))) (lift h d (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4))) (lift_head (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O v2) t4) h d)) (lift h (s (Flat Appl) d) (THead (Bind b) u1 t3)) 
+(lift_head (Bind b) u1 t3 h (s (Flat Appl) d))) (lift h d (THead (Flat Appl) 
+v1 (THead (Bind b) u1 t3))) (lift_head (Flat Appl) v1 (THead (Bind b) u1 t3) 
 h d)))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 
 u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d u1) 
-(lift h d u2)))))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (pr0 t0 
-t3)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t0) 
-(lift h d t3)))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t3 w)).(\lambda 
+(lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 
+t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) 
+(lift h d t4)))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t4 w)).(\lambda 
 (h: nat).(\lambda (d: nat).(eq_ind_r T (THead (Bind Abbr) (lift h d u1) (lift 
-h (s (Bind Abbr) d) t0)) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) 
+h (s (Bind Abbr) d) t3)) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) 
 u2 w)))) (eq_ind_r T (THead (Bind Abbr) (lift h d u2) (lift h (s (Bind Abbr) 
 d) w)) (\lambda (t: T).(pr0 (THead (Bind Abbr) (lift h d u1) (lift h (s (Bind 
-Abbr) d) t0)) t)) (pr0_delta (lift h d u1) (lift h d u2) (H1 h d) (lift h (S 
-d) t0) (lift h (S d) t3) (H3 h (S d)) (lift h (S d) w) (let d' \def (S d) in 
+Abbr) d) t3)) t)) (pr0_delta (lift h d u1) (lift h d u2) (H1 h d) (lift h (S 
+d) t3) (lift h (S d) t4) (H3 h (S d)) (lift h (S d) w) (let d' \def (S d) in 
 (eq_ind nat (minus (S d) (S O)) (\lambda (n: nat).(subst0 O (lift h n u2) 
-(lift h d' t3) (lift h d' w))) (subst0_lift_lt t3 w u2 O H4 (S d) (le_n_S O d 
+(lift h d' t4) (lift h d' w))) (subst0_lift_lt t4 w u2 O H4 (S d) (le_n_S O d 
 (le_O_n d)) h) d (eq_ind nat d (\lambda (n: nat).(eq nat n d)) (refl_equal 
 nat d) (minus d O) (minus_n_O d))))) (lift h d (THead (Bind Abbr) u2 w)) 
-(lift_head (Bind Abbr) u2 w h d)) (lift h d (THead (Bind Abbr) u1 t0)) 
-(lift_head (Bind Abbr) u1 t0 h d)))))))))))))) (\lambda (b: B).(\lambda (H0: 
-(not (eq B b Abst))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (pr0 t0 
-t3)).(\lambda (H2: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t0) 
-(lift h d t3)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: 
+(lift_head (Bind Abbr) u2 w h d)) (lift h d (THead (Bind Abbr) u1 t3)) 
+(lift_head (Bind Abbr) u1 t3 h d)))))))))))))) (\lambda (b: B).(\lambda (H0: 
+(not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 
+t4)).(\lambda (H2: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) 
+(lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: 
 nat).(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s (Bind b) d) (lift (S 
-O) O t0))) (\lambda (t: T).(pr0 t (lift h d t3))) (eq_ind nat (plus (S O) d) 
+O) O t3))) (\lambda (t: T).(pr0 t (lift h d t4))) (eq_ind nat (plus (S O) d) 
 (\lambda (n: nat).(pr0 (THead (Bind b) (lift h d u) (lift h n (lift (S O) O 
-t0))) (lift h d t3))) (eq_ind_r T (lift (S O) O (lift h d t0)) (\lambda (t: 
-T).(pr0 (THead (Bind b) (lift h d u) t) (lift h d t3))) (pr0_zeta b H0 (lift 
-h d t0) (lift h d t3) (H2 h d) (lift h d u)) (lift h (plus (S O) d) (lift (S 
-O) O t0)) (lift_d t0 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) 
-(lift h d (THead (Bind b) u (lift (S O) O t0))) (lift_head (Bind b) u (lift 
-(S O) O t0) h d))))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (_: 
-(pr0 t0 t3)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h 
-d t0) (lift h d t3)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: 
+t3))) (lift h d t4))) (eq_ind_r T (lift (S O) O (lift h d t3)) (\lambda (t: 
+T).(pr0 (THead (Bind b) (lift h d u) t) (lift h d t4))) (pr0_zeta b H0 (lift 
+h d t3) (lift h d t4) (H2 h d) (lift h d u)) (lift h (plus (S O) d) (lift (S 
+O) O t3)) (lift_d t3 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) 
+(lift h d (THead (Bind b) u (lift (S O) O t3))) (lift_head (Bind b) u (lift 
+(S O) O t3) h d))))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda (_: 
+(pr0 t3 t4)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h 
+d t3) (lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: 
 nat).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d) 
-t0)) (\lambda (t: T).(pr0 t (lift h d t3))) (pr0_epsilon (lift h (s (Flat 
-Cast) d) t0) (lift h d t3) (H1 h d) (lift h d u)) (lift h d (THead (Flat 
-Cast) u t0)) (lift_head (Flat Cast) u t0 h d))))))))) t1 t2 H))).
+t3)) (\lambda (t: T).(pr0 t (lift h d t4))) (pr0_epsilon (lift h (s (Flat 
+Cast) d) t3) (lift h d t4) (H1 h d) (lift h d u)) (lift h d (THead (Flat 
+Cast) u t3)) (lift_head (Flat Cast) u t3 h d))))))))) t1 t2 H))).
 
 theorem pr0_subst0_back:
  \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 
@@ -139,44 +139,44 @@ T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 u1 t) \to (ex2 T
 (\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 u1 
 v)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: 
 T).(pr0 t (lift (S i0) O v))) (lift (S i0) O u1) (subst0_lref u1 i0) 
-(pr0_lift u1 v H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u0: T).(\lambda 
-(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u0)).(\lambda (H1: 
-((\forall (u3: T).((pr0 u3 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u3 u1 t)) 
-(\lambda (t: T).(pr0 t u0))))))).(\lambda (t: T).(\lambda (k: K).(\lambda 
-(u3: T).(\lambda (H2: (pr0 u3 v)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u3 
-u1 t0)) (\lambda (t0: T).(pr0 t0 u0)) (ex2 T (\lambda (t0: T).(subst0 i0 u3 
-(THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 (THead k u0 t)))) (\lambda (x: 
-T).(\lambda (H3: (subst0 i0 u3 u1 x)).(\lambda (H4: (pr0 x u0)).(ex_intro2 T 
-(\lambda (t0: T).(subst0 i0 u3 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 
-(THead k u0 t))) (THead k x t) (subst0_fst u3 x u1 i0 H3 t k) (pr0_comp x u0 
-H4 t t (pr0_refl t) k))))) (H1 u3 H2)))))))))))) (\lambda (k: K).(\lambda (v: 
-T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (_: (subst0 
-(s k i0) v t3 t0)).(\lambda (H1: ((\forall (u1: T).((pr0 u1 v) \to (ex2 T 
-(\lambda (t: T).(subst0 (s k i0) u1 t3 t)) (\lambda (t: T).(pr0 t 
-t0))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 u1 v)).(ex2_ind 
-T (\lambda (t: T).(subst0 (s k i0) u1 t3 t)) (\lambda (t: T).(pr0 t t0)) (ex2 
-T (\lambda (t: T).(subst0 i0 u1 (THead k u t3) t)) (\lambda (t: T).(pr0 t 
-(THead k u t0)))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t3 
-x)).(\lambda (H4: (pr0 x t0)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 
-(THead k u t3) t)) (\lambda (t: T).(pr0 t (THead k u t0))) (THead k u x) 
-(subst0_snd k u1 x t3 i0 H3 u) (pr0_comp u u (pr0_refl u) x t0 H4 k))))) (H1 
-u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda 
-(i0: nat).(\lambda (_: (subst0 i0 v u1 u0)).(\lambda (H1: ((\forall (u3: 
-T).((pr0 u3 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u3 u1 t)) (\lambda (t: 
-T).(pr0 t u0))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (u3: 
-T).((pr0 u3 v) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u3 t0 t)) (\lambda 
-(t: T).(pr0 t t3))))))).(\lambda (u3: T).(\lambda (H4: (pr0 u3 v)).(ex2_ind T 
-(\lambda (t: T).(subst0 (s k i0) u3 t0 t)) (\lambda (t: T).(pr0 t t3)) (ex2 T 
-(\lambda (t: T).(subst0 i0 u3 (THead k u1 t0) t)) (\lambda (t: T).(pr0 t 
-(THead k u0 t3)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u3 t0 
-x)).(\lambda (H6: (pr0 x t3)).(ex2_ind T (\lambda (t: T).(subst0 i0 u3 u1 t)) 
-(\lambda (t: T).(pr0 t u0)) (ex2 T (\lambda (t: T).(subst0 i0 u3 (THead k u1 
-t0) t)) (\lambda (t: T).(pr0 t (THead k u0 t3)))) (\lambda (x0: T).(\lambda 
-(H7: (subst0 i0 u3 u1 x0)).(\lambda (H8: (pr0 x0 u0)).(ex_intro2 T (\lambda 
-(t: T).(subst0 i0 u3 (THead k u1 t0) t)) (\lambda (t: T).(pr0 t (THead k u0 
-t3))) (THead k x0 x) (subst0_both u3 u1 x0 i0 H7 k t0 x H5) (pr0_comp x0 u0 
-H8 x t3 H6 k))))) (H1 u3 H4))))) (H3 u3 H4))))))))))))))) i u2 t1 t2 H))))).
+(pr0_lift u1 v H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda 
+(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: 
+((\forall (u4: T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) 
+(\lambda (t: T).(pr0 t u3))))))).(\lambda (t: T).(\lambda (k: K).(\lambda 
+(u0: T).(\lambda (H2: (pr0 u0 v)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 
+u1 t0)) (\lambda (t0: T).(pr0 t0 u3)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 
+(THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 (THead k u3 t)))) (\lambda (x: 
+T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 x u3)).(ex_intro2 T 
+(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 
+(THead k u3 t))) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp x u3 
+H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda (v: 
+T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: (subst0 
+(s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 u1 v) \to (ex2 T 
+(\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t 
+t3))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 u1 v)).(ex2_ind 
+T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t t3)) (ex2 
+T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 t 
+(THead k u t3)))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 
+x)).(\lambda (H4: (pr0 x t3)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 
+(THead k u t4) t)) (\lambda (t: T).(pr0 t (THead k u t3))) (THead k u x) 
+(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) x t3 H4 k))))) (H1 
+u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda 
+(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: 
+T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: 
+T).(pr0 t u3))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: 
+T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda 
+(t: T).(pr0 t t4))))))).(\lambda (u0: T).(\lambda (H4: (pr0 u0 v)).(ex2_ind T 
+(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t t4)) (ex2 T 
+(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t 
+(THead k u3 t4)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 
+x)).(\lambda (H6: (pr0 x t4)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) 
+(\lambda (t: T).(pr0 t u3)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 
+t3) t)) (\lambda (t: T).(pr0 t (THead k u3 t4)))) (\lambda (x0: T).(\lambda 
+(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 x0 u3)).(ex_intro2 T (\lambda 
+(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t (THead k u3 
+t4))) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp x0 u3 
+H8 x t4 H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))).
 
 theorem pr0_subst0_fwd:
  \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 
@@ -190,44 +190,44 @@ T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 t u1) \to (ex2 T
 (\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 v 
 u1)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: 
 T).(pr0 (lift (S i0) O v) t)) (lift (S i0) O u1) (subst0_lref u1 i0) 
-(pr0_lift v u1 H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u0: T).(\lambda 
-(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u0)).(\lambda (H1: 
-((\forall (u3: T).((pr0 v u3) \to (ex2 T (\lambda (t: T).(subst0 i0 u3 u1 t)) 
-(\lambda (t: T).(pr0 u0 t))))))).(\lambda (t: T).(\lambda (k: K).(\lambda 
-(u3: T).(\lambda (H2: (pr0 v u3)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u3 
-u1 t0)) (\lambda (t0: T).(pr0 u0 t0)) (ex2 T (\lambda (t0: T).(subst0 i0 u3 
-(THead k u1 t) t0)) (\lambda (t0: T).(pr0 (THead k u0 t) t0))) (\lambda (x: 
-T).(\lambda (H3: (subst0 i0 u3 u1 x)).(\lambda (H4: (pr0 u0 x)).(ex_intro2 T 
-(\lambda (t0: T).(subst0 i0 u3 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 
-(THead k u0 t) t0)) (THead k x t) (subst0_fst u3 x u1 i0 H3 t k) (pr0_comp u0 
-x H4 t t (pr0_refl t) k))))) (H1 u3 H2)))))))))))) (\lambda (k: K).(\lambda 
-(v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (_: 
-(subst0 (s k i0) v t3 t0)).(\lambda (H1: ((\forall (u1: T).((pr0 v u1) \to 
-(ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t3 t)) (\lambda (t: T).(pr0 t0 
+(pr0_lift v u1 H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda 
+(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: 
+((\forall (u4: T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) 
+(\lambda (t: T).(pr0 u3 t))))))).(\lambda (t: T).(\lambda (k: K).(\lambda 
+(u0: T).(\lambda (H2: (pr0 v u0)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 
+u1 t0)) (\lambda (t0: T).(pr0 u3 t0)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 
+(THead k u1 t) t0)) (\lambda (t0: T).(pr0 (THead k u3 t) t0))) (\lambda (x: 
+T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 u3 x)).(ex_intro2 T 
+(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 
+(THead k u3 t) t0)) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp u3 
+x H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda 
+(v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: 
+(subst0 (s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 v u1) \to 
+(ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 
 t))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 v u1)).(ex2_ind 
-T (\lambda (t: T).(subst0 (s k i0) u1 t3 t)) (\lambda (t: T).(pr0 t0 t)) (ex2 
-T (\lambda (t: T).(subst0 i0 u1 (THead k u t3) t)) (\lambda (t: T).(pr0 
-(THead k u t0) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t3 
-x)).(\lambda (H4: (pr0 t0 x)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 
-(THead k u t3) t)) (\lambda (t: T).(pr0 (THead k u t0) t)) (THead k u x) 
-(subst0_snd k u1 x t3 i0 H3 u) (pr0_comp u u (pr0_refl u) t0 x H4 k))))) (H1 
-u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda 
-(i0: nat).(\lambda (_: (subst0 i0 v u1 u0)).(\lambda (H1: ((\forall (u3: 
-T).((pr0 v u3) \to (ex2 T (\lambda (t: T).(subst0 i0 u3 u1 t)) (\lambda (t: 
-T).(pr0 u0 t))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (u3: 
-T).((pr0 v u3) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u3 t0 t)) (\lambda 
-(t: T).(pr0 t3 t))))))).(\lambda (u3: T).(\lambda (H4: (pr0 v u3)).(ex2_ind T 
-(\lambda (t: T).(subst0 (s k i0) u3 t0 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T 
-(\lambda (t: T).(subst0 i0 u3 (THead k u1 t0) t)) (\lambda (t: T).(pr0 (THead 
-k u0 t3) t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u3 t0 
-x)).(\lambda (H6: (pr0 t3 x)).(ex2_ind T (\lambda (t: T).(subst0 i0 u3 u1 t)) 
-(\lambda (t: T).(pr0 u0 t)) (ex2 T (\lambda (t: T).(subst0 i0 u3 (THead k u1 
-t0) t)) (\lambda (t: T).(pr0 (THead k u0 t3) t))) (\lambda (x0: T).(\lambda 
-(H7: (subst0 i0 u3 u1 x0)).(\lambda (H8: (pr0 u0 x0)).(ex_intro2 T (\lambda 
-(t: T).(subst0 i0 u3 (THead k u1 t0) t)) (\lambda (t: T).(pr0 (THead k u0 t3) 
-t)) (THead k x0 x) (subst0_both u3 u1 x0 i0 H7 k t0 x H5) (pr0_comp u0 x0 H8 
-t3 x H6 k))))) (H1 u3 H4))))) (H3 u3 H4))))))))))))))) i u2 t1 t2 H))))).
+T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 
+T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 
+(THead k u t3) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 
+x)).(\lambda (H4: (pr0 t3 x)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 
+(THead k u t4) t)) (\lambda (t: T).(pr0 (THead k u t3) t)) (THead k u x) 
+(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) t3 x H4 k))))) (H1 
+u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda 
+(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: 
+T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: 
+T).(pr0 u3 t))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: 
+T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda 
+(t: T).(pr0 t4 t))))))).(\lambda (u0: T).(\lambda (H4: (pr0 v u0)).(ex2_ind T 
+(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t4 t)) (ex2 T 
+(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead 
+k u3 t4) t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 
+x)).(\lambda (H6: (pr0 t4 x)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) 
+(\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 
+t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) t))) (\lambda (x0: T).(\lambda 
+(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 u3 x0)).(ex_intro2 T (\lambda 
+(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) 
+t)) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp u3 x0 H8 
+t4 x H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))).
 
 theorem pr0_subst0:
  \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall 
@@ -438,223 +438,216 @@ w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))))
 Appl) i) v0 u u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda 
 (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
 w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead (Bind Abst) x0 
-t3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(eq_ind_r T (THead 
-(Flat Appl) v1 x) (\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 
-T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
-Abbr) v2 t4) w2))))) (eq_ind_r T (THead (Bind Abst) x0 t3) (\lambda (t: 
-T).(or (pr0 (THead (Flat Appl) v1 t) (THead (Bind Abbr) v2 t4)) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 t) w2)) (\lambda (w2: T).(subst0 
-i v3 (THead (Bind Abbr) v2 t4) w2))))) (or_introl (pr0 (THead (Flat Appl) v1 
-(THead (Bind Abst) x0 t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 t3 t4 
-H2)) x H10) w1 H7)))) H9)) (\lambda (H9: (ex2 T (\lambda (t5: T).(eq T x 
-(THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat 
-Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T x (THead (Bind Abst) 
-u t5))) (\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)) 
-(or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: 
-T).(\lambda (H10: (eq T x (THead (Bind Abst) u x0))).(\lambda (H11: (subst0 
-(s (Bind Abst) (s (Flat Appl) i)) v0 t3 x0)).(eq_ind_r T (THead (Flat Appl) 
-v1 x) (\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda 
-(w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
-w2))))) (eq_ind_r T (THead (Bind Abst) u x0) (\lambda (t: T).(or (pr0 (THead 
-(Flat Appl) v1 t) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) v1 t) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
-Abbr) v2 t4) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) 
-(\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or 
-(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 
-t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u 
-x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) 
-(\lambda (H12: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead 
-(Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 
-i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta u v1 v2 H0 x0 t4 H12))) 
-(\lambda (H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 
-(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: 
+t3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(let H12 \def (eq_ind 
+T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) 
+x0 t3) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) 
+(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
+w2))))) (or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) 
+(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
+(THead (Bind Abst) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
+Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 t3 t4 H2)) w1 H12))))) H9)) (\lambda 
+(H9: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: 
+T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda 
+(t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind 
+Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) 
+(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead 
+(Bind Abst) u x0))).(\lambda (H11: (subst0 (s (Bind Abst) (s (Flat Appl) i)) 
+v0 t3 x0)).(let H12 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat 
+Appl) v1 t))) H7 (THead (Bind Abst) u x0) H10) in (eq_ind_r T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u x0)) (\lambda (t: T).(or (pr0 t (THead (Bind 
+Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: 
 T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 
-t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead (Bind 
-Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind 
-Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
-w2)))) (\lambda (x1: T).(\lambda (H13: (pr0 x0 x1)).(\lambda (H14: (subst0 (s 
-(Bind Abst) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 (THead (Flat Appl) 
-v1 (THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: 
+t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead 
+(Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2)))) (\lambda (H13: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 
+(THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
 T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x1) 
-(pr0_beta u v1 v2 H0 x0 x1 H13) (subst0_snd (Bind Abbr) v3 x1 t4 i H14 
-v2)))))) H12)) (H3 v0 x0 (s (Bind Abst) (s (Flat Appl) i)) H11 v3 H5)) x H10) 
-w1 H7)))) H9)) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq 
-T x (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s 
-(Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind 
-Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T x (THead (Bind Abst) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: 
-T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))) (or 
-(pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H10: (eq T x (THead (Bind Abst) x0 
-x1))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(\lambda (H12: (subst0 
-(s (Bind Abst) (s (Flat Appl) i)) v0 t3 x1)).(eq_ind_r T (THead (Flat Appl) 
-v1 x) (\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda 
-(w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
-w2))))) (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t: T).(or (pr0 (THead 
-(Flat Appl) v1 t) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) v1 t) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
-Abbr) v2 t4) w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) 
-(\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or 
-(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 
-t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 
-x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) 
-(\lambda (H13: (pr0 x1 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead 
-(Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 x1 t4 
-H13))) (\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta u v1 v2 H0 x0 t4 
+H13))) (\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
 T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda 
-(w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) 
-i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) 
+(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) 
+i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) 
 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
-(THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
-Abbr) v2 t4) w2)))) (\lambda (x2: T).(\lambda (H14: (pr0 x1 x2)).(\lambda 
-(H15: (subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_intror (pr0 
-(THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) 
-(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) 
+(THead (Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
+Abbr) v2 t4) w2)))) (\lambda (x1: T).(\lambda (H14: (pr0 x0 x1)).(\lambda 
+(H15: (subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 
+(THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) 
+(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) 
 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 
-T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) 
+T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) 
 (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind 
-Abbr) v2 x2) (pr0_beta x0 v1 v2 H0 x1 x2 H14) (subst0_snd (Bind Abbr) v3 x2 
-t4 i H15 v2)))))) H13)) (H3 v0 x1 (s (Bind Abst) (s (Flat Appl) i)) H12 v3 
-H5)) x H10) w1 H7)))))) H9)) (subst0_gen_head (Bind Abst) v0 u t3 x (s (Flat 
-Appl) i) H8))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+Abbr) v2 x1) (pr0_beta u v1 v2 H0 x0 x1 H14) (subst0_snd (Bind Abbr) v3 x1 t4 
+i H15 v2)))))) H13)) (H3 v0 x0 (s (Bind Abst) (s (Flat Appl) i)) H11 v3 H5)) 
+w1 H12))))) H9)) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T x (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T x (THead (Bind Abst) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) 
+(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 
+t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
+w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T x (THead (Bind Abst) 
+x0 x1))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(\lambda (H12: 
+(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x1)).(let H13 \def (eq_ind T 
+x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) 
+x0 x1) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) 
+(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
+w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda 
+(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead 
+(Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H14: 
+(pr0 x1 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) 
+(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
+(THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
+Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 x1 t4 H14))) (\lambda (H14: (ex2 T 
+(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s 
+(Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) 
+(\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)) (or 
+(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 
+t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 
+x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) 
+(\lambda (x2: T).(\lambda (H15: (pr0 x1 x2)).(\lambda (H16: (subst0 (s (Bind 
+Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_intror (pr0 (THead (Flat Appl) v1 
+(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) 
+(pr0_beta x0 v1 v2 H0 x1 x2 H15) (subst0_snd (Bind Abbr) v3 x2 t4 i H16 
+v2)))))) H14)) (H3 v0 x1 (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 
+H13))))))) H9)) (subst0_gen_head (Bind Abst) v0 u t3 x (s (Flat Appl) i) 
+H8))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 
+v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead 
+(Bind Abst) u t3) t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: 
 T).(eq T w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
 T).(subst0 i v0 v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat 
-Appl) i) v0 (THead (Bind Abst) u t3) t5))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i v0 v1 u2))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) t5))) (or (pr0 w1 
-(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda 
-(x1: T).(\lambda (H7: (eq T w1 (THead (Flat Appl) x0 x1))).(\lambda (H8: 
-(subst0 i v0 v1 x0)).(\lambda (H9: (subst0 (s (Flat Appl) i) v0 (THead (Bind 
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-u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2))) (ex2 T 
-(\lambda (t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 
-(s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: 
-T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))) 
-(or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H10: 
-(ex2 T (\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) (\lambda (u2: 
+Appl) i) v0 (THead (Bind Abst) u t3) t5))) (or (pr0 w1 (THead (Bind Abbr) v2 
+t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
+(H7: (eq T w1 (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i v0 v1 
+x0)).(\lambda (H9: (subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) 
+x1)).(or3_ind (ex2 T (\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) 
+(\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2))) (ex2 T (\lambda (t5: 
+T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind 
+Abst) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))) (or (pr0 w1 (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H10: (ex2 T 
+(\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) (\lambda (u2: 
 T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T x1 
 (THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u 
 u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 
 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
 (x: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x t3))).(\lambda (_: 
-(subst0 (s (Flat Appl) i) v0 u x)).(eq_ind_r T (THead (Flat Appl) x0 x1) 
-(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
-w2))))) (eq_ind_r T (THead (Bind Abst) x t3) (\lambda (t: T).(or (pr0 (THead 
-(Flat Appl) x0 t) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) x0 t) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
-Abbr) v2 t4) w2))))) (or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) 
-(\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: T).(subst0 
-i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H13: (pr0 x0 v2)).(or_introl 
-(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 
-t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x 
-t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) 
-(pr0_beta x x0 v2 H13 t3 t4 H2))) (\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 
-x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: 
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat 
-Appl) x0 (THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda 
-(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda 
-(w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x2: 
-T).(\lambda (H14: (pr0 x0 x2)).(\lambda (H15: (subst0 i v3 v2 x2)).(or_intror 
-(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 
-t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x 
-t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) 
-(ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x 
-t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead 
-(Bind Abbr) x2 t4) (pr0_beta x x0 x2 H14 t3 t4 H2) (subst0_fst v3 x2 v2 i H15 
-t4 (Bind Abbr))))))) H13)) (H1 v0 x0 i H8 v3 H5)) x1 H11) w1 H7)))) H10)) 
-(\lambda (H10: (ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind Abst) u t5))) 
-(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 
-t5)))).(ex2_ind T (\lambda (t5: T).(eq T x1 (THead (Bind Abst) u t5))) 
-(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)) (or 
-(pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x: 
-T).(\lambda (H11: (eq T x1 (THead (Bind Abst) u x))).(\lambda (H12: (subst0 
-(s (Bind Abst) (s (Flat Appl) i)) v0 t3 x)).(eq_ind_r T (THead (Flat Appl) x0 
-x1) (\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda 
-(w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
-w2))))) (eq_ind_r T (THead (Bind Abst) u x) (\lambda (t: T).(or (pr0 (THead 
-(Flat Appl) x0 t) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) x0 t) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
-Abbr) v2 t4) w2))))) (or_ind (pr0 x t4) (ex2 T (\lambda (w2: T).(pr0 x w2)) 
-(\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or 
-(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 
-t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u 
-x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) 
-(\lambda (H13: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 
-x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 
-(THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H14: (pr0 x0 
+(subst0 (s (Flat Appl) i) v0 u x)).(let H13 \def (eq_ind T x1 (\lambda (t: 
+T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) x t3) H11) in 
+(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (\lambda (t: 
+T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 
+x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
+w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (THead (Bind 
+Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
+Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
+w2)))) (\lambda (H14: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat Appl) x0 
+(THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta x x0 v2 H14 t3 t4 
+H2))) (\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
+T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
+(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind 
+Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead 
+(Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 
+x2)).(\lambda (H16: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) 
+x0 (THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x2 t4) 
+(pr0_beta x x0 x2 H15 t3 t4 H2) (subst0_fst v3 x2 v2 i H16 t4 (Bind 
+Abbr))))))) H14)) (H1 v0 x0 i H8 v3 H5)) w1 H13))))) H10)) (\lambda (H10: 
+(ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: 
+T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda 
+(t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind 
+Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) 
+(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda (H11: (eq T x1 (THead 
+(Bind Abst) u x))).(\lambda (H12: (subst0 (s (Bind Abst) (s (Flat Appl) i)) 
+v0 t3 x)).(let H13 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat 
+Appl) x0 t))) H7 (THead (Bind Abst) u x) H11) in (eq_ind_r T (THead (Flat 
+Appl) x0 (THead (Bind Abst) u x)) (\lambda (t: T).(or (pr0 t (THead (Bind 
+Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x t4) (ex2 T (\lambda (w2: 
+T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 
+t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind 
+Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
+Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
+w2)))) (\lambda (H14: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: 
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat 
+Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda 
+(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) (\lambda 
+(w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: (pr0 x0 
 v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead 
 (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
 (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2))) (pr0_beta u x0 v2 H14 x t4 H13))) (\lambda (H14: (ex2 T (\lambda 
+t4) w2))) (pr0_beta u x0 v2 H15 x t4 H14))) (\lambda (H15: (ex2 T (\lambda 
 (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T 
 (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 
 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) 
 (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) 
 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
-(x2: T).(\lambda (H15: (pr0 x0 x2)).(\lambda (H16: (subst0 i v3 v2 
+(x2: T).(\lambda (H16: (pr0 x0 x2)).(\lambda (H17: (subst0 i v3 v2 
 x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead 
 (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
 (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
 (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2)) (THead (Bind Abbr) x2 t4) (pr0_beta u x0 x2 H15 x t4 H13) 
-(subst0_fst v3 x2 v2 i H16 t4 (Bind Abbr))))))) H14)) (H1 v0 x0 i H8 v3 H5))) 
-(\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 
+t4) w2)) (THead (Bind Abbr) x2 t4) (pr0_beta u x0 x2 H16 x t4 H14) 
+(subst0_fst v3 x2 v2 i H17 t4 (Bind Abbr))))))) H15)) (H1 v0 x0 i H8 v3 H5))) 
+(\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 
 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: 
 T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 
 t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind 
 Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
 Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
-w2)))) (\lambda (x2: T).(\lambda (H14: (pr0 x x2)).(\lambda (H15: (subst0 (s 
+w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x x2)).(\lambda (H16: (subst0 (s 
 (Bind Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda 
 (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead 
 (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T 
 (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H16: 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H17: 
 (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) 
 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
 (THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
 Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
 (THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
-Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) (pr0_beta u x0 v2 H16 x x2 H14) 
-(subst0_snd (Bind Abbr) v3 x2 t4 i H15 v2)))) (\lambda (H16: (ex2 T (\lambda 
+Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) (pr0_beta u x0 v2 H17 x x2 H15) 
+(subst0_snd (Bind Abbr) v3 x2 t4 i H16 v2)))) (\lambda (H17: (ex2 T (\lambda 
 (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T 
 (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 
 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) 
 (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) 
 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
-(x3: T).(\lambda (H17: (pr0 x0 x3)).(\lambda (H18: (subst0 i v3 v2 
+(x3: T).(\lambda (H18: (pr0 x0 x3)).(\lambda (H19: (subst0 i v3 v2 
 x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead 
 (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
 (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
 (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2)) (THead (Bind Abbr) x3 x2) (pr0_beta u x0 x3 H17 x x2 H14) 
-(subst0_both v3 v2 x3 i H18 (Bind Abbr) t4 x2 H15)))))) H16)) (H1 v0 x0 i H8 
-v3 H5))))) H13)) (H3 v0 x (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) x1 
-H11) w1 H7)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+t4) w2)) (THead (Bind Abbr) x3 x2) (pr0_beta u x0 x3 H18 x x2 H15) 
+(subst0_both v3 v2 x3 i H19 (Bind Abbr) t4 x2 H16)))))) H17)) (H1 v0 x0 i H8 
+v3 H5))))) H14)) (H3 v0 x (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 
+H13))))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
 T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
 T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: 
 T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T 
@@ -665,75 +658,74 @@ t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0
 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) 
 (\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) 
 x2 x3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x2)).(\lambda (H13: 
-(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x3)).(eq_ind_r T (THead (Flat 
-Appl) x0 x1) (\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T 
-(\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) 
-v2 t4) w2))))) (eq_ind_r T (THead (Bind Abst) x2 x3) (\lambda (t: T).(or (pr0 
-(THead (Flat Appl) x0 t) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 t) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 
-x3 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 
-w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind 
-Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
-Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
-w2)))) (\lambda (H14: (pr0 x3 t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: 
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat 
-Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T 
+(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x3)).(let H14 \def (eq_ind T 
+x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) 
+x2 x3) H11) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) 
+(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
+w2))))) (or_ind (pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda 
+(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead 
+(Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T 
 (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) 
 (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: 
-(pr0 x0 v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) 
-(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
-(THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
-Abbr) v2 t4) w2))) (pr0_beta x2 x0 v2 H15 x3 t4 H14))) (\lambda (H15: (ex2 T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
-v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead 
+(pr0 x3 t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) 
+(\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H16: (pr0 x0 
+v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead 
 (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
 (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2)))) (\lambda (x: T).(\lambda (H16: (pr0 x0 x)).(\lambda (H17: (subst0 
-i v3 v2 x)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) 
-(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
-(THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
-Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
-(THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
-Abbr) v2 t4) w2)) (THead (Bind Abbr) x t4) (pr0_beta x2 x0 x H16 x3 t4 H14) 
-(subst0_fst v3 x v2 i H17 t4 (Bind Abbr))))))) H15)) (H1 v0 x0 i H8 v3 H5))) 
-(\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 
+t4) w2))) (pr0_beta x2 x0 v2 H16 x3 t4 H15))) (\lambda (H16: (ex2 T (\lambda 
+(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T 
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 
+(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) 
+(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) 
+w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
+(x: T).(\lambda (H17: (pr0 x0 x)).(\lambda (H18: (subst0 i v3 v2 
+x)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2)) (THead (Bind Abbr) x t4) (pr0_beta x2 x0 x H17 x3 t4 H15) 
+(subst0_fst v3 x v2 i H18 t4 (Bind Abbr))))))) H16)) (H1 v0 x0 i H8 v3 H5))) 
+(\lambda (H15: (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 
 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: 
 T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 
 t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead 
 (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
 (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2)))) (\lambda (x: T).(\lambda (H15: (pr0 x3 x)).(\lambda (H16: (subst0 
+t4) w2)))) (\lambda (x: T).(\lambda (H16: (pr0 x3 x)).(\lambda (H17: (subst0 
 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 x)).(or_ind (pr0 x0 v2) (ex2 T 
 (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 
 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) 
 (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) 
 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
-(H17: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) 
+(H18: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) 
 x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat 
 Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
 (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat 
 Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x) (pr0_beta x2 x0 v2 H17 x3 x 
-H15) (subst0_snd (Bind Abbr) v3 x t4 i H16 v2)))) (\lambda (H17: (ex2 T 
+(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x) (pr0_beta x2 x0 v2 H18 x3 x 
+H16) (subst0_snd (Bind Abbr) v3 x t4 i H17 v2)))) (\lambda (H18: (ex2 T 
 (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead 
 (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
 (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2)))) (\lambda (x4: T).(\lambda (H18: (pr0 x0 x4)).(\lambda (H19: 
+t4) w2)))) (\lambda (x4: T).(\lambda (H19: (pr0 x0 x4)).(\lambda (H20: 
 (subst0 i v3 v2 x4)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) 
 x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat 
 Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
 (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat 
 Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x4 x) (pr0_beta x2 x0 x4 H18 x3 x 
-H15) (subst0_both v3 v2 x4 i H19 (Bind Abbr) t4 x H16)))))) H17)) (H1 v0 x0 i 
-H8 v3 H5))))) H14)) (H3 v0 x3 (s (Bind Abst) (s (Flat Appl) i)) H13 v3 H5)) 
-x1 H11) w1 H7)))))) H10)) (subst0_gen_head (Bind Abst) v0 u t3 x1 (s (Flat 
-Appl) i) H9))))))) H6)) (subst0_gen_head (Flat Appl) v0 v1 (THead (Bind Abst) 
-u t3) w1 i H4))))))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b 
+(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x4 x) (pr0_beta x2 x0 x4 H19 x3 x 
+H16) (subst0_both v3 v2 x4 i H20 (Bind Abbr) t4 x H17)))))) H18)) (H1 v0 x0 i 
+H8 v3 H5))))) H15)) (H3 v0 x3 (s (Bind Abst) (s (Flat Appl) i)) H13 v3 H5)) 
+w1 H14))))))) H10)) (subst0_gen_head (Bind Abst) v0 u t3 x1 (s (Flat Appl) i) 
+H9))))))) H6)) (subst0_gen_head (Flat Appl) v0 v1 (THead (Bind Abst) u t3) w1 
+i H4))))))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b 
 Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H1: (pr0 v1 v2)).(\lambda 
 (H2: ((\forall (v3: T).(\forall (w1: T).(\forall (i: nat).((subst0 i v3 v1 
 w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1 v2) (ex2 T (\lambda 
@@ -823,90 +815,85 @@ b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or (pr0 w1
 (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
 (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq 
 T x (THead (Bind b) x0 t3))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 
-x0)).(eq_ind_r T (THead (Flat Appl) v1 x) (\lambda (t: T).(or (pr0 t (THead 
+x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 
+t))) H10 (THead (Bind b) x0 t3) H13) in (eq_ind_r T (THead (Flat Appl) v1 
+(THead (Bind b) x0 t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2))))) (or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 
+w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or (pr0 (THead 
+(Flat Appl) v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
+(THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H16: (pr0 x0 
+u2)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) (THead 
 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2))))) (eq_ind_r T (THead (Bind b) x0 t3) 
-(\lambda (t: T).(or (pr0 (THead (Flat Appl) v1 t) (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat 
-Appl) v1 t) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x0 u2) (ex2 T 
+T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H16 t3 t4 H5))) (\lambda (H16: (ex2 T 
 (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 
-u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) (THead (Bind 
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (H15: (pr0 x0 u2)).(or_introl (pr0 (THead (Flat Appl) v1 (THead 
-(Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 
-t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H15 t3 t4 H5))) 
-(\lambda (H15: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 
-(s (Flat Appl) i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) 
-(\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat 
-Appl) v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead 
-(Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x1: T).(\lambda (H16: 
-(pr0 x0 x1)).(\lambda (H17: (subst0 (s (Flat Appl) i) v3 u2 x1)).(or_intror 
-(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat 
-Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T 
+u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 
+(s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) 
+x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
 (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) 
 (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O v2) 
-t4)) (pr0_upsilon b H0 v1 v2 H1 x0 x1 H16 t3 t4 H5) (subst0_fst v3 x1 u2 i 
-H17 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))))) H15)) (H4 v0 x0 
-(s (Flat Appl) i) H14 v3 H8)) x H13) w1 H10)))) H12)) (\lambda (H12: (ex2 T 
-(\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 (s 
-(Bind b) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T x 
-(THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) 
-i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) 
-O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (x0: T).(\lambda (H13: (eq T x (THead (Bind b) u1 x0))).(\lambda 
-(H14: (subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x0)).(eq_ind_r T (THead 
-(Flat Appl) v1 x) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda 
-(w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4)) w2))))) (eq_ind_r T (THead (Bind b) u1 x0) (\lambda (t: T).(or (pr0 
-(THead (Flat Appl) v1 t) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 t) w2)) (\lambda 
-(w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4)) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) 
-(\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 
-(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) 
-v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H15: (pr0 x0 
-t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead 
+O) O v2) t4)) w2)))) (\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda 
+(H18: (subst0 (s (Flat Appl) i) v3 u2 x1)).(or_intror (pr0 (THead (Flat Appl) 
+v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) 
+O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind 
+b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead 
+(Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead 
+(Bind b) x1 (THead (Flat Appl) (lift (S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 
+H1 x0 x1 H17 t3 t4 H5) (subst0_fst v3 x1 u2 i H18 (THead (Flat Appl) (lift (S 
+O) O v2) t4) (Bind b))))))) H16)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8)) w1 
+H15))))) H12)) (\lambda (H12: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind b) 
+u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 
+t5)))).(ex2_ind T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda 
+(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead 
 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2))) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 x0 t4 H15))) (\lambda (H15: (ex2 T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat 
-Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
-T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat 
-Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead 
-(Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x1: T).(\lambda (H16: 
-(pr0 x0 x1)).(\lambda (H17: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 
-x1)).(or_intror (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead 
+T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq T x 
+(THead (Bind b) u1 x0))).(\lambda (H14: (subst0 (s (Bind b) (s (Flat Appl) 
+i)) v0 t3 x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead 
+(Flat Appl) v1 t))) H10 (THead (Bind b) u1 x0) H13) in (eq_ind_r T (THead 
+(Flat Appl) v1 (THead (Bind b) u1 x0)) (\lambda (t: T).(or (pr0 t (THead 
 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind 
-b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) x1)) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 x0 x1 H16) (subst0_snd 
-(Bind b) v3 (THead (Flat Appl) (lift (S O) O v2) x1) (THead (Flat Appl) (lift 
-(S O) O v2) t4) i (subst0_snd (Flat Appl) v3 x1 t4 (s (Bind b) i) H17 (lift 
-(S O) O v2)) u2)))))) H15)) (H6 v0 x0 (s (Bind b) (s (Flat Appl) i)) H14 v3 
-H8)) x H13) w1 H10)))) H12)) (\lambda (H12: (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda (u3: 
-T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: 
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: 
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 
+w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
+(\lambda (H16: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead 
+(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 
+x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 x0 t4 H16))) 
+(\lambda (H16: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 
+(s (Bind b) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 
+x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 w2)) 
+(or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
+(\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda (H18: (subst0 (s (Bind 
+b) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 (THead (Flat Appl) v1 (THead 
+(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 
+x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat 
+Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) x1)) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 
+x0 x1 H17) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O v2) x1) 
+(THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl) v3 x1 t4 
+(s (Bind b) i) H18 (lift (S O) O v2)) u2)))))) H16)) (H6 v0 x0 (s (Bind b) (s 
+(Flat Appl) i)) H14 v3 H8)) w1 H15))))) H12)) (\lambda (H12: (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda 
+(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: 
 T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 
 t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind 
 b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 
@@ -916,177 +903,172 @@ v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3
 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda 
 (x0: T).(\lambda (x1: T).(\lambda (H13: (eq T x (THead (Bind b) x0 
 x1))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 x0)).(\lambda (H15: 
-(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x1)).(eq_ind_r T (THead (Flat 
-Appl) v1 x) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: 
+(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x1)).(let H16 \def (eq_ind T x 
+(\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H10 (THead (Bind b) x0 
+x1) H13) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) 
+(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) 
+O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind 
+(pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s 
+(Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead 
+(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 
+x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) w2)))) (\lambda (H17: (pr0 x1 t4)).(or_ind (pr0 x0 u2) 
+(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) 
+i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: 
 T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2))))) (eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t: T).(or (pr0 (THead 
-(Flat Appl) v1 t) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 t) w2)) (\lambda 
-(w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4)) w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) 
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 v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) 
 x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
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-(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i 
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-(lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
-(H10: (eq T w1 (THead (Flat Appl) x0 x1))).(\lambda (H11: (subst0 i v0 v1 
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-(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) 
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+(S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 H1 x0 x2 H19 x1 t4 H17) (subst0_fst 
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+(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) 
 (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
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-(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 
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-(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 
-w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
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-(Bind b) x t3))).(\lambda (H15: (subst0 (s (Flat Appl) i) v0 u1 x)).(eq_ind_r 
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+(Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i 
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+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 v1 v2 
+H1 x0 u2 H20 x1 x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S 
+O) O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat 
+Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: 
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+T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead 
+(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 
+x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) w2)))) (\lambda (x3: T).(\lambda (H21: (pr0 x0 
+x3)).(\lambda (H22: (subst0 (s (Flat Appl) i) v3 u2 x3)).(or_intror (pr0 
+(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) 
+v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2)) (THead (Bind b) x3 (THead (Flat Appl) (lift (S O) O v2) x2)) 
+(pr0_upsilon b H0 v1 v2 H1 x0 x3 H21 x1 x2 H18) (subst0_both v3 u2 x3 i H22 
+(Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S 
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+O v2)))))))) H20)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))))) H17)) (H6 v0 x1 
+(s (Bind b) (s (Flat Appl) i)) H15 v3 H8)) w1 H16))))))) H12)) 
+(subst0_gen_head (Bind b) v0 u1 t3 x (s (Flat Appl) i) H11))))) H9)) (\lambda 
+(H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead (Flat Appl) 
+u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) 
+t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead 
+(Flat Appl) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) 
+(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) 
+u1 t3) t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
+v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda 
+(x0: T).(\lambda (x1: T).(\lambda (H10: (eq T w1 (THead (Flat Appl) x0 
+x1))).(\lambda (H11: (subst0 i v0 v1 x0)).(\lambda (H12: (subst0 (s (Flat 
+Appl) i) v0 (THead (Bind b) u1 t3) x1)).(or3_ind (ex2 T (\lambda (u3: T).(eq 
+T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 
+u1 u3))) (ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind b) u1 t5))) (\lambda 
+(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda 
+(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or 
+(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H13: (ex2 T 
+(\lambda (u3: T).(eq T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 
+(s (Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x1 (THead 
+(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or 
+(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda 
+(H14: (eq T x1 (THead (Bind b) x t3))).(\lambda (H15: (subst0 (s (Flat Appl) 
+i) v0 u1 x)).(let H16 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat 
+Appl) x0 t))) H10 (THead (Bind b) x t3) H14) in (eq_ind_r T (THead (Flat 
+Appl) x0 (THead (Bind b) x t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 
 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t 
 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) w2))))) (eq_ind_r T (THead (Bind b) x t3) (\lambda (t: 
-T).(or (pr0 (THead (Flat Appl) x0 t) (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 t) 
-w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
 (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x u2) (ex2 T (\lambda (w2: 
 T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or 
 (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead 
 (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat 
 Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H16: 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H17: 
 (pr0 x u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
 (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) 
 x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
 (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) 
 (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2)))) (\lambda (H17: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat 
+O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat 
 Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
 (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
 (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 x0 v2 H17 x u2 H16 
-t3 t4 H5))) (\lambda (H17: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
+(Flat Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 x0 v2 H18 x u2 H17 
+t3 t4 H5))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
 T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
 (w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x 
 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
 (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) 
 (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x0 x2)).(\lambda 
-(H19: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind 
+O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda 
+(H20: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind 
 b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 
 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) 
 (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
 O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H18 x u2 H16 t3 t4 
+(Flat Appl) (lift (S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 x u2 H17 t3 t4 
 H5) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead 
 (Flat Appl) (lift (S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S 
-O) O v2) (s (Bind b) i) (subst0_lift_ge_s v2 x2 v3 i H19 O (le_O_n i) b) t4 
-(Flat Appl)) u2)))))) H17)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H16: (ex2 T 
+O) O v2) (s (Bind b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 
+(Flat Appl)) u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T 
 (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 
 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s 
 (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x 
 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
 (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) 
 (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H17: (pr0 x x2)).(\lambda 
-(H18: (subst0 (s (Flat Appl) i) v3 u2 x2)).(or_ind (pr0 x0 v2) (ex2 T 
+O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x x2)).(\lambda 
+(H19: (subst0 (s (Flat Appl) i) v3 u2 x2)).(or_ind (pr0 x0 v2) (ex2 T 
 (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 
 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat 
 Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) 
 x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H19: (pr0 x0 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 
 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead 
 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
 T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: 
@@ -1094,15 +1076,15 @@ T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
 w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
 b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
 Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift 
-(S O) O v2) t4)) (pr0_upsilon b H0 x0 v2 H19 x x2 H17 t3 t4 H5) (subst0_fst 
-v3 x2 u2 i H18 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))) (\lambda 
-(H19: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
+(S O) O v2) t4)) (pr0_upsilon b H0 x0 v2 H20 x x2 H18 t3 t4 H5) (subst0_fst 
+v3 x2 u2 i H19 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))) (\lambda 
+(H20: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) 
 u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i 
 v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (x3: T).(\lambda (H20: (pr0 x0 x3)).(\lambda (H21: (subst0 i v3 v2 
+(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 
 x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead 
 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
 T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: 
@@ -1110,93 +1092,91 @@ T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
 w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
 b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
 Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift 
-(S O) O x3) t4)) (pr0_upsilon b H0 x0 x3 H20 x x2 H17 t3 t4 H5) (subst0_both 
-v3 u2 x2 i H18 (Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat 
+(S O) O x3) t4)) (pr0_upsilon b H0 x0 x3 H21 x x2 H18 t3 t4 H5) (subst0_both 
+v3 u2 x2 i H19 (Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat 
 Appl) (lift (S O) O x3) t4) (subst0_fst v3 (lift (S O) O x3) (lift (S O) O 
-v2) (s (Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H21 O (le_O_n i) b) t4 (Flat 
-Appl)))))))) H19)) (H2 v0 x0 i H11 v3 H8))))) H16)) (H4 v0 x (s (Flat Appl) 
-i) H15 v3 H8)) x1 H14) w1 H10)))) H13)) (\lambda (H13: (ex2 T (\lambda (t5: 
-T).(eq T x1 (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s 
-(Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T x1 (THead 
-(Bind b) u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 
-t3 t5)) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda 
-(x: T).(\lambda (H14: (eq T x1 (THead (Bind b) u1 x))).(\lambda (H15: (subst0 
-(s (Bind b) (s (Flat Appl) i)) v0 t3 x)).(eq_ind_r T (THead (Flat Appl) x0 
-x1) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) 
-(eq_ind_r T (THead (Bind b) u1 x) (\lambda (t: T).(or (pr0 (THead (Flat Appl) 
-x0 t) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 t) w2)) (\lambda (w2: T).(subst0 
-i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) 
-(or_ind (pr0 x t4) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: 
-T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat 
+v2) (s (Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) t4 (Flat 
+Appl)))))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H4 v0 x (s (Flat Appl) 
+i) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex2 T (\lambda (t5: T).(eq T 
+x1 (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat 
+Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T x1 (THead (Bind b) 
+u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) 
+(or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
+(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: 
+T).(\lambda (H14: (eq T x1 (THead (Bind b) u1 x))).(\lambda (H15: (subst0 (s 
+(Bind b) (s (Flat Appl) i)) v0 t3 x)).(let H16 \def (eq_ind T x1 (\lambda (t: 
+T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) u1 x) H14) in 
+(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (\lambda (t: T).(or 
+(pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x t4) (ex2 T 
+(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat 
+Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda 
+(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2)))) (\lambda (H17: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: 
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat 
 Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
 (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
 (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H16: (pr0 x t4)).(or_ind 
+(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 
+v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2))) (pr0_upsilon b H0 x0 v2 H18 u1 u2 H3 x t4 H17))) (\lambda (H18: (ex2 T 
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
+v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
+(\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda (H20: (subst0 i v3 v2 
+x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
+b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 u1 u2 H3 x t4 H17) (subst0_snd 
+(Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead (Flat Appl) (lift 
+(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S O) O v2) (s (Bind 
+b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 (Flat Appl)) 
+u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: 
+T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s 
+(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead 
+(Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
+(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) 
+w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x 
+x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind 
 (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i 
 v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead 
 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
 T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: 
 T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2)))) (\lambda (H17: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat Appl) x0 
+w2)))) (\lambda (H20: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 
 (THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
 v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) 
 u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 x0 v2 H17 u1 u2 H3 x t4 
-H16))) (\lambda (H17: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
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-(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) 
-u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x0 x2)).(\lambda 
-(H19: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind 
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-T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
-(THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H18 u1 u2 H3 x t4 
-H16) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead 
-(Flat Appl) (lift (S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S 
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-(Flat Appl)) u2)))))) H17)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H16: (ex2 T 
-(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat 
-Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: 
-T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat 
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-(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H17: 
-(pr0 x x2)).(\lambda (H18: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 
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-T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 
-x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) 
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-O) O v2) t4)) w2)))) (\lambda (H19: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat 
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-(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead 
+Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead 
+(Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead 
 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 x0 v2 
-H19 u1 u2 H3 x x2 H17) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) 
+H20 u1 u2 H3 x x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) 
 O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl) 
-v3 x2 t4 (s (Bind b) i) H18 (lift (S O) O v2)) u2)))) (\lambda (H19: (ex2 T 
+v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: (ex2 T 
 (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) 
 u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i 
 v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (x3: T).(\lambda (H20: (pr0 x0 x3)).(\lambda (H21: (subst0 i v3 v2 
+(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 
 x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead 
 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
 T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: 
@@ -1204,15 +1184,15 @@ T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
 w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
 b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
 Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O x3) x2)) (pr0_upsilon b H0 x0 x3 H20 u1 u2 H3 x x2 H17) (subst0_snd 
+(S O) O x3) x2)) (pr0_upsilon b H0 x0 x3 H21 u1 u2 H3 x x2 H18) (subst0_snd 
 (Bind b) v3 (THead (Flat Appl) (lift (S O) O x3) x2) (THead (Flat Appl) (lift 
 (S O) O v2) t4) i (subst0_both v3 (lift (S O) O v2) (lift (S O) O x3) (s 
-(Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H21 O (le_O_n i) b) (Flat Appl) t4 
-x2 H18) u2)))))) H19)) (H2 v0 x0 i H11 v3 H8))))) H16)) (H6 v0 x (s (Bind b) 
-(s (Flat Appl) i)) H15 v3 H8)) x1 H14) w1 H10)))) H13)) (\lambda (H13: (ex3_2 
-T T (\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) 
-(\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) 
-(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 
+(Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) (Flat Appl) t4 
+x2 H19) u2)))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H6 v0 x (s (Bind b) 
+(s (Flat Appl) i)) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda 
+(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 
 t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead 
 (Bind b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) 
 v0 u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat 
@@ -1221,68 +1201,66 @@ Appl) i)) v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift
 T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
 w2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T x1 (THead (Bind 
 b) x2 x3))).(\lambda (H15: (subst0 (s (Flat Appl) i) v0 u1 x2)).(\lambda 
-(H16: (subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x3)).(eq_ind_r T (THead 
-(Flat Appl) x0 x1) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda 
-(w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4)) w2))))) (eq_ind_r T (THead (Bind b) x2 x3) (\lambda (t: T).(or (pr0 
-(THead (Flat Appl) x0 t) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 t) w2)) (\lambda 
-(w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4)) w2))))) (or_ind (pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) 
-(\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 
-(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) 
-x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H17: (pr0 x3 
-t4)).(or_ind (pr0 x2 u2) (ex2 T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: 
-T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) x0 
-(THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) 
-x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x2 u2)).(or_ind (pr0 
-x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
-w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (H19: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead 
+(H16: (subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x3)).(let H17 \def (eq_ind 
+T x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) 
+x2 x3) H14) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) 
+(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) 
+O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind 
+(pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s 
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 (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
 t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 
 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 x0 v2 H19 x2 u2 H18 x3 t4 
-H17))) (\lambda (H19: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
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-(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) 
-x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H20: (pr0 x0 x)).(\lambda 
-(H21: (subst0 i v3 v2 x)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind 
-b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 
-T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
-(THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O x) t4)) (pr0_upsilon b H0 x0 x H20 x2 u2 H18 x3 t4 
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-(Flat Appl) (lift (S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x) (lift (S 
-O) O v2) (s (Bind b) i) (subst0_lift_ge_s v2 x v3 i H21 O (le_O_n i) b) t4 
-(Flat Appl)) u2)))))) H19)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H18: (ex2 T 
-(\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 
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-(s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) 
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+(lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x3 t4)).(or_ind (pr0 x2 u2) 
+(ex2 T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) 
+i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2)))) (\lambda (H19: (pr0 x2 u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: 
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat 
+Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 
+v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2))) (pr0_upsilon b H0 x0 v2 H20 x2 u2 H19 x3 t4 H18))) (\lambda (H20: (ex2 
+T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
+v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind 
+b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
+(\lambda (x: T).(\lambda (H21: (pr0 x0 x)).(\lambda (H22: (subst0 i v3 v2 
+x)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
+b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O x) t4)) (pr0_upsilon b H0 x0 x H21 x2 u2 H19 x3 t4 H18) (subst0_snd 
+(Bind b) v3 (THead (Flat Appl) (lift (S O) O x) t4) (THead (Flat Appl) (lift 
+(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x) (lift (S O) O v2) (s (Bind 
+b) i) (subst0_lift_ge_s v2 x v3 i H22 O (le_O_n i) b) t4 (Flat Appl)) 
+u2)))))) H20)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H19: (ex2 T (\lambda (w2: 
+T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s 
+(Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 
+x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
 (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) 
 (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H19: (pr0 x2 x)).(\lambda 
-(H20: (subst0 (s (Flat Appl) i) v3 u2 x)).(or_ind (pr0 x0 v2) (ex2 T (\lambda 
+O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H20: (pr0 x2 x)).(\lambda 
+(H21: (subst0 (s (Flat Appl) i) v3 u2 x)).(or_ind (pr0 x0 v2) (ex2 T (\lambda 
 (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead 
 (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) 
 (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H21: (pr0 x0 
+(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H22: (pr0 x0 
 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead 
 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
 T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: 
@@ -1290,15 +1268,15 @@ T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
 w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
 b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
 Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x (THead (Flat Appl) (lift 
-(S O) O v2) t4)) (pr0_upsilon b H0 x0 v2 H21 x2 x H19 x3 t4 H17) (subst0_fst 
-v3 x u2 i H20 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))) (\lambda 
-(H21: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
+(S O) O v2) t4)) (pr0_upsilon b H0 x0 v2 H22 x2 x H20 x3 t4 H18) (subst0_fst 
+v3 x u2 i H21 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))) (\lambda 
+(H22: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind 
 b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i 
 v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (x4: T).(\lambda (H22: (pr0 x0 x4)).(\lambda (H23: (subst0 i v3 v2 
+(\lambda (x4: T).(\lambda (H23: (pr0 x0 x4)).(\lambda (H24: (subst0 i v3 v2 
 x4)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead 
 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
 T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: 
@@ -1306,31 +1284,31 @@ T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
 w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
 b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
 Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x (THead (Flat Appl) (lift 
-(S O) O x4) t4)) (pr0_upsilon b H0 x0 x4 H22 x2 x H19 x3 t4 H17) (subst0_both 
-v3 u2 x i H20 (Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat 
+(S O) O x4) t4)) (pr0_upsilon b H0 x0 x4 H23 x2 x H20 x3 t4 H18) (subst0_both 
+v3 u2 x i H21 (Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat 
 Appl) (lift (S O) O x4) t4) (subst0_fst v3 (lift (S O) O x4) (lift (S O) O 
-v2) (s (Bind b) i) (subst0_lift_ge_s v2 x4 v3 i H23 O (le_O_n i) b) t4 (Flat 
-Appl)))))))) H21)) (H2 v0 x0 i H11 v3 H8))))) H18)) (H4 v0 x2 (s (Flat Appl) 
-i) H15 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda 
+v2) (s (Bind b) i) (subst0_lift_ge_s v2 x4 v3 i H24 O (le_O_n i) b) t4 (Flat 
+Appl)))))))) H22)) (H2 v0 x0 i H11 v3 H8))))) H19)) (H4 v0 x2 (s (Flat Appl) 
+i) H15 v3 H8))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda 
 (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T 
 (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat 
 Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) 
 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda 
 (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: 
 T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2)))) (\lambda (x: T).(\lambda (H18: (pr0 x3 x)).(\lambda (H19: (subst0 (s 
+w2)))) (\lambda (x: T).(\lambda (H19: (pr0 x3 x)).(\lambda (H20: (subst0 (s 
 (Bind b) (s (Flat Appl) i)) v3 t4 x)).(or_ind (pr0 x2 u2) (ex2 T (\lambda 
 (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) 
 (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 
 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i 
 v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (H20: (pr0 x2 u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 
+(\lambda (H21: (pr0 x2 u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 
 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 
 (THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
 v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) 
 x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H21: (pr0 x0 v2)).(or_intror 
+Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H22: (pr0 x0 v2)).(or_intror 
 (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead 
 (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat 
 Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
@@ -1338,59 +1316,59 @@ Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead
 (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) 
 (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
 O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-x)) (pr0_upsilon b H0 x0 v2 H21 x2 u2 H20 x3 x H18) (subst0_snd (Bind b) v3 
+x)) (pr0_upsilon b H0 x0 v2 H22 x2 u2 H21 x3 x H19) (subst0_snd (Bind b) v3 
 (THead (Flat Appl) (lift (S O) O v2) x) (THead (Flat Appl) (lift (S O) O v2) 
-t4) i (subst0_snd (Flat Appl) v3 x t4 (s (Bind b) i) H19 (lift (S O) O v2)) 
-u2)))) (\lambda (H21: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
+t4) i (subst0_snd (Flat Appl) v3 x t4 (s (Bind b) i) H20 (lift (S O) O v2)) 
+u2)))) (\lambda (H22: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
 T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
 (w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) 
 x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
 (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) 
 (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2)))) (\lambda (x4: T).(\lambda (H22: (pr0 x0 x4)).(\lambda 
-(H23: (subst0 i v3 v2 x4)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind 
+O) O v2) t4)) w2)))) (\lambda (x4: T).(\lambda (H23: (pr0 x0 x4)).(\lambda 
+(H24: (subst0 i v3 v2 x4)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind 
 b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 
 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) 
 (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
 O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O x4) x)) (pr0_upsilon b H0 x0 x4 H22 x2 u2 H20 x3 x 
-H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O x4) x) (THead 
+(Flat Appl) (lift (S O) O x4) x)) (pr0_upsilon b H0 x0 x4 H23 x2 u2 H21 x3 x 
+H19) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O x4) x) (THead 
 (Flat Appl) (lift (S O) O v2) t4) i (subst0_both v3 (lift (S O) O v2) (lift 
-(S O) O x4) (s (Bind b) i) (subst0_lift_ge_s v2 x4 v3 i H23 O (le_O_n i) b) 
-(Flat Appl) t4 x H19) u2)))))) H21)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H20: 
+(S O) O x4) (s (Bind b) i) (subst0_lift_ge_s v2 x4 v3 i H24 O (le_O_n i) b) 
+(Flat Appl) t4 x H20) u2)))))) H22)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H21: 
 (ex2 T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) 
 i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: 
 T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead 
 (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
 t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 
 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) w2)))) (\lambda (x4: T).(\lambda (H21: (pr0 x2 
-x4)).(\lambda (H22: (subst0 (s (Flat Appl) i) v3 u2 x4)).(or_ind (pr0 x0 v2) 
+(lift (S O) O v2) t4)) w2)))) (\lambda (x4: T).(\lambda (H22: (pr0 x2 
+x4)).(\lambda (H23: (subst0 (s (Flat Appl) i) v3 u2 x4)).(or_ind (pr0 x0 v2) 
 (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) 
 (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 
 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i 
 v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (H23: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead 
+(\lambda (H24: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead 
 (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
 t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 
 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
 (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat 
 Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x4 
-(THead (Flat Appl) (lift (S O) O v2) x)) (pr0_upsilon b H0 x0 v2 H23 x2 x4 
-H21 x3 x H18) (subst0_both v3 u2 x4 i H22 (Bind b) (THead (Flat Appl) (lift 
+(THead (Flat Appl) (lift (S O) O v2) x)) (pr0_upsilon b H0 x0 v2 H24 x2 x4 
+H22 x3 x H19) (subst0_both v3 u2 x4 i H23 (Bind b) (THead (Flat Appl) (lift 
 (S O) O v2) t4) (THead (Flat Appl) (lift (S O) O v2) x) (subst0_snd (Flat 
-Appl) v3 x t4 (s (Bind b) i) H19 (lift (S O) O v2)))))) (\lambda (H23: (ex2 T 
+Appl) v3 x t4 (s (Bind b) i) H20 (lift (S O) O v2)))))) (\lambda (H24: (ex2 T 
 (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind 
 b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i 
 v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (x5: T).(\lambda (H24: (pr0 x0 x5)).(\lambda (H25: (subst0 i v3 v2 
+(\lambda (x5: T).(\lambda (H25: (pr0 x0 x5)).(\lambda (H26: (subst0 i v3 v2 
 x5)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead 
 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
 T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: 
@@ -1398,14 +1376,14 @@ T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
 w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
 b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
 Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x4 (THead (Flat Appl) (lift 
-(S O) O x5) x)) (pr0_upsilon b H0 x0 x5 H24 x2 x4 H21 x3 x H18) (subst0_both 
-v3 u2 x4 i H22 (Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat 
+(S O) O x5) x)) (pr0_upsilon b H0 x0 x5 H25 x2 x4 H22 x3 x H19) (subst0_both 
+v3 u2 x4 i H23 (Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat 
 Appl) (lift (S O) O x5) x) (subst0_both v3 (lift (S O) O v2) (lift (S O) O 
-x5) (s (Bind b) i) (subst0_lift_ge_s v2 x5 v3 i H25 O (le_O_n i) b) (Flat 
-Appl) t4 x H19))))))) H23)) (H2 v0 x0 i H11 v3 H8))))) H20)) (H4 v0 x2 (s 
-(Flat Appl) i) H15 v3 H8))))) H17)) (H6 v0 x3 (s (Bind b) (s (Flat Appl) i)) 
-H16 v3 H8)) x1 H14) w1 H10)))))) H13)) (subst0_gen_head (Bind b) v0 u1 t3 x1 
-(s (Flat Appl) i) H12))))))) H9)) (subst0_gen_head (Flat Appl) v0 v1 (THead 
+x5) (s (Bind b) i) (subst0_lift_ge_s v2 x5 v3 i H26 O (le_O_n i) b) (Flat 
+Appl) t4 x H20))))))) H24)) (H2 v0 x0 i H11 v3 H8))))) H21)) (H4 v0 x2 (s 
+(Flat Appl) i) H15 v3 H8))))) H18)) (H6 v0 x3 (s (Bind b) (s (Flat Appl) i)) 
+H16 v3 H8)) w1 H17))))))) H13)) (subst0_gen_head (Bind b) v0 u1 t3 x1 (s 
+(Flat Appl) i) H12))))))) H9)) (subst0_gen_head (Flat Appl) v0 v1 (THead 
 (Bind b) u1 t3) w1 i H7)))))))))))))))))))))) (\lambda (u1: T).(\lambda (u2: 
 T).(\lambda (H0: (pr0 u1 u2)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: 
 T).(\forall (i: nat).((subst0 i v1 u1 w1) \to (\forall (v2: T).((pr0 v1 v2) 
@@ -1627,33 +1605,32 @@ w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6:
 (\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or (pr0 w1 
 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 
 w2)))) (\lambda (x0: T).(\lambda (H8: (eq T x (lift (S O) O x0))).(\lambda 
-(H9: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x0)).(eq_ind_r T (THead (Bind 
-b) u x) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) 
-(\lambda (w2: T).(subst0 i v2 t4 w2))))) (eq_ind_r T (lift (S O) O x0) 
-(\lambda (t: T).(or (pr0 (THead (Bind b) u t) t4) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Bind b) u t) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) 
-(let H10 \def (eq_ind_r nat (minus i O) (\lambda (n: nat).(subst0 n v1 t3 
-x0)) H9 i (minus_n_O i)) in (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: T).(pr0 
-x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or (pr0 (THead (Bind b) u 
-(lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift 
-(S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H11: (pr0 
-x0 t4)).(or_introl (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x0 t4 H11 u))) (\lambda (H11: (ex2 T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 
+(H9: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x0)).(let H10 \def (eq_ind T 
+x (\lambda (t: T).(eq T w1 (THead (Bind b) u t))) H6 (lift (S O) O x0) H8) in 
+(eq_ind_r T (THead (Bind b) u (lift (S O) O x0)) (\lambda (t: T).(or (pr0 t 
+t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 
+w2))))) (let H11 \def (eq_ind_r nat (minus i O) (\lambda (n: nat).(subst0 n 
+v1 t3 x0)) H9 i (minus_n_O i)) in (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: 
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or (pr0 (THead (Bind 
+b) u (lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u 
+(lift (S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda 
+(H12: (pr0 x0 t4)).(or_introl (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) 
+(ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) 
+(\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x0 t4 H12 u))) (\lambda 
+(H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 
 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 
 t4 w2)) (or (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T (\lambda 
 (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2)))) (\lambda (x1: T).(\lambda (H12: (pr0 x0 
-x1)).(\lambda (H13: (subst0 i v2 t4 x1)).(or_intror (pr0 (THead (Bind b) u 
+T).(subst0 i v2 t4 w2)))) (\lambda (x1: T).(\lambda (H13: (pr0 x0 
+x1)).(\lambda (H14: (subst0 i v2 t4 x1)).(or_intror (pr0 (THead (Bind b) u 
 (lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift 
 (S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T 
 (\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2)) x1 (pr0_zeta b H0 x0 x1 H12 u) H13))))) H11)) (H2 v1 
-x0 i H10 v2 H4))) x H8) w1 H6)))) (subst0_gen_lift_ge v1 t3 x (s (Bind b) i) 
-(S O) O H7 (le_n_S O i (le_O_n i))))))) H5)) (\lambda (H5: (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda 
-(u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 i v2 t4 w2)) x1 (pr0_zeta b H0 x0 x1 H13 u) H14))))) H12)) (H2 v1 
+x0 i H11 v2 H4))) w1 H10))))) (subst0_gen_lift_ge v1 t3 x (s (Bind b) i) (S 
+O) O H7 (le_n_S O i (le_O_n i))))))) H5)) (\lambda (H5: (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
 T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))))).(ex3_2_ind T T 
 (\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda 
 (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
@@ -1665,91 +1642,90 @@ v1 (lift (S O) O t3) x1)).(ex2_ind T (\lambda (t5: T).(eq T x1 (lift (S O) O
 t5))) (\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or 
 (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i 
 v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (eq T x1 (lift (S O) O 
-x))).(\lambda (H10: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x)).(eq_ind_r 
-T (THead (Bind b) x0 x1) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: 
-T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (eq_ind_r T (lift (S 
-O) O x) (\lambda (t: T).(or (pr0 (THead (Bind b) x0 t) t4) (ex2 T (\lambda 
-(w2: T).(pr0 (THead (Bind b) x0 t) w2)) (\lambda (w2: T).(subst0 i v2 t4 
-w2))))) (let H11 \def (eq_ind_r nat (minus i O) (\lambda (n: nat).(subst0 n 
-v1 t3 x)) H10 i (minus_n_O i)) in (or_ind (pr0 x t4) (ex2 T (\lambda (w2: 
-T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or (pr0 (THead (Bind 
-b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 
-(lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H12: 
-(pr0 x t4)).(or_introl (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T 
+x))).(\lambda (H10: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x)).(let H11 
+\def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Bind b) x0 t))) H6 (lift 
+(S O) O x) H9) in (eq_ind_r T (THead (Bind b) x0 (lift (S O) O x)) (\lambda 
+(t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2))))) (let H12 \def (eq_ind_r nat (minus i O) (\lambda 
+(n: nat).(subst0 n v1 t3 x)) H10 i (minus_n_O i)) in (or_ind (pr0 x t4) (ex2 
+T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or 
+(pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 
+w2)))) (\lambda (H13: (pr0 x t4)).(or_introl (pr0 (THead (Bind b) x0 (lift (S 
+O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O 
+x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x t4 H13 x0))) 
+(\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i 
+v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 
+i v2 t4 w2)) (or (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T 
 (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x t4 H12 x0))) (\lambda (H12: (ex2 T 
-(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))).(ex2_ind 
-T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)) (or (pr0 
-(THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead 
-(Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
-(\lambda (x2: T).(\lambda (H13: (pr0 x x2)).(\lambda (H14: (subst0 i v2 t4 
-x2)).(or_intror (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda 
+T).(subst0 i v2 t4 w2)))) (\lambda (x2: T).(\lambda (H14: (pr0 x 
+x2)).(\lambda (H15: (subst0 i v2 t4 x2)).(or_intror (pr0 (THead (Bind b) x0 
+(lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift 
+(S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda 
 (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Bind b) 
-x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)) x2 (pr0_zeta 
-b H0 x x2 H13 x0) H14))))) H12)) (H2 v1 x i H11 v2 H4))) x1 H9) w1 H6)))) 
-(subst0_gen_lift_ge v1 t3 x1 (s (Bind b) i) (S O) O H8 (le_n_S O i (le_O_n 
-i))))))))) H5)) (subst0_gen_head (Bind b) v1 u (lift (S O) O t3) w1 i 
-H3))))))))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 
-t4)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: T).(\forall (i: 
-nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 
-t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 
-w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: 
-nat).(\lambda (H2: (subst0 i v1 (THead (Flat Cast) u t3) w1)).(\lambda (v2: 
-T).(\lambda (H3: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda (u2: T).(eq T w1 
-(THead (Flat Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u u2))) (ex2 T 
-(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0 
-(s (Flat Cast) i) v1 t3 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
-T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat 
-Cast) i) v1 t3 t5)))) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) 
-(\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H4: (ex2 T (\lambda (u2: 
-T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u 
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) 
-(\lambda (u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: 
-T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: 
-T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x t3))).(\lambda (_: (subst0 i 
-v1 u x)).(eq_ind_r T (THead (Flat Cast) x t3) (\lambda (t: T).(or (pr0 t t4) 
-(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) 
-(or_introl (pr0 (THead (Flat Cast) x t3) t4) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Cast) x t3) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) 
-(pr0_epsilon t3 t4 H0 x)) w1 H5)))) H4)) (\lambda (H4: (ex2 T (\lambda (t5: 
-T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0 (s (Flat 
-Cast) i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) 
-u t5))) (\lambda (t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5)) (or (pr0 w1 t4) 
-(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
-(\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) u x))).(\lambda 
-(H6: (subst0 (s (Flat Cast) i) v1 t3 x)).(eq_ind_r T (THead (Flat Cast) u x) 
-(\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda 
-(w2: T).(subst0 i v2 t4 w2))))) (or_ind (pr0 x t4) (ex2 T (\lambda (w2: 
-T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2))) (or 
-(pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat 
-Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H7: (pr0 x 
-t4)).(or_introl (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) 
-(pr0_epsilon x t4 H7 u))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 x w2)) 
-(\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T (\lambda 
-(w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)) (or 
-(pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat 
-Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: 
-T).(\lambda (H8: (pr0 x x0)).(\lambda (H9: (subst0 (s (Flat Cast) i) v2 t4 
-x0)).(or_intror (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) 
-(ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2)) x0 (pr0_epsilon x x0 H8 u) H9))))) H7)) (H1 v1 x (s 
-(Flat Cast) i) H6 v2 H3)) w1 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s (Flat Cast) i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u2: 
+T).(subst0 i v2 t4 w2)) x2 (pr0_zeta b H0 x x2 H14 x0) H15))))) H13)) (H2 v1 
+x i H12 v2 H4))) w1 H11))))) (subst0_gen_lift_ge v1 t3 x1 (s (Bind b) i) (S 
+O) O H8 (le_n_S O i (le_O_n i))))))))) H5)) (subst0_gen_head (Bind b) v1 u 
+(lift (S O) O t3) w1 i H3))))))))))))))) (\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (H0: (pr0 t3 t4)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: 
+T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) 
+\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda 
+(w1: T).(\lambda (i: nat).(\lambda (H2: (subst0 i v1 (THead (Flat Cast) u t3) 
+w1)).(\lambda (v2: T).(\lambda (H3: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda 
+(u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u 
+u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda 
+(t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (ex3_2 T T (\lambda (u2: 
 T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
 T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (or (pr0 w1 t4) (ex2 T (\lambda (w2: 
-T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x0 
-x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H7: (subst0 (s (Flat Cast) 
-i) v1 t3 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (pr0 
-t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 
-w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda 
+T).(subst0 (s (Flat Cast) i) v1 t3 t5)))) (or (pr0 w1 t4) (ex2 T (\lambda 
+(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H4: 
+(ex2 T (\lambda (u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: 
+T).(subst0 i v1 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat 
+Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T 
+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
+(\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x t3))).(\lambda 
+(_: (subst0 i v1 u x)).(eq_ind_r T (THead (Flat Cast) x t3) (\lambda (t: 
+T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2))))) (or_introl (pr0 (THead (Flat Cast) x t3) t4) (ex2 
+T (\lambda (w2: T).(pr0 (THead (Flat Cast) x t3) w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2))) (pr0_epsilon t3 t4 H0 x)) w1 H5)))) H4)) (\lambda 
+(H4: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda 
+(t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: 
+T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0 (s (Flat 
+Cast) i) v1 t3 t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) 
+(\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H5: (eq T 
+w1 (THead (Flat Cast) u x))).(\lambda (H6: (subst0 (s (Flat Cast) i) v1 t3 
+x)).(eq_ind_r T (THead (Flat Cast) u x) (\lambda (t: T).(or (pr0 t t4) (ex2 T 
+(\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (or_ind 
+(pr0 x t4) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s 
+(Flat Cast) i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i 
+v2 t4 w2)))) (\lambda (H7: (pr0 x t4)).(or_introl (pr0 (THead (Flat Cast) u 
+x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda 
+(w2: T).(subst0 i v2 t4 w2))) (pr0_epsilon x t4 H7 u))) (\lambda (H7: (ex2 T 
+(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s 
+(Flat Cast) i) v2 t4 w2)) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i 
+v2 t4 w2)))) (\lambda (x0: T).(\lambda (H8: (pr0 x x0)).(\lambda (H9: (subst0 
+(s (Flat Cast) i) v2 t4 x0)).(or_intror (pr0 (THead (Flat Cast) u x) t4) (ex2 
+T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 
+i v2 t4 w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) 
+(\lambda (w2: T).(subst0 i v2 t4 w2)) x0 (pr0_epsilon x x0 H8 u) H9))))) H7)) 
+(H1 v1 x (s (Flat Cast) i) H6 v2 H3)) w1 H5)))) H4)) (\lambda (H4: (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (or (pr0 w1 t4) 
+(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) 
+x0 x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H7: (subst0 (s (Flat 
+Cast) i) v1 t3 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or 
+(pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 
+t4 w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda 
 (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) x0 
 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda 
 (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H8: (pr0 x1 t4)).(or_introl (pr0 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/subst1.ma
index c7efff402..091348890 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/subst1.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/subst1.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr0/props.ma".
+include "LambdaDelta-1/pr0/props.ma".
 
-include "subst1/defs.ma".
+include "LambdaDelta-1/subst1/defs.ma".
 
 theorem pr0_delta1:
  \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr1/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr1/defs.ma
index bfa5c9d32..7452fc5cf 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr1/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr1/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr0/defs.ma".
+include "LambdaDelta-1/pr0/defs.ma".
 
 inductive pr1: T \to (T \to Prop) \def
 | pr1_refl: \forall (t: T).(pr1 t t)
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr1/pr1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr1/pr1.ma
index 9b935f805..96a3bbf03 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr1/pr1.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr1/pr1.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr1/props.ma".
+include "LambdaDelta-1/pr1/props.ma".
 
-include "pr0/pr0.ma".
+include "LambdaDelta-1/pr0/pr0.ma".
 
 theorem pr1_strip:
  \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr0 t0 
@@ -32,12 +32,13 @@ T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda
 (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: 
 T).(\lambda (H3: (pr0 t3 t5)).(ex2_ind T (\lambda (t: T).(pr0 t5 t)) (\lambda 
 (t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 
-t))) (\lambda (x: T).(\lambda (H4: (pr0 t5 x)).(\lambda (H5: (pr0 t2 
-x)).(ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 x t)) (ex2 T 
-(\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) (\lambda (x0: 
-T).(\lambda (H6: (pr1 t4 x0)).(\lambda (H7: (pr1 x x0)).(ex_intro2 T (\lambda 
-(t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 H6 (pr1_sing x t5 H4 x0 
-H7))))) (H2 x H5))))) (pr0_confluence t3 t5 H3 t2 H0)))))))))) t0 t1 H))).
+t))) (\lambda (x: T).(\lambda (H4: (pr0 t5 x)).(\lambda (H5: (pr0 t2 x)).(let 
+H6 \def (H2 x H5) in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: 
+T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) 
+(\lambda (x0: T).(\lambda (H7: (pr1 t4 x0)).(\lambda (H8: (pr1 x 
+x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 
+H7 (pr1_t x t5 (pr1_pr0 t5 x H4) x0 H8))))) H6))))) (pr0_confluence t3 t5 H3 
+t2 H0)))))))))) t0 t1 H))).
 
 theorem pr1_confluence:
  \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr1 t0 
@@ -51,12 +52,13 @@ T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda
 T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda (_: 
 (pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t2 t5) \to (ex2 T (\lambda 
 (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: T).(\lambda 
-(H3: (pr1 t3 t5)).(ex2_ind T (\lambda (t: T).(pr1 t5 t)) (\lambda (t: T).(pr1 
-t2 t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) 
-(\lambda (x: T).(\lambda (H4: (pr1 t5 x)).(\lambda (H5: (pr1 t2 x)).(ex2_ind 
-T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 x t)) (ex2 T (\lambda (t: 
-T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) (\lambda (x0: T).(\lambda (H6: 
-(pr1 t4 x0)).(\lambda (H7: (pr1 x x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 
-t)) (\lambda (t: T).(pr1 t5 t)) x0 H6 (pr1_t x t5 H4 x0 H7))))) (H2 x H5))))) 
-(pr1_strip t3 t5 H3 t2 H0)))))))))) t0 t1 H))).
+(H3: (pr1 t3 t5)).(let H_x \def (pr1_strip t3 t5 H3 t2 H0) in (let H4 \def 
+H_x in (ex2_ind T (\lambda (t: T).(pr1 t5 t)) (\lambda (t: T).(pr1 t2 t)) 
+(ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) (\lambda (x: 
+T).(\lambda (H5: (pr1 t5 x)).(\lambda (H6: (pr1 t2 x)).(let H_x0 \def (H2 x 
+H6) in (let H7 \def H_x0 in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda 
+(t: T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 
+t))) (\lambda (x0: T).(\lambda (H8: (pr1 t4 x0)).(\lambda (H9: (pr1 x 
+x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 
+H8 (pr1_t x t5 H5 x0 H9))))) H7)))))) H4))))))))))) t0 t1 H))).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr1/props.ma
index a5d606b06..acb36f90f 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr1/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr1/props.ma
@@ -14,13 +14,13 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr1/defs.ma".
+include "LambdaDelta-1/pr1/defs.ma".
 
-include "pr0/subst1.ma".
+include "LambdaDelta-1/pr0/subst1.ma".
 
-include "subst1/props.ma".
+include "LambdaDelta-1/subst1/props.ma".
 
-include "T/props.ma".
+include "LambdaDelta-1/T/props.ma".
 
 theorem pr1_pr0:
  \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr1 t1 t2)))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/clen.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/clen.ma
index b935b2616..09a9a3d89 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/clen.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/clen.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr2/props.ma".
+include "LambdaDelta-1/pr2/props.ma".
 
-include "clen/getl.ma".
+include "LambdaDelta-1/clen/getl.ma".
 
 theorem pr2_gen_ctail:
  \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall 
@@ -82,100 +82,71 @@ theorem pr2_gen_cbind:
 (THead (Bind b) v t2)))))))
 \def
  \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(let H0 \def (match H 
-in pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: 
-(pr2 c0 t t0)).((eq C c0 (CHead c (Bind b) v)) \to ((eq T t t1) \to ((eq T t0 
-t2) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))))))))) with 
-[(pr2_free c0 t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c0 (CHead c (Bind b) 
-v))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead 
-c (Bind b) v) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 
-t3) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)))))) (\lambda (H4: 
-(eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to 
-(pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))))) (\lambda (H5: (eq T t3 
-t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Bind b) v 
-t1) (THead (Bind b) v t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead 
-(Bind b) v t1) (THead (Bind b) v t2) (pr0_comp v v (pr0_refl v) t1 t2 H6 
-(Bind b)))) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 
-(CHead c (Bind b) v) H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t3 H1 t H2) 
-\Rightarrow (\lambda (H3: (eq C c0 (CHead c (Bind b) v))).(\lambda (H4: (eq T 
-t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Bind b) v) (\lambda 
-(c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) 
-u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c (THead (Bind b) v t1) 
-(THead (Bind b) v t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 
-(\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c (Bind b) v) (CHead d 
-(Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c (THead 
-(Bind b) v t1) (THead (Bind b) v t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind 
-T t2 (\lambda (t4: T).((getl i (CHead c (Bind b) v) (CHead d (Bind Abbr) u)) 
-\to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c (THead (Bind b) v t1) 
-(THead (Bind b) v t2)))))) (\lambda (H8: (getl i (CHead c (Bind b) v) (CHead 
-d (Bind Abbr) u))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u t3 
-t2)).(let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u) v i H8) in (let 
-H11 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d (Bind Abbr) u) 
-(CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda 
-(j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead (Bind b) v t1) 
-(THead (Bind b) v t2)) (\lambda (H12: (land (eq nat i O) (eq C (CHead d (Bind 
-Abbr) u) (CHead c (Bind b) v)))).(and_ind (eq nat i O) (eq C (CHead d (Bind 
-Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t1) (THead (Bind b) v 
-t2)) (\lambda (H13: (eq nat i O)).(\lambda (H14: (eq C (CHead d (Bind Abbr) 
-u) (CHead c (Bind b) v))).(let H15 \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) v) H14) in ((let 
-H16 \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) v) H14) in 
-((let H17 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: 
-C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead 
-d (Bind Abbr) u) (CHead c (Bind b) v) H14) in (\lambda (H18: (eq B Abbr 
-b)).(\lambda (_: (eq C d c)).(let H20 \def (eq_ind nat i (\lambda (n: 
-nat).(subst0 n u t3 t2)) H10 O H13) in (let H21 \def (eq_ind T u (\lambda 
-(t4: T).(subst0 O t4 t3 t2)) H20 v H17) in (eq_ind B Abbr (\lambda (b0: 
-B).(pr2 c (THead (Bind b0) v t1) (THead (Bind b0) v t2))) (pr2_free c (THead 
-(Bind Abbr) v t1) (THead (Bind Abbr) v t2) (pr0_delta v v (pr0_refl v) t1 t3 
-H9 t2 H21)) b H18)))))) H16)) H15)))) H12)) (\lambda (H12: (ex2 nat (\lambda 
-(j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) 
-u))))).(ex2_ind nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: 
-nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c (THead (Bind b) v t1) (THead 
-(Bind b) v t2)) (\lambda (x: nat).(\lambda (H13: (eq nat i (S x))).(\lambda 
-(H14: (getl x c (CHead d (Bind Abbr) u))).(let H15 \def (f_equal nat nat 
-(\lambda (e: nat).e) i (S x) H13) in (let H16 \def (eq_ind nat i (\lambda (n: 
-nat).(subst0 n u t3 t2)) H10 (S x) H15) in (pr2_head_2 c v t1 t2 (Bind b) 
-(pr2_delta (CHead c (Bind b) v) d u (S x) (getl_clear_bind b (CHead c (Bind 
-b) v) c v (clear_bind b c v) (CHead d (Bind Abbr) u) x H14) t1 t3 H9 t2 
-H16))))))) H12)) H11)))))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 
-(sym_eq C c0 (CHead c (Bind b) v) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal 
-C (CHead c (Bind b) v)) (refl_equal T t1) (refl_equal T t2)))))))).
+(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(insert_eq C (CHead c 
+(Bind b) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead 
+(Bind b) v t1) (THead (Bind b) v t2))) (\lambda (y: C).(\lambda (H0: (pr2 y 
+t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 
+(CHead c (Bind b) v)) \to (pr2 c (THead (Bind b) v t) (THead (Bind b) v 
+t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
+(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) v))).(pr2_free c (THead 
+(Bind b) v t3) (THead (Bind b) v t4) (pr0_comp v v (pr0_refl v) t3 t4 H1 
+(Bind b)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
+nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: 
+(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) v))).(let H5 \def 
+(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead 
+c (Bind b) v) H4) in (let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u) 
+v i H5) in (let H6 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d 
+(Bind Abbr) u) (CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S 
+j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead 
+(Bind b) v t3) (THead (Bind b) v t)) (\lambda (H7: (land (eq nat i O) (eq C 
+(CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(and_ind (eq nat i O) (eq C 
+(CHead d (Bind Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t3) 
+(THead (Bind b) v t)) (\lambda (H8: (eq nat i O)).(\lambda (H9: (eq C (CHead 
+d (Bind Abbr) u) (CHead c (Bind b) v))).(let H10 \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) 
+v) H9) in ((let H11 \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) v) H9) in ((let H12 \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) v) 
+H9) in (\lambda (H13: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H15 \def 
+(eq_ind nat i (\lambda (n: nat).(subst0 n u t4 t)) H3 O H8) in (let H16 \def 
+(eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H15 v H12) in (eq_ind B Abbr 
+(\lambda (b0: B).(pr2 c (THead (Bind b0) v t3) (THead (Bind b0) v t))) 
+(pr2_free c (THead (Bind Abbr) v t3) (THead (Bind Abbr) v t) (pr0_delta v v 
+(pr0_refl v) t3 t4 H2 t H16)) b H13)))))) H11)) H10)))) H7)) (\lambda (H7: 
+(ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c 
+(CHead d (Bind Abbr) u))))).(ex2_ind nat (\lambda (j: nat).(eq nat i (S j))) 
+(\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c (THead (Bind b) 
+v t3) (THead (Bind b) v t)) (\lambda (x: nat).(\lambda (H8: (eq nat i (S 
+x))).(\lambda (H9: (getl x c (CHead d (Bind Abbr) u))).(let H10 \def (f_equal 
+nat nat (\lambda (e: nat).e) i (S x) H8) in (let H11 \def (eq_ind nat i 
+(\lambda (n: nat).(subst0 n u t4 t)) H3 (S x) H10) in (pr2_head_2 c v t3 t 
+(Bind b) (pr2_delta (CHead c (Bind b) v) d u (S x) (getl_clear_bind b (CHead 
+c (Bind b) v) c v (clear_bind b c v) (CHead d (Bind Abbr) u) x H9) t3 t4 H2 t 
+H11))))))) H7)) H6))))))))))))))) y t1 t2 H0))) H)))))).
 
 theorem pr2_gen_cflat:
  \forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall 
 (t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2))))))
 \def
  \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(let H0 \def (match H 
-in pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: 
-(pr2 c0 t t0)).((eq C c0 (CHead c (Flat f) v)) \to ((eq T t t1) \to ((eq T t0 
-t2) \to (pr2 c t1 t2)))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow 
-(\lambda (H1: (eq C c0 (CHead c (Flat f) v))).(\lambda (H2: (eq T t0 
-t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Flat f) v) (\lambda (_: 
-C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c t1 t2))))) 
-(\lambda (H4: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to 
-((pr0 t t3) \to (pr2 c t1 t2)))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 
-(\lambda (t: T).((pr0 t1 t) \to (pr2 c t1 t2))) (\lambda (H6: (pr0 t1 
-t2)).(pr2_free c t1 t2 H6)) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) 
-c0 (sym_eq C c0 (CHead c (Flat f) v) H1) H2 H3 H0)))) | (pr2_delta c0 d u i 
-H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 (CHead c (Flat f) 
-v))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead 
-c (Flat f) v) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 
-(CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c 
-t1 t2))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T 
-t t2) \to ((getl i (CHead c (Flat f) v) (CHead d (Bind Abbr) u)) \to ((pr0 t4 
-t3) \to ((subst0 i u t3 t) \to (pr2 c t1 t2)))))) (\lambda (H7: (eq T t 
-t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c (Flat f) v) (CHead d 
-(Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c t1 
-t2))))) (\lambda (H8: (getl i (CHead c (Flat f) v) (CHead d (Bind Abbr) 
-u))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u t3 t2)).(let H_y 
-\def (getl_gen_flat f c (CHead d (Bind Abbr) u) v i H8) in (pr2_delta c d u i 
-H_y t1 t3 H9 t2 H10))))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 
-(sym_eq C c0 (CHead c (Flat f) v) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal 
-C (CHead c (Flat f) v)) (refl_equal T t1) (refl_equal T t2)))))))).
+(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(insert_eq C (CHead c 
+(Flat f) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c t1 t2)) 
+(\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: 
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) v)) \to (pr2 
+c t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
+(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Flat f) v))).(pr2_free c t3 t4 
+H1)))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
+nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: 
+(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Flat f) v))).(let H5 \def 
+(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead 
+c (Flat f) v) H4) in (let H_y \def (getl_gen_flat f c (CHead d (Bind Abbr) u) 
+v i H5) in (pr2_delta c d u i H_y t3 t4 H2 t H3)))))))))))))) y t1 t2 H0))) 
+H)))))).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/defs.ma
index d712689b2..0db7dcaea 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/defs.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr0/defs.ma".
+include "LambdaDelta-1/pr0/defs.ma".
 
-include "getl/defs.ma".
+include "LambdaDelta-1/getl/defs.ma".
 
 inductive pr2: C \to (T \to (T \to Prop)) \def
 | pr2_free: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/fwd.ma
index b2f5dff45..c332a5ffb 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/fwd.ma
@@ -14,48 +14,35 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr2/defs.ma".
+include "LambdaDelta-1/pr2/defs.ma".
 
-include "pr0/fwd.ma".
+include "LambdaDelta-1/pr0/fwd.ma".
 
-include "getl/drop.ma".
+include "LambdaDelta-1/getl/drop.ma".
 
-include "getl/clear.ma".
+include "LambdaDelta-1/getl/clear.ma".
 
 theorem pr2_gen_sort:
  \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to 
 (eq T x (TSort n)))))
 \def
  \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TSort 
-n) x)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t 
-(TSort n)) \to ((eq T t0 x) \to (eq T x (TSort n))))))))) with [(pr2_free c0 
-t1 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1 
-(TSort n))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t1 
-(TSort n)) \to ((eq T t2 x) \to ((pr0 t1 t2) \to (eq T x (TSort n)))))) 
-(\lambda (H4: (eq T t1 (TSort n))).(eq_ind T (TSort n) (\lambda (t: T).((eq T 
-t2 x) \to ((pr0 t t2) \to (eq T x (TSort n))))) (\lambda (H5: (eq T t2 
-x)).(eq_ind T x (\lambda (t: T).((pr0 (TSort n) t) \to (eq T x (TSort n)))) 
-(\lambda (H6: (pr0 (TSort n) x)).(let H7 \def (eq_ind T x (\lambda (t: 
-T).(pr2 c (TSort n) t)) H (TSort n) (pr0_gen_sort x n H6)) in (eq_ind_r T 
-(TSort n) (\lambda (t: T).(eq T t (TSort n))) (refl_equal T (TSort n)) x 
-(pr0_gen_sort x n H6)))) t2 (sym_eq T t2 x H5))) t1 (sym_eq T t1 (TSort n) 
-H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t1 t2 H1 t 
-H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t1 (TSort 
-n))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t1 (TSort 
-n)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1 t2) 
-\to ((subst0 i u t2 t) \to (eq T x (TSort n)))))))) (\lambda (H6: (eq T t1 
-(TSort n))).(eq_ind T (TSort n) (\lambda (t0: T).((eq T t x) \to ((getl i c 
-(CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (eq T x 
-(TSort n))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t0: T).((getl 
-i c (CHead d (Bind Abbr) u)) \to ((pr0 (TSort n) t2) \to ((subst0 i u t2 t0) 
-\to (eq T x (TSort n)))))) (\lambda (_: (getl i c (CHead d (Bind Abbr) 
-u))).(\lambda (H9: (pr0 (TSort n) t2)).(\lambda (H10: (subst0 i u t2 x)).(let 
-H11 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 x)) H10 (TSort n) 
-(pr0_gen_sort t2 n H9)) in (subst0_gen_sort u x i n H11 (eq T x (TSort 
-n))))))) t (sym_eq T t x H7))) t1 (sym_eq T t1 (TSort n) H6))) c0 (sym_eq C 
-c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T (TSort 
-n)) (refl_equal T x)))))).
+n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr2 c t x)) (\lambda (t: 
+T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda 
+(_: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 
+t))))) (\lambda (_: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 
+t1 t2)).(\lambda (H2: (eq T t1 (TSort n))).(let H3 \def (eq_ind T t1 (\lambda 
+(t: T).(pr0 t t2)) H1 (TSort n) H2) in (eq_ind_r T (TSort n) (\lambda (t: 
+T).(eq T t2 t)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T t (TSort n))) 
+(refl_equal T (TSort n)) t2 (pr0_gen_sort t2 n H3)) t1 H2))))))) (\lambda 
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl 
+i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H2: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda 
+(H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0 
+t2)) H2 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t t0)) 
+(let H6 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 t)) H3 (TSort n) 
+(pr0_gen_sort t2 n H5)) in (subst0_gen_sort u t i n H6 (eq T t (TSort n)))) 
+t1 H4))))))))))))) c y x H0))) H)))).
 
 theorem pr2_gen_lref:
  \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TLRef n) x) \to 
@@ -64,68 +51,48 @@ theorem pr2_gen_lref:
 n) O u)))))))))
 \def
  \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TLRef 
-n) x)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t 
-(TLRef n)) \to ((eq T t0 x) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda 
-(d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: 
-C).(\lambda (u: T).(eq T x (lift (S n) O u))))))))))))) with [(pr2_free c0 t1 
-t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1 (TLRef 
-n))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t1 (TLRef 
-n)) \to ((eq T t2 x) \to ((pr0 t1 t2) \to (or (eq T x (TLRef n)) (ex2_2 C T 
-(\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda 
-(_: C).(\lambda (u: T).(eq T x (lift (S n) O u)))))))))) (\lambda (H4: (eq T 
-t1 (TLRef n))).(eq_ind T (TLRef n) (\lambda (t: T).((eq T t2 x) \to ((pr0 t 
-t2) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: 
-T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T 
-x (lift (S n) O u))))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda 
-(t: T).((pr0 (TLRef n) t) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: 
-C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: 
-C).(\lambda (u: T).(eq T x (lift (S n) O u)))))))) (\lambda (H6: (pr0 (TLRef 
-n) x)).(let H7 \def (eq_ind T x (\lambda (t: T).(pr2 c (TLRef n) t)) H (TLRef 
-n) (pr0_gen_lref x n H6)) in (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T 
-t (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d 
-(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t (lift (S n) O 
-u))))))) (or_introl (eq T (TLRef n) (TLRef n)) (ex2_2 C T (\lambda (d: 
-C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: 
-C).(\lambda (u: T).(eq T (TLRef n) (lift (S n) O u))))) (refl_equal T (TLRef 
-n))) x (pr0_gen_lref x n H6)))) t2 (sym_eq T t2 x H5))) t1 (sym_eq T t1 
-(TLRef n) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 
-t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t1 
-(TLRef n))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t1 
-(TLRef n)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 
-t1 t2) \to ((subst0 i u t2 t) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda 
-(d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: 
-C).(\lambda (u0: T).(eq T x (lift (S n) O u0)))))))))))) (\lambda (H6: (eq T 
-t1 (TLRef n))).(eq_ind T (TLRef n) (\lambda (t0: T).((eq T t x) \to ((getl i 
-c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (or 
-(eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c 
-(CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T x (lift 
-(S n) O u0))))))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t0: 
-T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (TLRef n) t2) \to ((subst0 i 
-u t2 t0) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: 
-T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: 
-T).(eq T x (lift (S n) O u0)))))))))) (\lambda (H8: (getl i c (CHead d (Bind 
-Abbr) u))).(\lambda (H9: (pr0 (TLRef n) t2)).(\lambda (H10: (subst0 i u t2 
-x)).(let H11 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 x)) H10 (TLRef 
-n) (pr0_gen_lref t2 n H9)) in (and_ind (eq nat n i) (eq T x (lift (S n) O u)) 
-(or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c 
-(CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T x (lift 
-(S n) O u0)))))) (\lambda (H12: (eq nat n i)).(\lambda (H13: (eq T x (lift (S 
-n) O u))).(let H14 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead 
-d (Bind Abbr) u))) H8 n H12) in (let H15 \def (eq_ind T x (\lambda (t0: 
-T).(pr2 c (TLRef n) t0)) H (lift (S n) O u) H13) in (eq_ind_r T (lift (S n) O 
-u) (\lambda (t0: T).(or (eq T t0 (TLRef n)) (ex2_2 C T (\lambda (d0: 
-C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: 
-C).(\lambda (u0: T).(eq T t0 (lift (S n) O u0))))))) (or_intror (eq T (lift 
-(S n) O u) (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c 
-(CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S 
-n) O u) (lift (S n) O u0))))) (ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: 
-T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: 
-T).(eq T (lift (S n) O u) (lift (S n) O u0)))) d u H14 (refl_equal T (lift (S 
-n) O u)))) x H13))))) (subst0_gen_lref u x i n H11)))))) t (sym_eq T t x 
-H7))) t1 (sym_eq T t1 (TLRef n) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 
-H2))))]) in (H0 (refl_equal C c) (refl_equal T (TLRef n)) (refl_equal T 
-x)))))).
+n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr2 c t x)) (\lambda (t: 
+T).(or (eq T x t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead 
+d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O 
+u))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: 
+C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (eq T t0 t) 
+(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr) 
+u)))) (\lambda (_: C).(\lambda (u: T).(eq T t0 (lift (S n) O u)))))))))) 
+(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 
+t2)).(\lambda (H2: (eq T t1 (TLRef n))).(let H3 \def (eq_ind T t1 (\lambda 
+(t: T).(pr0 t t2)) H1 (TLRef n) H2) in (eq_ind_r T (TLRef n) (\lambda (t: 
+T).(or (eq T t2 t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 
+(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S 
+n) O u))))))) (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t (TLRef n)) 
+(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr) 
+u)))) (\lambda (_: C).(\lambda (u: T).(eq T t (lift (S n) O u))))))) 
+(or_introl (eq T (TLRef n) (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: 
+T).(getl n c0 (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq 
+T (TLRef n) (lift (S n) O u))))) (refl_equal T (TLRef n))) t2 (pr0_gen_lref 
+t2 n H3)) t1 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) 
+u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H2: (pr0 t1 t2)).(\lambda 
+(t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t1 (TLRef 
+n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0 t2)) H2 (TLRef n) H4) 
+in (eq_ind_r T (TLRef n) (\lambda (t0: T).(or (eq T t t0) (ex2_2 C T (\lambda 
+(d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (_: 
+C).(\lambda (u0: T).(eq T t (lift (S n) O u0))))))) (let H6 \def (eq_ind T t2 
+(\lambda (t0: T).(subst0 i u t0 t)) H3 (TLRef n) (pr0_gen_lref t2 n H5)) in 
+(and_ind (eq nat n i) (eq T t (lift (S n) O u)) (or (eq T t (TLRef n)) (ex2_2 
+C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) 
+(\lambda (_: C).(\lambda (u0: T).(eq T t (lift (S n) O u0)))))) (\lambda (H7: 
+(eq nat n i)).(\lambda (H8: (eq T t (lift (S n) O u))).(eq_ind_r T (lift (S 
+n) O u) (\lambda (t0: T).(or (eq T t0 (TLRef n)) (ex2_2 C T (\lambda (d0: 
+C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (_: 
+C).(\lambda (u0: T).(eq T t0 (lift (S n) O u0))))))) (let H9 \def (eq_ind_r 
+nat i (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind Abbr) u))) H1 n H7) in 
+(or_intror (eq T (lift (S n) O u) (TLRef n)) (ex2_2 C T (\lambda (d0: 
+C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (_: 
+C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0))))) (ex2_2_intro 
+C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) 
+(\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0)))) 
+d u H9 (refl_equal T (lift (S n) O u))))) t H8))) (subst0_gen_lref u t i n 
+H6))) t1 H4))))))))))))) c y x H0))) H)))).
 
 theorem pr2_gen_abst:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
@@ -135,154 +102,119 @@ c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u:
 T).(pr2 (CHead c (Bind b) u) t1 t2))))))))))
 \def
  \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr2 c (THead (Bind Abst) u1 t1) x)).(let H0 \def (match H in pr2 return 
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t 
-t0)).((eq C c0 c) \to ((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).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-t1 t2))))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq 
-C c0 c)).(\lambda (H2: (eq T t0 (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq 
-T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Abst) u1 t1)) \to 
-((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).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) t1 t3)))))))))) (\lambda (H4: (eq T t0 (THead 
-(Bind Abst) u1 t1))).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t: T).((eq 
-T t2 x) \to ((pr0 t 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).(pr2 c u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 
-(CHead c (Bind b) u) t1 t3))))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x 
-(\lambda (t: T).((pr0 (THead (Bind Abst) u1 t1) 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).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))))) (\lambda (H6: 
-(pr0 (THead (Bind Abst) u1 t1) x)).(ex3_2_ind 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))) (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind Abst) x0 
-x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def 
-(eq_ind T x (\lambda (t: T).(pr2 c (THead (Bind Abst) u1 t1) t)) H (THead 
-(Bind Abst) x0 x1) H7) in (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t: 
-T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-t1 t3))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead 
-(Bind Abst) x0 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind 
-Abst) x0 x1)) (pr2_free c u1 x0 H8) (\lambda (b: B).(\lambda (u: T).(pr2_free 
-(CHead c (Bind b) u) t1 x1 H9)))) x H7))))))) (pr0_gen_abst u1 t1 x H6))) t2 
-(sym_eq T t2 x H5))) t0 (sym_eq T t0 (THead (Bind Abst) u1 t1) H4))) c0 
-(sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) 
-\Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Bind 
-Abst) u1 t1))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T 
-t0 (THead (Bind Abst) u1 t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind 
-Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 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).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))))))))) (\lambda 
-(H6: (eq T t0 (THead (Bind Abst) u1 t1))).(eq_ind T (THead (Bind Abst) u1 t1) 
-(\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to 
-((pr0 t3 t2) \to ((subst0 i u t2 t) \to (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t4: T).(eq T x (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))))))))))) (\lambda (H7: (eq T t 
-x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to 
-((pr0 (THead (Bind Abst) u1 t1) t2) \to ((subst0 i u t2 t3) \to (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Abst) u2 t4)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 
-t4)))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: 
-(pr0 (THead (Bind Abst) u1 t1) t2)).(\lambda (H10: (subst0 i u t2 
-x)).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+(H: (pr2 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 
+t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y: 
+T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda (t: 
+T).(\lambda (t0: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 
+t2)))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: 
+(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abst) u1 t1))).(let H3 \def 
+(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abst) u1 t1) H2) in 
+(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
 T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: 
-T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H12: (pr0 u1 
-x0)).(\lambda (H13: (pr0 t1 x1)).(let H14 \def (eq_ind T t2 (\lambda (t3: 
-T).(subst0 i u t3 x)) H10 (THead (Bind Abst) x0 x1) H11) in (or3_ind (ex2 T 
-(\lambda (u2: T).(eq T x (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 
-i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind Abst) x0 t3))) 
-(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3))) (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Bind Abst) i) u x1 t3)))) (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (H15: (ex2 T (\lambda 
-(u2: T).(eq T x (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 x1))) 
-(\lambda (u2: T).(subst0 i u x0 u2)) (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: T).(\lambda 
-(H16: (eq T x (THead (Bind Abst) x2 x1))).(\lambda (H17: (subst0 i u x0 
-x2)).(let H18 \def (eq_ind T x (\lambda (t3: T).(pr2 c (THead (Bind Abst) u1 
-t1) t3)) H (THead (Bind Abst) x2 x1) H16) in (eq_ind_r T (THead (Bind Abst) 
-x2 x1) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 
-(THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
+T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c0 (Bind b) u) t1 t3)))))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H4: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr0 u1 
+x0)).(\lambda (H6: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) 
+(\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
+(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead 
+c0 (Bind b) u) t1 t3))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))))) x0 x1 
+(refl_equal T (THead (Bind Abst) x0 x1)) (pr2_free c0 u1 x0 H5) (\lambda (b: 
+B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1 H6)))) t2 H4)))))) 
+(pr0_gen_abst u1 t1 t2 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda 
+(u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) 
+u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda 
+(t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Bind 
+Abst) u1 t1))).(let H5 \def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 
+(THead (Bind Abst) u1 t1) H4) in (ex3_2_ind T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
+t3)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t2 (THead 
+(Bind Abst) x0 x1))).(\lambda (H7: (pr0 u1 x0)).(\lambda (H8: (pr0 t1 
+x1)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead 
+(Bind Abst) x0 x1) H6) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead 
+(Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda 
+(t3: T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind 
+Abst) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
+(THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)))) 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) t1 t3)))))) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind 
+Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda 
+(u2: T).(eq T t (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 
+u2)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) t1 t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x2 
+x1))).(\lambda (H12: (subst0 i u x0 x2)).(eq_ind_r T (THead (Bind Abst) x2 
+x1) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 
+(THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T (THead (Bind Abst) x2 x1) (THead (Bind Abst) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) t1 t3))))) x2 x1 (refl_equal T (THead (Bind Abst) x2 x1)) (pr2_delta c0 d 
+u i H1 u1 x0 H7 x2 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 
+(Bind b) u0) t1 x1 H8)))) t H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: 
+T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind 
+Abst) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abst) 
+x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)) (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
+t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x0 
+x2))).(\lambda (H12: (subst0 (s (Bind Abst) i) u x1 x2)).(eq_ind_r T (THead 
+(Bind Abst) x0 x2) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: 
+T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x0 x2) (THead (Bind Abst) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) t1 t3))))) x0 x2 (refl_equal T (THead (Bind Abst) x0 x2)) (pr2_free c0 u1 
+x0 H7) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u 
+(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x2 
+H12)))) t H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind 
+Abst) i) u x1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u 
+x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 
+t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) t1 t3)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T t 
+(THead (Bind Abst) x2 x3))).(\lambda (H12: (subst0 i u x0 x2)).(\lambda (H13: 
+(subst0 (s (Bind Abst) i) u x1 x3)).(eq_ind_r T (THead (Bind Abst) x2 x3) 
+(\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
+(Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
 (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead 
-c (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind Abst) x2 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x2 x1 
-(refl_equal T (THead (Bind Abst) x2 x1)) (pr2_delta c d u i H8 u1 x0 H12 x2 
-H17) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) t1 x1 
-H13)))) x H16))))) H15)) (\lambda (H15: (ex2 T (\lambda (t3: T).(eq T x 
-(THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind Abst) x0 t3))) 
-(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)) (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: 
-T).(\lambda (H16: (eq T x (THead (Bind Abst) x0 x2))).(\lambda (H17: (subst0 
-(s (Bind Abst) i) u x1 x2)).(let H18 \def (eq_ind T x (\lambda (t3: T).(pr2 c 
-(THead (Bind Abst) u1 t1) t3)) H (THead (Bind Abst) x0 x2) H16) in (eq_ind_r 
-T (THead (Bind Abst) x0 x2) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: 
-T).(\lambda (t4: T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))))))) (ex3_2_intro 
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x0 x2) (THead 
-(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead 
-c (Bind b) u0) t1 t3))))) x0 x2 (refl_equal T (THead (Bind Abst) x0 x2)) 
-(pr2_free c u1 x0 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c 
-(Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) 
-t1 x1 H13 x2 H17)))) x H16))))) H15)) (\lambda (H15: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Bind Abst) i) u x1 t3))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Bind Abst) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H16: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (H17: (subst0 
-i u x0 x2)).(\lambda (H18: (subst0 (s (Bind Abst) i) u x1 x3)).(let H19 \def 
-(eq_ind T x (\lambda (t3: T).(pr2 c (THead (Bind Abst) u1 t1) t3)) H (THead 
-(Bind Abst) x2 x3) H16) in (eq_ind_r T (THead (Bind Abst) x2 x3) (\lambda 
-(t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
-Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
-u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-(THead (Bind Abst) x2 x3) (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x2 x3 
-(refl_equal T (THead (Bind Abst) x2 x3)) (pr2_delta c d u i H8 u1 x0 H12 x2 
-H17) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S 
-i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H13 x3 
-H18)))) x H16))))))) H15)) (subst0_gen_head (Bind Abst) u x0 x1 x i 
-H14)))))))) (pr0_gen_abst u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T 
-t0 (THead (Bind Abst) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) 
-in (H0 (refl_equal C c) (refl_equal T (THead (Bind Abst) u1 t1)) (refl_equal 
-T x))))))).
+c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind Abst) x2 x3) (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))))) x2 x3 
+(refl_equal T (THead (Bind Abst) x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H7 x2 
+H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u 
+(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x3 
+H13)))) t H11)))))) H10)) (subst0_gen_head (Bind Abst) u x0 x1 t i H9)))))))) 
+(pr0_gen_abst u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
 
 theorem pr2_gen_cast:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
@@ -292,185 +224,146 @@ T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c
 t1 x))))))
 \def
  \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr2 c (THead (Flat Cast) u1 t1) x)).(let H0 \def (match H in pr2 return 
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t 
-t0)).((eq C c0 c) \to ((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).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c t1 x))))))))) with [(pr2_free c0 
-t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0 
-(THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda 
-(_: C).((eq T t0 (THead (Flat Cast) u1 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).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)))))) (\lambda (H4: (eq T t0 
-(THead (Flat Cast) u1 t1))).(eq_ind T (THead (Flat Cast) u1 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).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c 
-t1 x))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead 
-(Flat Cast) u1 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).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)))) 
-(\lambda (H6: (pr0 (THead (Flat Cast) u1 t1) x)).(or_ind (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) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x 
-(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (H7: 
-(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))))).(ex3_2_ind 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))) (or (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H8: (eq T x (THead (Flat Cast) x0 x1))).(\lambda (H9: (pr0 u1 
-x0)).(\lambda (H10: (pr0 t1 x1)).(let H11 \def (eq_ind T x (\lambda (t: 
-T).(pr2 c (THead (Flat Cast) u1 t1) t)) H (THead (Flat Cast) x0 x1) H8) in 
-(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 
-t3)))) (pr2 c t1 t))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 
-t3)))) (pr2 c t1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x1 (refl_equal T (THead (Flat Cast) x0 
-x1)) (pr2_free c u1 x0 H9) (pr2_free c t1 x1 H10))) x H8))))))) H7)) (\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).(pr2 c u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) 
-(pr2_free c t1 x H7))) (pr0_gen_cast u1 t1 x H6))) t2 (sym_eq T t2 x H5))) t0 
-(sym_eq T t0 (THead (Flat Cast) u1 t1) H4))) c0 (sym_eq C c0 c H1) H2 H3 
-H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq 
-C c0 c)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(\lambda (H5: (eq 
-T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Flat Cast) u1 t1)) \to 
-((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to 
-((subst0 i u t2 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).(pr2 c u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)))))))) 
-(\lambda (H6: (eq T t0 (THead (Flat Cast) u1 t1))).(eq_ind T (THead (Flat 
-Cast) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) 
-u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t4: T).(eq T x (THead (Flat Cast) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 
-t4)))) (pr2 c t1 x))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t3: 
-T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (THead (Flat Cast) u1 t1) 
-t2) \to ((subst0 i u t2 t3) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4: 
-T).(eq T x (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 t4)))) (pr2 c t1 
-x)))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 
-(THead (Flat Cast) u1 t1) t2)).(\lambda (H10: (subst0 i u t2 x)).(or_ind 
+(H: (pr2 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 
+t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 
+t2)))) (pr2 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind 
+(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Cast) 
+u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead 
+(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (pr2 c0 t1 t0)))))) 
+(\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 
+t2)).(\lambda (H2: (eq T t0 (THead (Flat Cast) u1 t1))).(let H3 \def (eq_ind 
+T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Cast) u1 t1) H2) in (or_ind 
 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 
 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
 T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 
-t3)))) (pr2 c t1 x)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3)))) (pr2 c0 t1 t2)) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
 T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind 
 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) 
 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x 
-(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Flat Cast) x0 
-x1))).(\lambda (H13: (pr0 u1 x0)).(\lambda (H14: (pr0 t1 x1)).(let H15 \def 
-(eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 x)) H10 (THead (Flat Cast) x0 
-x1) H12) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Flat Cast) u2 
-x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x 
-(THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 
-t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
+T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t2)) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead (Flat Cast) 
+x0 x1))).(\lambda (H6: (pr0 u1 x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T 
+(THead (Flat Cast) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3)))) (pr2 c0 t1 t))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t3)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(pr2 c0 t1 t3)))) (pr2 c0 t1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead 
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x1 (refl_equal T (THead 
+(Flat Cast) x0 x1)) (pr2_free c0 u1 x0 H6) (pr2_free c0 t1 x1 H7))) t2 
+H5)))))) H4)) (\lambda (H4: (pr0 t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3)))) (pr2 c0 t1 t2) (pr2_free c0 t1 t2 H4))) (pr0_gen_cast u1 t1 t2 
+H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
+nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: 
+T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: 
+(subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(let H5 
+\def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Cast) u1 t1) 
+H4) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3)))) (pr2 c0 t1 t)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind 
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T t2 (THead (Flat Cast) 
+x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def 
+(eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Flat Cast) x0 x1) 
+H7) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1))) 
+(\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead 
+(Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
 T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))) (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (H16: (ex2 T (\lambda (u2: 
-T).(eq T x (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0 
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Flat Cast) u2 x1))) 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (H11: (ex2 T (\lambda (u2: 
+T).(eq T t (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0 
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1))) 
 (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c 
-t1 x)) (\lambda (x2: T).(\lambda (H17: (eq T x (THead (Flat Cast) x2 
-x1))).(\lambda (H18: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 
-t3)))) (pr2 c t1 x) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x2 x1 H17 (pr2_delta c 
-d u i H8 u1 x0 H13 x2 H18) (pr2_free c t1 x1 H14)))))) H16)) (\lambda (H16: 
-(ex2 T (\lambda (t3: T).(eq T x (THead (Flat Cast) x0 t3))) (\lambda (t3: 
-T).(subst0 (s (Flat Cast) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x 
-(THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 
-t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat 
-Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x2: T).(\lambda 
-(H17: (eq T x (THead (Flat Cast) x0 x2))).(\lambda (H18: (subst0 (s (Flat 
-Cast) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) 
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x2 H17 (pr2_free c u1 x0 H13) 
-(pr2_delta c d u i H8 t1 x1 H14 x2 H18)))))) H16)) (\lambda (H16: (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H17: (eq T x (THead (Flat Cast) x2 x3))).(\lambda (H18: (subst0 
-i u x0 x2)).(\lambda (H19: (subst0 (s (Flat Cast) i) u x1 x3)).(or_introl 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) (ex3_2_intro T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 
-t3))) x2 x3 H17 (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (pr2_delta c d u i H8 
-t1 x1 H14 x3 H19)))))))) H16)) (subst0_gen_head (Flat Cast) u x0 x1 x i 
-H15)))))))) H11)) (\lambda (H11: (pr0 t1 t2)).(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).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 
-t3)))) (pr2 c t1 x) (pr2_delta c d u i H8 t1 t2 H11 x H10))) (pr0_gen_cast u1 
-t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Flat Cast) u1 t1) 
-H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) 
-(refl_equal T (THead (Flat Cast) u1 t1)) (refl_equal T x))))))).
+(t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 
+c0 t1 t)) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2 
+x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x1 H12 
+(pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9)))))) H11)) 
+(\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Cast) x0 t3))) 
+(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))).(ex2_ind T (\lambda 
+(t3: T).(eq T t (THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat 
+Cast) i) u x1 t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) 
+(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x0 x2))).(\lambda 
+(H13: (subst0 (s (Flat Cast) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 H12 
+(pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13)))))) H11)) 
+(\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat 
+Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2 
+x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat 
+Cast) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t) 
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 
+H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1 H9 x3 H14)))))))) H11)) 
+(subst0_gen_head (Flat Cast) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0 
+t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t) 
+(pr2_delta c0 d u i H1 t1 t2 H6 t H3))) (pr0_gen_cast u1 t1 t2 
+H5)))))))))))))) c y x H0))) H))))).
 
 theorem pr2_gen_csort:
  \forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2) 
 \to (pr0 t1 t2))))
 \def
  \lambda (t1: T).(\lambda (t2: T).(\lambda (n: nat).(\lambda (H: (pr2 (CSort 
-n) t1 t2)).(let H0 \def (match H in pr2 return (\lambda (c: C).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (_: (pr2 c t t0)).((eq C c (CSort n)) \to ((eq T 
-t t1) \to ((eq T t0 t2) \to (pr0 t1 t2)))))))) with [(pr2_free c t0 t3 H0) 
-\Rightarrow (\lambda (H1: (eq C c (CSort n))).(\lambda (H2: (eq T t0 
-t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CSort n) (\lambda (_: C).((eq T 
-t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr0 t1 t2))))) (\lambda (H4: 
-(eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to 
-(pr0 t1 t2)))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 
-t1 t) \to (pr0 t1 t2))) (\lambda (H6: (pr0 t1 t2)).H6) t3 (sym_eq T t3 t2 
-H5))) t0 (sym_eq T t0 t1 H4))) c (sym_eq C c (CSort n) H1) H2 H3 H0)))) | 
-(pr2_delta c d u i H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c (CSort 
-n))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CSort 
-n) (\lambda (c0: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c0 (CHead d 
-(Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr0 t1 t2))))))) 
-(\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to 
-((getl i (CSort n) (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u 
-t3 t) \to (pr0 t1 t2)))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda 
-(t4: T).((getl i (CSort n) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to 
-((subst0 i u t3 t4) \to (pr0 t1 t2))))) (\lambda (H8: (getl i (CSort n) 
-(CHead d (Bind Abbr) u))).(\lambda (_: (pr0 t1 t3)).(\lambda (_: (subst0 i u 
-t3 t2)).(getl_gen_sort n i (CHead d (Bind Abbr) u) H8 (pr0 t1 t2))))) t 
-(sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c (sym_eq C c (CSort n) H3) H4 
-H5 H0 H1 H2))))]) in (H0 (refl_equal C (CSort n)) (refl_equal T t1) 
-(refl_equal T t2)))))).
+n) t1 t2)).(insert_eq C (CSort n) (\lambda (c: C).(pr2 c t1 t2)) (\lambda (_: 
+C).(pr0 t1 t2)) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind 
+(\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CSort n)) \to (pr0 
+t t0))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
+(pr0 t3 t4)).(\lambda (_: (eq C c (CSort n))).H1))))) (\lambda (c: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c 
+(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 
+t3 t4)).(\lambda (t: T).(\lambda (_: (subst0 i u t4 t)).(\lambda (H4: (eq C c 
+(CSort n))).(let H5 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead d 
+(Bind Abbr) u))) H1 (CSort n) H4) in (getl_gen_sort n i (CHead d (Bind Abbr) 
+u) H5 (pr0 t3 t)))))))))))))) y t1 t2 H0))) H)))).
 
 theorem pr2_gen_appl:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
@@ -496,147 +389,17 @@ T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))))
 \def
  \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr2 c (THead (Flat Appl) u1 t1) x)).(let H0 \def (match H in pr2 return 
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t 
-t0)).((eq C c0 c) \to ((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).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr2 c 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).(pr2 c u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
-(u: T).(pr2 (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).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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)))))))))))))))) with [(pr2_free c0 
-t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0 
-(THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda 
-(_: C).((eq T t0 (THead (Flat Appl) u1 t1)) \to ((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).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c 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).(pr2 c u1 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 t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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))))))))))))) (\lambda (H4: (eq T t0 
-(THead (Flat Appl) u1 t1))).(eq_ind T (THead (Flat Appl) u1 t1) (\lambda (t: 
-T).((eq T t2 x) \to ((pr0 t 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).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 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).(pr2 c u1 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 t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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)))))))))))) (\lambda 
-(H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead (Flat Appl) u1 t1) 
-t) \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).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c 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).(pr2 c u1 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 t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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))))))))))) (\lambda (H6: (pr0 (THead 
-(Flat Appl) u1 t1) x)).(or3_ind (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)))))))) (or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c 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).(pr2 c u1 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 t1 (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 
-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))))))))) (\lambda (H7: (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))))).(ex3_2_ind 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))) (or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+(H: (pr2 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 
+t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c 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 (t3: T).(eq T x (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\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).(pr2 c u1 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 (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (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).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
@@ -647,262 +410,352 @@ T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq T x 
-(THead (Flat Appl) x0 x1))).(\lambda (H9: (pr0 u1 x0)).(\lambda (H10: (pr0 t1 
-x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c 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 t 
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 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 t1 (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (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 u1 
-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)))))))))) (or3_intro0 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda 
+y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind 
+(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl) 
+u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead 
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr2 c0 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 (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 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 t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T (THead (Flat Appl) x0 x1) (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 u1 u2))))))) 
+T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c0 (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).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+t0 (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 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
+z2))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda 
+(H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Flat Appl) u1 t1))).(let H3 
+\def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Appl) u1 t1) 
+H2) 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).(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 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 t1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (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)))))))) (or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+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 t2 (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (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 t1 (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 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+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).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+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 t2 (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (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 t1 (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 c0 u1 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 (y2: T).(pr2 c0 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)))))))) (ex3_2_intro T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x1 (refl_equal T (THead (Flat Appl) x0 
-x1)) (pr2_free c u1 x0 H9) (pr2_free c t1 x1 H10))) x H8)))))) H7)) (\lambda 
-(H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H5: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H6: (pr0 u1 
+x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) 
+(\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
+(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 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 t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c0 (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 t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t 
+(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 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
+z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+(THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+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 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) 
+x0 x1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c0 (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 t1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Flat 
+Appl) x0 x1) (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 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+(THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3))) x0 x1 (refl_equal T (THead (Flat Appl) x0 x1)) (pr2_free c0 u1 x0 
+H6) (pr2_free c0 t1 x1 H7))) t2 H5)))))) H4)) (\lambda (H4: (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 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))))))).(ex4_4_ind 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 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))))))).(ex4_4_ind 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))))) (or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 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 t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c0 (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 t1 (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 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
+z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H5: (eq T t1 (THead (Bind Abst) x0 x1))).(\lambda (H6: (eq T t2 
+(THead (Bind Abbr) x2 x3))).(\lambda (H7: (pr0 u1 x2)).(\lambda (H8: (pr0 x1 
+x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 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 
+(\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).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(pr2 c0 u1 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: 
+(CHead c0 (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 t1 (THead (Bind 
 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (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 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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))))))))) (\lambda (x0: T).(\lambda 
-(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t1 (THead (Bind 
-Abst) x0 x1))).(\lambda (H9: (eq T x (THead (Bind Abbr) x2 x3))).(\lambda 
-(H10: (pr0 u1 x2)).(\lambda (H11: (pr0 x1 x3)).(eq_ind_r T (THead (Bind Abbr) 
-x2 x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
-T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 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 t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (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 u1 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)))))))))) (eq_ind_r 
-T (THead (Bind Abst) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c 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 (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (eq_ind_r T (THead (Bind 
+Abst) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+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 (THead (Bind Abbr) 
+x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c0 (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 t 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind 
 Abbr) x2 x3) (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 u1 u2))))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
 y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
 T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 
 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 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 (THead (Bind Abst) x0 x1) (THead (Bind b) y1 
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
-T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind Abbr) x2 x3) (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 u1 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)))))))) (ex4_4_intro 
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq 
-T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) 
-x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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))))))) x0 x1 x2 x3 (refl_equal T 
-(THead (Bind Abst) x0 x1)) (refl_equal T (THead (Bind Abbr) x2 x3)) (pr2_free 
-c u1 x2 H10) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c (Bind b) u) 
-x1 x3 H11))))) t1 H8) x H9))))))))) H7)) (\lambda (H7: (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))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (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 (THead (Bind Abst) x0 x1) (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind 
+Abbr) x2 x3) (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 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) 
+u) z1 t3))))))) x0 x1 x2 x3 (refl_equal T (THead (Bind Abst) x0 x1)) 
+(refl_equal T (THead (Bind Abbr) x2 x3)) (pr2_free c0 u1 x2 H7) (\lambda (b: 
+B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) x1 x3 H8))))) t1 H5) t2 
+H6))))))))) 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).(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 
+T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (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))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c 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).(pr2 c u1 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 t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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))))))))) (\lambda 
-(x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (x5: T).(\lambda (H8: (not (eq B x0 Abst))).(\lambda (H9: (eq T 
-t1 (THead (Bind x0) x1 x2))).(\lambda (H10: (eq T x (THead (Bind x0) x4 
-(THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda (H11: (pr0 u1 
-x3)).(\lambda (H12: (pr0 x1 x4)).(\lambda (H13: (pr0 x2 x5)).(eq_ind_r T 
-(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t: 
-T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c 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 t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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 t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) 
+t3))))))))).(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 t1 (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(v2: T).(\lambda (t3: T).(eq T t2 (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))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 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 t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c0 (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 t1 (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 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
+z2))))))))) (\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: (eq T t1 (THead (Bind x0) x1 x2))).(\lambda (H7: (eq T 
+t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda 
+(H8: (pr0 u1 x3)).(\lambda (H9: (pr0 x1 x4)).(\lambda (H10: (pr0 x2 
+x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) 
+x5)) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 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 t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c0 (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 t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t 
+(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 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
+z2)))))))))) (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t: T).(or3 (ex3_2 
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat 
+Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+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 (THead (Bind x0) 
+x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (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 t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (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 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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)))))))))) (eq_ind_r T (THead (Bind 
-x0) x1 x2) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c t t3)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat 
+Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
 (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) 
 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 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 t (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead 
-(Flat Appl) (lift (S O) O x3) x5)) (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 u1 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)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) 
-O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) 
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead 
-(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(pr2 c0 u1 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: 
+(CHead c0 (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 (THead (Bind 
@@ -911,10 +764,10 @@ T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (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 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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_intro B T T T T T 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro 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 
@@ -923,211 +776,145 @@ B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
 (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) 
 (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 u1 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))))))) 
-x0 x1 x2 x5 x3 x4 H8 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T 
-(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))) (pr2_free c u1 
-x3 H11) (pr2_free c x1 x4 H12) (pr2_free (CHead c (Bind x0) x4) x2 x5 H13))) 
-t1 H9) x H10))))))))))))) H7)) (pr0_gen_appl u1 t1 x H6))) t2 (sym_eq T t2 x 
-H5))) t0 (sym_eq T t0 (THead (Flat Appl) u1 t1) H4))) c0 (sym_eq C c0 c H1) 
-H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda 
-(H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(\lambda 
-(H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Flat Appl) u1 
-t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 
-t2) \to ((subst0 i u t2 t) \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).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 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).(pr2 c u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) 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 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x 
-(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 u1 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))))))))))))))) (\lambda (H6: (eq T t0 (THead (Flat Appl) u1 t1))).(eq_ind 
-T (THead (Flat Appl) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c 
-(CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2 
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) 
+x0 x1 x2 x5 x3 x4 H5 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T 
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))) (pr2_free c0 
+u1 x3 H8) (pr2_free c0 x1 x4 H9) (pr2_free (CHead c0 (Bind x0) x4) x2 x5 
+H10))) t1 H6) t2 H7))))))))))))) H4)) (pr0_gen_appl u1 t1 t2 H3)))))))) 
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 
+t)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(let H5 \def (eq_ind T 
+t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Appl) u1 t1) H4) 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).(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 (t4: 
-T).(eq T x (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+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).(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 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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)))))))))))))) (\lambda (H7: (eq T t 
-x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to 
-((pr0 (THead (Flat Appl) u1 t1) t2) \to ((subst0 i u t2 t3) \to (or3 (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2 t4)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t4: T).(pr2 c t1 t4)))) (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 (t4: T).(eq T x 
-(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c (Bind b) u0) 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 t1 (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 
-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))))))))))))) (\lambda (H8: (getl i c 
-(CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (THead (Flat Appl) u1 t1) 
-t2)).(\lambda (H10: (subst0 i u t2 x)).(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).(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 (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (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)))))))) (or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+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 t2 (THead (Bind 
+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 t1 (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (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)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 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).(pr2 c u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) 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 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x 
-(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 u1 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))))))))) 
-(\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind T T (\lambda (u2: 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) 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 (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(eq T t (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 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: 
 T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
 T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c 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).(pr2 c u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) 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 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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))))))))) (\lambda (x0: T).(\lambda 
-(x1: T).(\lambda (H12: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H13: 
-(pr0 u1 x0)).(\lambda (H14: (pr0 t1 x1)).(let H15 \def (eq_ind T t2 (\lambda 
-(t3: T).(subst0 i u t3 x)) H10 (THead (Flat Appl) x0 x1) H12) in (or3_ind 
-(ex2 T (\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1))) (\lambda (u2: 
-T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Flat Appl) x0 
-t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3)))) (or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c 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).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c (Bind b) u0) 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 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 
-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))))))))) (\lambda (H16: (ex2 T 
-(\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 
-i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1))) 
+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).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+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 t (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) 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 (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(eq T t (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 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr0 u1 
+x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3: 
+T).(subst0 i u t3 t)) H3 (THead (Flat Appl) x0 x1) H7) in (or3_ind (ex2 T 
+(\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 
+i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3))) 
+(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s (Flat Appl) i) u x1 t3)))) (or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+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 t (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) 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 (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(eq T t (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 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H11: (ex2 T (\lambda (u2: 
+T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 i u x0 
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1))) 
 (\lambda (u2: T).(subst0 i u x0 u2)) (or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+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 
+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).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) 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 (z2: T).(\lambda 
-(u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(u2: T).(\lambda (y2: T).(eq T t (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 u1 u2))))))) 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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 (y2: T).(pr2 c0 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))))))))) (\lambda (x2: T).(\lambda (H17: (eq T x 
-(THead (Flat Appl) x2 x1))).(\lambda (H18: (subst0 i u x0 x2)).(eq_ind_r T 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda (H12: (eq T t 
+(THead (Flat Appl) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(eq_ind_r T 
 (THead (Flat Appl) x2 x1) (\lambda (t3: T).(or3 (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 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 
-t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 
+t1 t4)))) (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 (t4: T).(eq T t3 (THead (Bind 
 Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) 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 (_: 
@@ -1135,70 +922,70 @@ T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (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 u1 u2))))))) 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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 (y2: T).(pr2 c0 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)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: 
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: 
 T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c 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 (THead (Flat Appl) x2 x1) (THead (Bind Abbr) u2 t3)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) 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: 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr2 c0 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 (THead (Flat Appl) x2 x1) (THead (Bind Abbr) u2 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
 (THead (Flat Appl) x2 x1) (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 u1 u2))))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
 y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
 (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 
-t3))) x2 x1 (refl_equal T (THead (Flat Appl) x2 x1)) (pr2_delta c d u i H8 u1 
-x0 H13 x2 H18) (pr2_free c t1 x1 H14))) x H17)))) H16)) (\lambda (H16: (ex2 T 
-(\lambda (t3: T).(eq T x (THead (Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 
-(s (Flat Appl) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead 
-(Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3)) 
-(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c 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).(pr2 c u1 u2))))) (\lambda (_: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3))) x2 x1 (refl_equal T (THead (Flat Appl) x2 x1)) (pr2_delta c0 d u i 
+H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9))) t H12)))) H11)) (\lambda (H11: 
+(ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3))) (\lambda (t3: 
+T).(subst0 (s (Flat Appl) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t 
+(THead (Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 
+t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat 
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr2 c0 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 t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) 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 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (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 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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))))))))) (\lambda (x2: T).(\lambda 
-(H17: (eq T x (THead (Flat Appl) x0 x2))).(\lambda (H18: (subst0 (s (Flat 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda 
+(H12: (eq T t (THead (Flat Appl) x0 x2))).(\lambda (H13: (subst0 (s (Flat 
 Appl) i) u x1 x2)).(eq_ind_r T (THead (Flat Appl) x0 x2) (\lambda (t3: 
 T).(or3 (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 u1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr2 c t1 t4)))) (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 (t4: 
-T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: 
+Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
+(_: T).(\lambda (t4: T).(pr2 c0 t1 t4)))) (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 
+(t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) 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 t1 
@@ -1206,19 +993,19 @@ T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
 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 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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)))))))))) 
+(_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) 
 (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat 
 Appl) x0 x2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 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 
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 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 (THead (Flat Appl) x0 x2) (THead (Bind Abbr) 
 u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) 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 (_: 
@@ -1226,73 +1013,73 @@ T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
 (u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x0 x2) (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 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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)))))))) (ex3_2_intro T T (\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda 
 (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Flat Appl) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x2 (refl_equal T (THead (Flat Appl) x0 
-x2)) (pr2_free c u1 x0 H13) (pr2_delta c d u i H8 t1 x1 H14 x2 H18))) x 
-H17)))) H16)) (\lambda (H16: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
-T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 (refl_equal T (THead (Flat Appl) 
+x0 x2)) (pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13))) t 
+H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
+T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u 
 x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 
-t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat 
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat 
 Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) 
 (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c 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).(pr2 c u1 u2))))) (\lambda (_: 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr2 c0 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 t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) 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 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (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 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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))))))))) (\lambda (x2: T).(\lambda 
-(x3: T).(\lambda (H17: (eq T x (THead (Flat Appl) x2 x3))).(\lambda (H18: 
-(subst0 i u x0 x2)).(\lambda (H19: (subst0 (s (Flat Appl) i) u x1 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (H12: (eq T t (THead (Flat Appl) x2 x3))).(\lambda (H13: 
+(subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat Appl) i) u x1 
 x3)).(eq_ind_r T (THead (Flat Appl) x2 x3) (\lambda (t3: T).(or3 (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 u1 u2))) (\lambda (_: T).(\lambda 
-(t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t4: T).(pr2 c0 t1 t4)))) (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 (t4: T).(eq T t3 
 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: 
+(CHead c0 (Bind b) u0) 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 t1 (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 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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)))))))))) (or3_intro0 (ex3_2 T T 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T 
 (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 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 (THead (Flat Appl) x2 x3) (THead (Bind Abbr) u2 
 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) 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 (_: 
@@ -1300,133 +1087,133 @@ T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
 (u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x2 x3) (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 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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)))))))) (ex3_2_intro T T (\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda 
 (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Flat Appl) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c t1 t3))) x2 x3 (refl_equal T (THead (Flat Appl) x2 
-x3)) (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (pr2_delta c d u i H8 t1 x1 H14 
-x3 H19))) x H17)))))) H16)) (subst0_gen_head (Flat Appl) u x0 x1 x i 
-H15)))))))) H11)) (\lambda (H11: (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 t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 (refl_equal T (THead (Flat Appl) 
+x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1 
+H9 x3 H14))) t H12)))))) H11)) (subst0_gen_head (Flat Appl) u x0 x1 t i 
+H10)))))))) H6)) (\lambda (H6: (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 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))))))).(ex4_4_ind 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 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))))))).(ex4_4_ind 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 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))))) (or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c 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).(pr2 c u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) 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 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 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 t 
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) 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 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (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 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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))))))))) (\lambda (x0: T).(\lambda 
-(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq T t1 (THead 
-(Bind Abst) x0 x1))).(\lambda (H13: (eq T t2 (THead (Bind Abbr) x2 
-x3))).(\lambda (H14: (pr0 u1 x2)).(\lambda (H15: (pr0 x1 x3)).(let H16 \def 
-(eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 x)) H10 (THead (Bind Abbr) x2 
-x3) H13) in (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2 
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr2 c t3 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: 
-T).(eq T x (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) 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 t3 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda 
+(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T t1 (THead (Bind 
+Abst) x0 x1))).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda 
+(H9: (pr0 u1 x2)).(\lambda (H10: (pr0 x1 x3)).(let H11 \def (eq_ind T t2 
+(\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Bind Abbr) x2 x3) H8) in 
+(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2 t4)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t 
+(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) 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 t3 (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (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 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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)))))))))) (or3_ind (ex2 T (\lambda 
-(u2: T).(eq T x (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 
-u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind Abbr) x2 t3))) (\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda 
+(u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 
+u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3))) (\lambda 
 (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
 T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: T).(\lambda (t3: 
 T).(subst0 (s (Bind Abbr) i) u x3 t3)))) (or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
 (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).(pr2 c u1 u2))))) 
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T 
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda 
 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) 
+T).(\lambda (y2: T).(eq T t (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 u1 u2))))))) (\lambda (_: 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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))))))))) (\lambda (H17: (ex2 T (\lambda (u2: T).(eq T x (THead 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (H12: (ex2 T (\lambda (u2: T).(eq T t (THead 
 (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2)))).(ex2_ind T 
-(\lambda (u2: T).(eq T x (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 
-i u x2 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) 
+(\lambda (u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 
+i u x2 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3)))) 
 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
 T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind 
+(_: 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).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) 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) x0 x1) (THead 
 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (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 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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))))))))) (\lambda (x4: T).(\lambda 
-(H18: (eq T x (THead (Bind Abbr) x4 x3))).(\lambda (H19: (subst0 i u x2 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x4: T).(\lambda 
+(H13: (eq T t (THead (Bind Abbr) x4 x3))).(\lambda (H14: (subst0 i u x2 
 x4)).(eq_ind_r T (THead (Bind Abbr) x4 x3) (\lambda (t3: T).(or3 (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 u1 u2))) (\lambda (_: T).(\lambda 
-(t4: T).(pr2 c (THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t4: T).(pr2 c0 (THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) 
 x0 x1) (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 u1 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: 
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) 
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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 
@@ -1434,20 +1221,20 @@ B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\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 u1 u2))))))) (\lambda (_: 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
 y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
 T).(eq T (THead (Bind Abbr) x4 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2 
 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) 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 (_: 
@@ -1456,77 +1243,77 @@ T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind 
 Abbr) x4 x3) (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 u1 u2))))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
 y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2 
 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) z1 t3))))))) x0 x1 x4 x3 (refl_equal T (THead (Bind Abst) x0 x1)) 
-(refl_equal T (THead (Bind Abbr) x4 x3)) (pr2_delta c d u i H8 u1 x2 H14 x4 
-H19) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) x1 x3 
-H15))))) x H18)))) H17)) (\lambda (H17: (ex2 T (\lambda (t3: T).(eq T x 
+(refl_equal T (THead (Bind Abbr) x4 x3)) (pr2_delta c0 d u i H1 u1 x2 H9 x4 
+H14) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) x1 x3 
+H10))))) t H13)))) H12)) (\lambda (H12: (ex2 T (\lambda (t3: T).(eq T t 
 (THead (Bind Abbr) x2 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind Abbr) x2 t3))) 
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3))) 
 (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3)) (or3 (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
+(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
 (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).(pr2 c u1 u2))))) 
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T 
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda 
 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) 
+T).(\lambda (y2: T).(eq T t (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 u1 u2))))))) (\lambda (_: 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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))))))))) (\lambda (x4: T).(\lambda (H18: (eq T x (THead (Bind Abbr) 
-x2 x4))).(\lambda (H19: (subst0 (s (Bind Abbr) i) u x3 x4)).(eq_ind_r T 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (H13: (eq T t (THead (Bind Abbr) 
+x2 x4))).(\lambda (H14: (subst0 (s (Bind Abbr) i) u x3 x4)).(eq_ind_r T 
 (THead (Bind Abbr) x2 x4) (\lambda (t3: T).(or3 (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 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 
 (THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
 (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 u1 u2))))) 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T 
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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) x0 x1) (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 u1 u2))))))) (\lambda (_: 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
 y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
 T).(eq T (THead (Bind Abbr) x2 x4) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2 
 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) 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 (_: 
@@ -1535,82 +1322,82 @@ T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind 
 Abbr) x2 x4) (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 u1 u2))))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
 y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2 
 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) z1 t3))))))) x0 x1 x2 x4 (refl_equal T (THead (Bind Abst) x0 x1)) 
-(refl_equal T (THead (Bind Abbr) x2 x4)) (pr2_free c u1 x2 H14) (\lambda (b: 
-B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) 
-(getl_clear_bind b (CHead c (Bind b) u0) c u0 (clear_bind b c u0) (CHead d 
-(Bind Abbr) u) i H8) x1 x3 H15 x4 H19))))) x H18)))) H17)) (\lambda (H17: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 
+(refl_equal T (THead (Bind Abbr) x2 x4)) (pr2_free c0 u1 x2 H9) (\lambda (b: 
+B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) 
+(getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0) (CHead d 
+(Bind Abbr) u) i H1) x1 x3 H10 x4 H14))))) t H13)))) H12)) (\lambda (H12: 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 
 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: 
 T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) 
 (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: 
 T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) 
 x0 x1) (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).(pr2 c u1 
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) 
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda 
 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) 
+T).(\lambda (y2: T).(eq T t (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 u1 u2))))))) (\lambda (_: 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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))))))))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H18: (eq T x 
-(THead (Bind Abbr) x4 x5))).(\lambda (H19: (subst0 i u x2 x4)).(\lambda (H20: 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H13: (eq T t 
+(THead (Bind Abbr) x4 x5))).(\lambda (H14: (subst0 i u x2 x4)).(\lambda (H15: 
 (subst0 (s (Bind Abbr) i) u x3 x5)).(eq_ind_r T (THead (Bind Abbr) x4 x5) 
 (\lambda (t3: T).(or3 (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 u1 u2))) 
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) x0 x1) t4)))) 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (THead (Bind Abst) x0 x1) (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 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c (Bind b) u0) 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) x0 x1) (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 u1 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)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-(THead (Bind Abbr) x4 x5) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
+(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind Abst) x0 x1) 
+t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (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 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) 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) x0 x1) (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 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind Abbr) x4 x5) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 
 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) 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 (_: 
@@ -1619,33 +1406,33 @@ T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind 
 Abbr) x4 x5) (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 u1 u2))))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
 y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
 T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 
 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) z1 t3))))))) x0 x1 x4 x5 (refl_equal T (THead (Bind Abst) x0 x1)) 
-(refl_equal T (THead (Bind Abbr) x4 x5)) (pr2_delta c d u i H8 u1 x2 H14 x4 
-H19) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S 
-i) (getl_clear_bind b (CHead c (Bind b) u0) c u0 (clear_bind b c u0) (CHead d 
-(Bind Abbr) u) i H8) x1 x3 H15 x5 H20))))) x H18)))))) H17)) (subst0_gen_head 
-(Bind Abbr) u x2 x3 x i H16)) t1 H12)))))))))) H11)) (\lambda (H11: (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 
-t2 (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 (_: 
+(refl_equal T (THead (Bind Abbr) x4 x5)) (pr2_delta c0 d u i H1 u1 x2 H9 x4 
+H14) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u 
+(S i) (getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0) 
+(CHead d (Bind Abbr) u) i H1) x1 x3 H10 x5 H15))))) t H13)))))) H12)) 
+(subst0_gen_head (Bind Abbr) u x2 x3 t i H11)) t1 H7)))))))))) H6)) (\lambda 
+(H6: (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 t2 (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))))))))).(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 
@@ -1658,139 +1445,139 @@ B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 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))))))) (or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+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 
+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).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) 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 (z2: T).(\lambda 
-(u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(u2: T).(\lambda (y2: T).(eq T t (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 u1 u2))))))) 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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 (y2: T).(pr2 c0 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))))))))) (\lambda (x0: B).(\lambda (x1: 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: B).(\lambda (x1: 
 T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: 
-T).(\lambda (H12: (not (eq B x0 Abst))).(\lambda (H13: (eq T t1 (THead (Bind 
-x0) x1 x2))).(\lambda (H14: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) 
-(lift (S O) O x3) x5)))).(\lambda (H15: (pr0 u1 x3)).(\lambda (H16: (pr0 x1 
-x4)).(\lambda (H17: (pr0 x2 x5)).(let H18 \def (eq_ind T t2 (\lambda (t3: 
-T).(subst0 i u t3 x)) H10 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) 
-O x3) x5)) H14) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2 
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr2 c t3 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: 
-T).(eq T x (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: 
+T).(\lambda (H7: (not (eq B x0 Abst))).(\lambda (H8: (eq T t1 (THead (Bind 
+x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) 
+(lift (S O) O x3) x5)))).(\lambda (H10: (pr0 u1 x3)).(\lambda (H11: (pr0 x1 
+x4)).(\lambda (H12: (pr0 x2 x5)).(let H13 \def (eq_ind T t2 (\lambda (t3: 
+T).(subst0 i u t3 t)) H3 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O 
+x3) x5)) H9) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(or3 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2 
+t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t4: T).(eq T t (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) 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 t3 
 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (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 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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)))))))))) (or3_ind (ex2 T (\lambda 
-(u2: T).(eq T x (THead (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) 
-x5)))) (\lambda (u2: T).(subst0 i u x4 u2))) (ex2 T (\lambda (t3: T).(eq T x 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda 
+(u2: T).(eq T t (THead (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) 
+x5)))) (\lambda (u2: T).(subst0 i u x4 u2))) (ex2 T (\lambda (t3: T).(eq T t 
 (THead (Bind x0) x4 t3))) (\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead 
 (Flat Appl) (lift (S O) O x3) x5) t3))) (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T x (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+(t3: T).(eq T t (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
 T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) 
 i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3)))) (or3 (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
+(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (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).(pr2 c u1 u2))))) (\lambda (_: 
+(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) 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 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+(y2: T).(eq T t (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 u1 u2))))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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))))))))) (\lambda (H19: (ex2 T (\lambda (u2: T).(eq T x (THead 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (H14: (ex2 T (\lambda (u2: T).(eq T t (THead 
 (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: 
-T).(subst0 i u x4 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Bind x0) 
+T).(subst0 i u x4 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Bind x0) 
 u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: T).(subst0 i u 
-x4 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq 
-T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind 
+x4 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T (THead (Bind x0) x1 x2) (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).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) 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 x0) x1 x2) (THead (Bind 
 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (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 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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))))))))) (\lambda (x6: T).(\lambda 
-(H20: (eq T x (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) 
-x5)))).(\lambda (H21: (subst0 i u x4 x6)).(eq_ind_r T (THead (Bind x0) x6 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda 
+(H15: (eq T t (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) 
+x5)))).(\lambda (H16: (subst0 i u x4 x6)).(eq_ind_r T (THead (Bind x0) x6 
 (THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(or3 (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 u1 u2))) (\lambda (_: T).(\lambda 
-(t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
 x2) (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 u1 u2))))) 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T 
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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 x0) x1 x2) (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 u1 u2))))))) (\lambda (_: 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
 y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
 T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq 
-T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
 x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) 
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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 
@@ -1799,10 +1586,10 @@ B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) 
 O x3) x5)) (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 u1 u2))))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
 y2) z1 z2)))))))) (ex6_6_intro 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 (_: 
@@ -1811,51 +1598,51 @@ b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
 (Flat Appl) (lift (S O) O x3) x5)) (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 u1 u2))))))) 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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 (y2: T).(pr2 c0 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))))))) x0 x1 x2 x5 x3 x6 H12 (refl_equal T (THead 
+(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x3 x6 H7 (refl_equal T (THead 
 (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift 
-(S O) O x3) x5))) (pr2_free c u1 x3 H15) (pr2_delta c d u i H8 x1 x4 H16 x6 
-H21) (pr2_free (CHead c (Bind x0) x6) x2 x5 H17))) x H20)))) H19)) (\lambda 
-(H19: (ex2 T (\lambda (t3: T).(eq T x (THead (Bind x0) x4 t3))) (\lambda (t3: 
+(S O) O x3) x5))) (pr2_free c0 u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6 
+H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) t H15)))) H14)) (\lambda 
+(H14: (ex2 T (\lambda (t3: T).(eq T t (THead (Bind x0) x4 t3))) (\lambda (t3: 
 T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) 
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind x0) x4 t3))) (\lambda 
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind x0) x4 t3))) (\lambda 
 (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) 
-t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
+t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat 
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind 
+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).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) 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 x0) x1 x2) (THead (Bind 
 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (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 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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))))))))) (\lambda (x6: T).(\lambda 
-(H20: (eq T x (THead (Bind x0) x4 x6))).(\lambda (H21: (subst0 (s (Bind x0) 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda 
+(H15: (eq T t (THead (Bind x0) x4 x6))).(\lambda (H16: (subst0 (s (Bind x0) 
 i) u (THead (Flat Appl) (lift (S O) O x3) x5) x6)).(eq_ind_r T (THead (Bind 
 x0) x4 x6) (\lambda (t3: T).(or3 (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 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0) 
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) 
 x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
 T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (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 u1 u2))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) 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 
@@ -1863,10 +1650,10 @@ T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
 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 u1 u2))))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
 y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x6 (THead (Flat 
 Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) 
 u2))) (ex2 T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S O) O x3) 
@@ -1876,40 +1663,40 @@ t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O)
 O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind 
 x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
 (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
 (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T 
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (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 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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))))))))) (\lambda (H22: (ex2 T 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H17: (ex2 T 
 (\lambda (u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2: 
 T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda 
 (u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s 
 (Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2: 
 T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
 x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) 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 
@@ -1917,23 +1704,23 @@ T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
 (y2: T).(eq T (THead (Bind x0) x4 x6) (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 u1 u2))))))) 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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 (y2: T).(pr2 c0 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))))))))) (\lambda (x7: T).(\lambda (H23: (eq T x6 
-(THead (Flat Appl) x7 x5))).(\lambda (H24: (subst0 (s (Bind x0) i) u (lift (S 
-O) O x3) x7)).(eq_ind_r T (THead (Flat Appl) x7 x5) (\lambda (t3: T).(or3 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (H18: (eq T 
+x6 (THead (Flat Appl) x7 x5))).(\lambda (H19: (subst0 (s (Bind x0) i) u (lift 
+(S O) O x3) x7)).(eq_ind_r T (THead (Flat Appl) x7 x5) (\lambda (t3: T).(or3 
 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3) 
-(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq 
-T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
 x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) 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 
@@ -1941,22 +1728,22 @@ T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
 (y2: T).(eq T (THead (Bind x0) x4 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 u1 u2))))))) 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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 (y2: T).(pr2 c0 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)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x7 
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x7 
 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u 
 x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind 
 x0) x4 (THead (Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
 (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind 
 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) 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 (_: 
@@ -1965,24 +1752,24 @@ b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead 
 (Flat Appl) x7 x5)) (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 u1 u2))))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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))))))))) (\lambda (x8: T).(\lambda (H25: (eq T x7 (lift (S O) O 
-x8))).(\lambda (H26: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x8)).(let H27 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H20: (eq T x7 (lift (S O) O 
+x8))).(\lambda (H21: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x8)).(let H22 
 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u 
-x3 x8)) H26 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x8) (\lambda (t3: 
+x3 x8)) H21 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x8) (\lambda (t3: 
 T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
 x4 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 
 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
 (t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Bind 
 Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) 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 (_: 
@@ -1991,21 +1778,21 @@ b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead 
 (Flat Appl) t3 x5)) (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 u1 u2))))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
 y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
 T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq 
-T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
 x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) 
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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 
@@ -2014,10 +1801,10 @@ B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) 
 O x8) x5)) (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 u1 u2))))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
 y2) z1 z2)))))))) (ex6_6_intro 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 (_: 
@@ -2026,54 +1813,54 @@ b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead 
 (Flat Appl) (lift (S O) O x8) x5)) (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 u1 u2))))))) 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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 (y2: T).(pr2 c0 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))))))) x0 x1 x2 x5 x8 x4 H12 (refl_equal T (THead 
+(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x8 x4 H7 (refl_equal T (THead 
 (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift 
-(S O) O x8) x5))) (pr2_delta c d u i H8 u1 x3 H15 x8 H27) (pr2_free c x1 x4 
-H16) (pr2_free (CHead c (Bind x0) x4) x2 x5 H17))) x7 H25))))) 
-(subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O H24 (le_n_S O i (le_O_n 
-i)))) x6 H23)))) H22)) (\lambda (H22: (ex2 T (\lambda (t3: T).(eq T x6 (THead 
+(S O) O x8) x5))) (pr2_delta c0 d u i H1 u1 x3 H10 x8 H22) (pr2_free c0 x1 x4 
+H11) (pr2_free (CHead c0 (Bind x0) x4) x2 x5 H12))) x7 H20))))) 
+(subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O H19 (le_n_S O i (le_O_n 
+i)))) x6 H18)))) H17)) (\lambda (H17: (ex2 T (\lambda (t3: T).(eq T x6 (THead 
 (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) 
 (s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead 
 (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) 
 (s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
 T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
 (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T 
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (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 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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))))))))) (\lambda (x7: T).(\lambda 
-(H23: (eq T x6 (THead (Flat Appl) (lift (S O) O x3) x7))).(\lambda (H24: 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda 
+(H18: (eq T x6 (THead (Flat Appl) (lift (S O) O x3) x7))).(\lambda (H19: 
 (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x7)).(eq_ind_r T (THead (Flat 
 Appl) (lift (S O) O x3) x7) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: 
 T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Flat Appl) u2 
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T 
+t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T 
 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
 x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) 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 
@@ -2081,22 +1868,22 @@ T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
 (y2: T).(eq T (THead (Bind x0) x4 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 u1 u2))))))) 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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 (y2: T).(pr2 c0 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)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: 
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: 
 T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) 
 O x3) x7)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) 
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) 
 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
 T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
 T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead 
 (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
+(CHead c0 (Bind b) u0) 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 
@@ -2105,10 +1892,10 @@ T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (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 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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_intro B T T T T T 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro 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 
@@ -2117,32 +1904,32 @@ B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
 (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) 
 (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 u1 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))))))) 
-x0 x1 x2 x7 x3 x4 H12 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T 
-(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))) (pr2_free c u1 
-x3 H15) (pr2_free c x1 x4 H16) (pr2_delta (CHead c (Bind x0) x4) d u (S i) 
-(getl_clear_bind x0 (CHead c (Bind x0) x4) c x4 (clear_bind x0 c x4) (CHead d 
-(Bind Abbr) u) i H8) x2 x5 H17 x7 H24))) x6 H23)))) H22)) (\lambda (H22: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) 
-O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind 
-x0) i)) u x5 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-x6 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s 
-(Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (or3 (ex3_2 T T 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) 
+x0 x1 x2 x7 x3 x4 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T 
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))) (pr2_free c0 
+u1 x3 H10) (pr2_free c0 x1 x4 H11) (pr2_delta (CHead c0 (Bind x0) x4) d u (S 
+i) (getl_clear_bind x0 (CHead c0 (Bind x0) x4) c0 x4 (clear_bind x0 c0 x4) 
+(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x7 H19))) x6 H18)))) H17)) (\lambda 
+(H17: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat 
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u 
+(lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat 
+Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T x6 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda 
+(t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (or3 (ex3_2 T T 
 (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
 x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) 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 
@@ -2150,48 +1937,48 @@ T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
 (y2: T).(eq T (THead (Bind x0) x4 x6) (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 u1 u2))))))) 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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 (y2: T).(pr2 c0 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))))))))) (\lambda (x7: T).(\lambda (x8: 
-T).(\lambda (H23: (eq T x6 (THead (Flat Appl) x7 x8))).(\lambda (H24: (subst0 
-(s (Bind x0) i) u (lift (S O) O x3) x7)).(\lambda (H25: (subst0 (s (Flat 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (x8: 
+T).(\lambda (H18: (eq T x6 (THead (Flat Appl) x7 x8))).(\lambda (H19: (subst0 
+(s (Bind x0) i) u (lift (S O) O x3) x7)).(\lambda (H20: (subst0 (s (Flat 
 Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat Appl) x7 x8) 
 (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T 
 (THead (Bind x0) x4 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 
 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
 (t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T 
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 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 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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)))))))))) (ex2_ind T (\lambda (t3: 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: 
 T).(eq T x7 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) 
 i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
 (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
 (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) 
 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
+(CHead c0 (Bind b) u0) 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 
@@ -2199,26 +1986,26 @@ x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
 (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (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 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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))))))))) (\lambda (x9: T).(\lambda 
-(H26: (eq T x7 (lift (S O) O x9))).(\lambda (H27: (subst0 (minus (s (Bind x0) 
-i) (S O)) u x3 x9)).(let H28 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) 
-(\lambda (n: nat).(subst0 n u x3 x9)) H27 i (s_arith1 x0 i)) in (eq_ind_r T 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x9: T).(\lambda 
+(H21: (eq T x7 (lift (S O) O x9))).(\lambda (H22: (subst0 (minus (s (Bind x0) 
+i) (S O)) u x3 x9)).(let H23 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) 
+(\lambda (n: nat).(subst0 n u x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T 
 (lift (S O) O x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
 (t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Flat 
-Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T 
+Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
+(_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T 
 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
 x4 (THead (Flat Appl) t3 x8)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T 
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
@@ -2226,21 +2013,21 @@ B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) 
 (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 u1 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 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
 x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) 
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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 
@@ -2249,10 +2036,10 @@ B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
 T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) 
 O x9) x8)) (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 u1 u2))))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
 y2) z1 z2)))))))) (ex6_6_intro 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 (_: 
@@ -2261,69 +2048,69 @@ b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead 
 (Flat Appl) (lift (S O) O x9) x8)) (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 u1 u2))))))) 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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 (y2: T).(pr2 c0 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))))))) x0 x1 x2 x8 x9 x4 H12 (refl_equal T (THead 
+(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x8 x9 x4 H7 (refl_equal T (THead 
 (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift 
-(S O) O x9) x8))) (pr2_delta c d u i H8 u1 x3 H15 x9 H28) (pr2_free c x1 x4 
-H16) (pr2_delta (CHead c (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead c 
-(Bind x0) x4) c x4 (clear_bind x0 c x4) (CHead d (Bind Abbr) u) i H8) x2 x5 
-H17 x8 H25))) x7 H26))))) (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O 
-H24 (le_n_S O i (le_O_n i)))) x6 H23)))))) H22)) (subst0_gen_head (Flat Appl) 
-u (lift (S O) O x3) x5 x6 (s (Bind x0) i) H21)) x H20)))) H19)) (\lambda 
-(H19: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind x0) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) 
-O x3) x5) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x 
-(THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 
-u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat 
-Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) 
-x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (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).(pr2 c u1 u2))))) (\lambda (_: 
+(S O) O x9) x8))) (pr2_delta c0 d u i H1 u1 x3 H10 x9 H23) (pr2_free c0 x1 x4 
+H11) (pr2_delta (CHead c0 (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead 
+c0 (Bind x0) x4) c0 x4 (clear_bind x0 c0 x4) (CHead d (Bind Abbr) u) i H1) x2 
+x5 H12 x8 H20))) x7 H21))))) (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S 
+O) O H19 (le_n_S O i (le_O_n i)))) x6 H18)))))) H17)) (subst0_gen_head (Flat 
+Appl) u (lift (S O) O x3) x5 x6 (s (Bind x0) i) H16)) t H15)))) H14)) 
+(\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
+(Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) 
+(lift (S O) O x3) x5) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T t (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) 
+i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (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).(pr2 c0 u1 u2))))) (\lambda (_: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) 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 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+(y2: T).(eq T t (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 u1 u2))))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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))))))))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (eq T x 
-(THead (Bind x0) x6 x7))).(\lambda (H21: (subst0 i u x4 x6)).(\lambda (H22: 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq T t 
+(THead (Bind x0) x6 x7))).(\lambda (H16: (subst0 i u x4 x6)).(\lambda (H17: 
 (subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) 
 x7)).(eq_ind_r T (THead (Bind x0) x6 x7) (\lambda (t3: T).(or3 (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 u1 u2))) (\lambda (_: T).(\lambda 
-(t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
 x2) (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 u1 u2))))) 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T 
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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 x0) x1 x2) (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 u1 u2))))))) (\lambda (_: 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
 y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x7 (THead (Flat 
 Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) 
 u2))) (ex2 T (\lambda (t3: T).(eq T x7 (THead (Flat Appl) (lift (S O) O x3) 
@@ -2333,40 +2120,40 @@ t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O)
 O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind 
 x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
 (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
 (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T 
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (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 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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))))))))) (\lambda (H23: (ex2 T 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H18: (ex2 T 
 (\lambda (u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2: 
 T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda 
 (u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s 
 (Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2: 
 T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
 x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) 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 
@@ -2374,23 +2161,23 @@ T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
 (y2: T).(eq T (THead (Bind x0) x6 x7) (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 u1 u2))))))) 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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 (y2: T).(pr2 c0 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))))))))) (\lambda (x8: T).(\lambda (H24: (eq T x7 
-(THead (Flat Appl) x8 x5))).(\lambda (H25: (subst0 (s (Bind x0) i) u (lift (S 
-O) O x3) x8)).(eq_ind_r T (THead (Flat Appl) x8 x5) (\lambda (t3: T).(or3 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H19: (eq T 
+x7 (THead (Flat Appl) x8 x5))).(\lambda (H20: (subst0 (s (Bind x0) i) u (lift 
+(S O) O x3) x8)).(eq_ind_r T (THead (Flat Appl) x8 x5) (\lambda (t3: T).(or3 
 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 t3) 
-(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq 
-T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
 x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) 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 
@@ -2398,22 +2185,22 @@ T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
 (y2: T).(eq T (THead (Bind x0) x6 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 u1 u2))))))) 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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 (y2: T).(pr2 c0 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)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x8 
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x8 
 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u 
 x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind 
 x0) x6 (THead (Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
 (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind 
 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) 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 (_: 
@@ -2422,24 +2209,24 @@ b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
 (Flat Appl) x8 x5)) (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 u1 u2))))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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))))))))) (\lambda (x9: T).(\lambda (H26: (eq T x8 (lift (S O) O 
-x9))).(\lambda (H27: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x9)).(let H28 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H21: (eq T x8 (lift (S O) O 
+x9))).(\lambda (H22: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x9)).(let H23 
 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u 
-x3 x9)) H27 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x9) (\lambda (t3: 
+x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x9) (\lambda (t3: 
 T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
 x6 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 
 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
 (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Bind 
 Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) 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 (_: 
@@ -2448,21 +2235,21 @@ b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
 (Flat Appl) t3 x5)) (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 u1 u2))))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
 y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
 T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq 
-T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
 x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) 
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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 
@@ -2471,10 +2258,10 @@ B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) 
 O x9) x5)) (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 u1 u2))))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
 y2) z1 z2)))))))) (ex6_6_intro 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 (_: 
@@ -2483,29 +2270,29 @@ b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
 (Flat Appl) (lift (S O) O x9) x5)) (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 u1 u2))))))) 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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 (y2: T).(pr2 c0 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))))))) x0 x1 x2 x5 x9 x6 H12 (refl_equal T (THead 
+(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x9 x6 H7 (refl_equal T (THead 
 (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift 
-(S O) O x9) x5))) (pr2_delta c d u i H8 u1 x3 H15 x9 H28) (pr2_delta c d u i 
-H8 x1 x4 H16 x6 H21) (pr2_free (CHead c (Bind x0) x6) x2 x5 H17))) x8 
-H26))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H25 (le_n_S O i 
-(le_O_n i)))) x7 H24)))) H23)) (\lambda (H23: (ex2 T (\lambda (t3: T).(eq T 
+(S O) O x9) x5))) (pr2_delta c0 d u i H1 u1 x3 H10 x9 H23) (pr2_delta c0 d u 
+i H1 x1 x4 H11 x6 H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) x8 
+H21))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i 
+(le_O_n i)))) x7 H19)))) H18)) (\lambda (H18: (ex2 T (\lambda (t3: T).(eq T 
 x7 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s 
 (Flat Appl) (s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7 
 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat 
 Appl) (s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
 (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda 
-(u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 
-c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 
+(u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 
 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) 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 (_: 
@@ -2514,46 +2301,46 @@ b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) 
 (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 u1 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))))))))) 
-(\lambda (x8: T).(\lambda (H24: (eq T x7 (THead (Flat Appl) (lift (S O) O x3) 
-x8))).(\lambda (H25: (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
+z2))))))))) (\lambda (x8: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) (lift 
+(S O) O x3) x8))).(\lambda (H20: (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 
 x8)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O x3) x8) (\lambda (t3: 
 T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
-x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c 
-u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) 
+x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) 
 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
 T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: 
 T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T 
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 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 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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)))))))))) (or3_intro2 (ex3_2 T T 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T 
 (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat 
 Appl) (lift (S O) O x3) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
 (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) 
 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
+(CHead c0 (Bind b) u0) 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 
@@ -2562,10 +2349,10 @@ T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
 (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) (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 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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_intro B T T T T T 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro 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 
@@ -2574,81 +2361,81 @@ B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
 (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) 
 (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 u1 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))))))) 
-x0 x1 x2 x8 x3 x6 H12 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T 
-(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))) (pr2_free c u1 
-x3 H15) (pr2_delta c d u i H8 x1 x4 H16 x6 H21) (pr2_delta (CHead c (Bind x0) 
-x6) d u (S i) (getl_clear_bind x0 (CHead c (Bind x0) x6) c x6 (clear_bind x0 
-c x6) (CHead d (Bind Abbr) u) i H8) x2 x5 H17 x8 H25))) x7 H24)))) H23)) 
-(\lambda (H23: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) 
-i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s 
-(Flat Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda 
-(_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) 
-(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 
-x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
-x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) 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 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T (THead (Bind x0) x6 x7) (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 u1 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))))))))) (\lambda (x8: T).(\lambda (x9: 
-T).(\lambda (H24: (eq T x7 (THead (Flat Appl) x8 x9))).(\lambda (H25: (subst0 
-(s (Bind x0) i) u (lift (S O) O x3) x8)).(\lambda (H26: (subst0 (s (Flat 
-Appl) (s (Bind x0) i)) u x5 x9)).(eq_ind_r T (THead (Flat Appl) x8 x9) 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) 
+x0 x1 x2 x8 x3 x6 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T 
+(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))) (pr2_free c0 
+u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 
+(Bind x0) x6) d u (S i) (getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 
+(clear_bind x0 c0 x6) (CHead d (Bind Abbr) u) i H1) x2 x5 H12 x8 H20))) x7 
+H19)))) H18)) (\lambda (H18: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
+T x7 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s 
+(Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) 
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) 
+i)) u x5 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead 
+(Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) 
+x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (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 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda 
+(x9: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) x8 x9))).(\lambda (H20: 
+(subst0 (s (Bind x0) i) u (lift (S O) O x3) x8)).(\lambda (H21: (subst0 (s 
+(Flat Appl) (s (Bind x0) i)) u x5 x9)).(eq_ind_r T (THead (Flat Appl) x8 x9) 
 (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T 
 (THead (Bind x0) x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 
 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
 (t4: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T 
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 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 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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)))))))))) (ex2_ind T (\lambda (t3: 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: 
 T).(eq T x8 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) 
 i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
 (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
 (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) 
 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
+(CHead c0 (Bind b) u0) 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 
@@ -2656,26 +2443,26 @@ x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
 (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (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 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 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))))))))) (\lambda (x10: T).(\lambda 
-(H27: (eq T x8 (lift (S O) O x10))).(\lambda (H28: (subst0 (minus (s (Bind 
-x0) i) (S O)) u x3 x10)).(let H29 \def (eq_ind nat (minus (s (Bind x0) i) (S 
-O)) (\lambda (n: nat).(subst0 n u x3 x10)) H28 i (s_arith1 x0 i)) in 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x10: T).(\lambda 
+(H22: (eq T x8 (lift (S O) O x10))).(\lambda (H23: (subst0 (minus (s (Bind 
+x0) i) (S O)) u x3 x10)).(let H24 \def (eq_ind nat (minus (s (Bind x0) i) (S 
+O)) (\lambda (n: nat).(subst0 n u x3 x10)) H23 i (s_arith1 x0 i)) in 
 (eq_ind_r T (lift (S O) O x10) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: 
 T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) 
-(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq 
-T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
 x6 (THead (Flat Appl) t3 x9)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T 
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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 x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
@@ -2683,21 +2470,21 @@ B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) 
 (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 u1 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 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
 (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
 x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) 
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 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 
@@ -2706,10 +2493,10 @@ B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
 T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) 
 O x10) x9)) (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 u1 u2))))))) (\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
 y2) z1 z2)))))))) (ex6_6_intro 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 (_: 
@@ -2718,23 +2505,21 @@ b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
 (Flat Appl) (lift (S O) O x10) x9)) (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 u1 u2))))))) 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 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 (y2: T).(pr2 c0 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))))))) x0 x1 x2 x9 x10 x6 H12 (refl_equal T 
+(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x9 x10 x6 H7 (refl_equal T 
 (THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) 
-(lift (S O) O x10) x9))) (pr2_delta c d u i H8 u1 x3 H15 x10 H29) (pr2_delta 
-c d u i H8 x1 x4 H16 x6 H21) (pr2_delta (CHead c (Bind x0) x6) d u (S i) 
-(getl_clear_bind x0 (CHead c (Bind x0) x6) c x6 (clear_bind x0 c x6) (CHead d 
-(Bind Abbr) u) i H8) x2 x5 H17 x9 H26))) x8 H27))))) (subst0_gen_lift_ge u x3 
-x8 (s (Bind x0) i) (S O) O H25 (le_n_S O i (le_O_n i)))) x7 H24)))))) H23)) 
-(subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 x7 (s (Bind x0) i) H22)) 
-x H20)))))) H19)) (subst0_gen_head (Bind x0) u x4 (THead (Flat Appl) (lift (S 
-O) O x3) x5) x i H18)) t1 H13)))))))))))))) H11)) (pr0_gen_appl u1 t1 t2 
-H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Flat Appl) u1 t1) H6))) 
-c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) 
-(refl_equal T (THead (Flat Appl) u1 t1)) (refl_equal T x))))))).
+(lift (S O) O x10) x9))) (pr2_delta c0 d u i H1 u1 x3 H10 x10 H24) (pr2_delta 
+c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 (Bind x0) x6) d u (S i) 
+(getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 (clear_bind x0 c0 x6) 
+(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x9 H21))) x8 H22))))) 
+(subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i (le_O_n 
+i)))) x7 H19)))))) H18)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 
+x7 (s (Bind x0) i) H17)) t H15)))))) H14)) (subst0_gen_head (Bind x0) u x4 
+(THead (Flat Appl) (lift (S O) O x3) x5) t i H13)) t1 H8)))))))))))))) H6)) 
+(pr0_gen_appl u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
 
 theorem pr2_gen_abbr:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
@@ -2749,591 +2534,542 @@ T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u:
 T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
 \def
  \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(let H0 \def (match H in pr2 return 
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t 
-t0)).((eq C c0 c) \to ((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).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind 
-b) u) t1 t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead 
-c (Bind Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 
-(CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) 
-(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) 
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O 
-x)))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 
-c)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T t2 
-x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Abbr) u1 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 (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) 
-(\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: 
-T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: 
-T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c 
-(Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c 
-(Bind b) u) t1 (lift (S O) O x))))))))) (\lambda (H4: (eq T t0 (THead (Bind 
-Abbr) u1 t1))).(eq_ind T (THead (Bind Abbr) u1 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 (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) 
-(\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: 
-T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: 
-T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c 
-(Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c 
-(Bind b) u) t1 (lift (S O) O x)))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x 
-(\lambda (t: T).((pr0 (THead (Bind Abbr) u1 t1) 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).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 
-t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind 
-Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c 
-(Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda 
-(_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))) 
-(\lambda (H6: (pr0 (THead (Bind Abbr) u1 t1) x)).(or_ind (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)) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: 
-T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T 
-T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) 
-(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: 
-T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda (H7: (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)))))))).(ex3_2_ind 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)))))) (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T 
-(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 
-t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) 
-u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda 
-(x0: T).(\lambda (x1: T).(\lambda (H8: (eq T x (THead (Bind Abbr) x0 
-x1))).(\lambda (H9: (pr0 u1 x0)).(\lambda (H_x: (or (pr0 t1 x1) (ex2 T 
-(\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O x0 y x1))))).(or_ind 
-(pr0 t1 x1) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O x0 y 
-x1))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind 
-Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind 
-b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead 
-c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 
-(CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) 
-(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) 
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O 
-x))))) (\lambda (H10: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) 
-(\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
-(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 
-(CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: 
-T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda 
-(_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: 
-T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) 
-z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 
-(lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T 
-(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 
-t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) 
-u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O (THead (Bind 
-Abbr) x0 x1))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-(THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T 
-(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 
-t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) 
-u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))) x0 x1 
-(refl_equal T (THead (Bind Abbr) x0 x1)) (pr2_free c u1 x0 H9) (or3_intro0 
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 x1))) (ex2 T 
+(H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 
+t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T 
 (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 
-x1))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) 
+t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) 
 u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z x1)))) (\lambda (b: 
-B).(\lambda (u: T).(pr2_free (CHead c (Bind b) u) t1 x1 H10)))))) x H8)) 
-(\lambda (H_x0: (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O 
-x0 y x1)))).(ex2_ind T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O 
-x0 y x1)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead 
-(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 
-(CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: 
-T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda 
-(_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: 
-T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) 
-z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 
-(lift (S O) O x))))) (\lambda (x2: T).(\lambda (H10: (pr0 t1 x2)).(\lambda 
-(H11: (subst0 O x0 x2 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: 
-T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind 
-Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind 
+T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))) 
+(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda 
+(t: T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 
+t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind 
+Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 
+(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) 
+(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t2)))))))) 
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O 
+t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: 
+(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(let H3 \def 
+(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abbr) u1 t1) H2) in 
+(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2)) (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind 
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+(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T 
+(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 
+t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) 
+u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2)))) 
+(\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 (lift (S 
+O) O t2) H4))))) (pr0_gen_abbr u1 t1 t2 H3)))))))) (\lambda (c0: C).(\lambda 
+(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d 
+(Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 
+t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0 
+(THead (Bind Abbr) u1 t1))).(let H5 \def (eq_ind T t0 (\lambda (t3: T).(pr0 
+t3 t2)) H2 (THead (Bind Abbr) u1 t1) H4) in (or_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (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 (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 
+t3))))))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: 
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: 
+T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) 
+(ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 
+y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda 
+(z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (H6: 
 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 t2)) (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind 
-b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 
-(CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: 
-T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 
-y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z 
-t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 
-(lift (S O) O x))))) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T t2 (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)))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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)))))) (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T 
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) 
-t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) 
-u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Bind Abbr) 
-x0 x1))).(\lambda (H13: (pr0 u1 x0)).(\lambda (H_x: (or (pr0 t1 x1) (ex2 T 
-(\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O x0 y x1))))).(or_ind 
-(pr0 t1 x1) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O x0 y 
-x1))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind 
-Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind 
-b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 
-(CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: 
-T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 
-y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z 
-t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 
-(lift (S O) O x))))) (\lambda (H14: (pr0 t1 x1)).(let H15 \def (eq_ind T t2 
-(\lambda (t3: T).(subst0 i u t3 x)) H10 (THead (Bind Abbr) x0 x1) H12) in 
-(or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abbr) u2 x1))) (\lambda 
-(u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind 
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t3)))))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (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 (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 
+t3)))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
+(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda 
+(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: 
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
+c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead 
+c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H7: (eq T t2 (THead (Bind Abbr) x0 x1))).(\lambda (H8: (pr0 u1 
+x0)).(\lambda (H_x: (or (pr0 t1 x1) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O x0 y0 x1))))).(or_ind (pr0 t1 x1) (ex2 T (\lambda 
+(y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0 x1))) (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
+t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 
+(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 
+(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z 
+t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
+(lift (S O) O t))))) (\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 
+(\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Bind Abbr) x0 x1) H7) in 
+(or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead (Bind Abbr) u2 x1))) (\lambda 
+(u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind 
 Abbr) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))) (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) 
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) 
 (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
 T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3)))) (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 
-t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c 
-(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 
-(CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) 
-(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O 
-x))))) (\lambda (H16: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abbr) u2 
-x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq 
-T x (THead (Bind Abbr) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind 
-b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 
-(CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: 
-T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 
-y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z 
-t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 
-(lift (S O) O x))))) (\lambda (x2: T).(\lambda (H17: (eq T x (THead (Bind 
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-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 
-t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c 
-(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 
-(CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) 
-(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O 
-x)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind 
-Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind 
-b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 
-(CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: 
-T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 
-y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z 
-t3))))))) x2 x1 H17 (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (or3_intro0 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 x1))) (ex2 T 
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) 
-t1 x1))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) 
-u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z x1)))) (\lambda (b: 
-B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) t1 x1 H14))))))))) H16)) 
-(\lambda (H16: (ex2 T (\lambda (t3: T).(eq T x (THead (Bind Abbr) x0 t3))) 
-(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3)))).(ex2_ind T (\lambda 
-(t3: T).(eq T x (THead (Bind Abbr) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind 
-Abbr) i) u x1 t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x 
-(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
+t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 
+(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 
+(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z 
+t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
+(lift (S O) O t))))) (\lambda (H11: (ex2 T (\lambda (u2: T).(eq T t (THead 
+(Bind Abbr) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T 
+(\lambda (u2: T).(eq T t (THead (Bind Abbr) u2 x1))) (\lambda (u2: T).(subst0 
+i u x0 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
+(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
 (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda 
-(u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: 
-T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: 
-T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c 
-(Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c 
-(Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x2: T).(\lambda (H17: (eq T x 
-(THead (Bind Abbr) x0 x2))).(\lambda (H18: (subst0 (s (Bind Abbr) i) u x1 
-x2)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead 
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+(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda 
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+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
+c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead 
+c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x2: T).(\lambda (H12: (eq T 
+t (THead (Bind Abbr) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 
+(Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: 
+T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: 
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
+c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead 
+c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T 
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) 
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x2 x1 H12 
+(pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (or3_intro0 (\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 x1))) (ex2 T (\lambda (u0: T).(pr0 u1 
+u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x1))) (ex3_2 T T 
+(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) 
+(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: 
+T).(pr2 (CHead c0 (Bind Abbr) u1) z x1)))) (\lambda (b: B).(\lambda (u0: 
+T).(pr2_free (CHead c0 (Bind b) u0) t1 x1 H9))))))))) H11)) (\lambda (H11: 
+(ex2 T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x0 t3))) (\lambda (t3: 
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+(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z 
+t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
+(lift (S O) O t))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H13: (eq T t 
+(THead (Bind Abbr) x3 x4))).(\lambda (H14: (subst0 i u x0 x3)).(\lambda (H15: 
 (subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t3: T).(subst0 O u1 
 x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T 
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) 
-t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) 
-u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) 
-(\lambda (x5: T).(\lambda (H21: (subst0 O u1 x2 x5)).(\lambda (H22: (pr0 x5 
-x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead 
-(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T 
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) 
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: 
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) 
+(\lambda (x5: T).(\lambda (H16: (subst0 O u1 x2 x5)).(\lambda (H17: (pr0 x5 
+x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
+(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
 (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda 
-(u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: 
-T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: 
-T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c 
-(Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c 
-(Bind b) u0) t1 (lift (S O) O x)))) (ex3_2_intro T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T 
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) 
-t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) 
-u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))) x3 x4 H18 
-(pr2_delta c d u i H8 u1 x0 H13 x3 H19) (or3_intro2 (\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1 
-u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 x4))) (ex3_2 T T 
-(\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) 
-(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: 
-T).(pr2 (CHead c (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y: 
-T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: 
-T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c 
-(Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O 
-(getl_refl Abbr c u1) t1 x2 H14 x5 H21) H22 (pr2_delta (CHead c (Bind Abbr) 
-u1) d u (S i) (getl_head (Bind Abbr) i c (CHead d (Bind Abbr) u) H8 u1) x1 x1 
-(pr0_refl x1) x4 H20)))))))) (pr0_subst0_back x0 x2 x1 O H15 u1 H13))))))) 
-H17)) (subst0_gen_head (Bind Abbr) u x0 x1 x i H16)))))) H_x0)) H_x)))))) 
-H11)) (\lambda (H11: (pr0 t1 (lift (S O) O t2))).(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).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 
-t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c 
-(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 
-(CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) 
-(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O 
-x)))) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u 
-(S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 (lift (S O) O 
-t2) H11 (lift (S O) O x) (subst0_lift_ge_S t2 x u i H10 O (le_O_n i))))))) 
-(pr0_gen_abbr u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead 
-(Bind Abbr) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 
-(refl_equal C c) (refl_equal T (THead (Bind Abbr) u1 t1)) (refl_equal T 
-x))))))).
+(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda 
+(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: 
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
+c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead 
+c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T 
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) 
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x3 x4 H13 
+(pr2_delta c0 d u i H1 u1 x0 H8 x3 H14) (or3_intro2 (\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1 
+u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x4))) (ex3_2 T T 
+(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) 
+(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: 
+T).(pr2 (CHead c0 (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y0: 
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
+c0 (Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O 
+(getl_refl Abbr c0 u1) t1 x2 H9 x5 H16) H17 (pr2_delta (CHead c0 (Bind Abbr) 
+u1) d u (S i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 
+x1 (pr0_refl x1) x4 H15)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))))) 
+H12)) (subst0_gen_head (Bind Abbr) u x0 x1 t i H11)))))) H_x0)) H_x)))))) 
+H6)) (\lambda (H6: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T 
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) 
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: 
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) 
+(\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) 
+(getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) 
+H6 (lift (S O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) 
+(pr0_gen_abbr u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
 
 theorem pr2_gen_void:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
@@ -3344,201 +3080,168 @@ T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
 T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
 \def
  \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr2 c (THead (Bind Void) u1 t1) x)).(let H0 \def (match H in pr2 return 
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t 
-t0)).((eq C c0 c) \to ((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).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-t1 t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 
-(lift (S O) O x)))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda 
-(H1: (eq C c0 c)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1 t1))).(\lambda 
-(H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Void) u1 
-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 (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) 
-(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(eq_ind T (THead (Bind 
-Void) u1 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 (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) 
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O 
-x)))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead 
-(Bind Void) u1 t1) 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).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 
-(CHead c (Bind b) u) t1 (lift (S O) O x))))))) (\lambda (H6: (pr0 (THead 
-(Bind Void) u1 t1) x)).(or_ind (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)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind 
-Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 
-(lift (S O) O x))))) (\lambda (H7: (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))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) 
+(H: (pr2 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 
+t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))) 
+(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda 
+(t: T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 
+t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift 
+(S O) O t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1 
+t1))).(let H3 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind 
+Void) u1 t1) H2) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
+T t2 (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 
+t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind 
+b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) 
+t1 (lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind 
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) 
 (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x 
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead 
-c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind 
-b) u) t1 (lift (S O) O x))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: 
-(eq T x (THead (Bind Void) x0 x1))).(\lambda (H9: (pr0 u1 x0)).(\lambda (H10: 
-(pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1) (\lambda (t: T).(or (ex3_2 
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) 
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O 
-t)))))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead 
-(Bind Void) x0 x1) (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O (THead (Bind Void) x0 x1))))) 
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Void) 
-x0 x1) (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c 
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind 
-Void) x0 x1)) (pr2_free c u1 x0 H9) (\lambda (b: B).(\lambda (u: T).(pr2_free 
-(CHead c (Bind b) u) t1 x1 H10))))) x H8)))))) H7)) (\lambda (H7: (pr0 t1 
-(lift (S O) O 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).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 
-(CHead c (Bind b) u) t1 (lift (S O) O x)))) (\lambda (b: B).(\lambda (u: 
-T).(pr2_free (CHead c (Bind b) u) t1 (lift (S O) O x) H7))))) (pr0_gen_void 
-u1 t1 x H6))) t2 (sym_eq T t2 x H5))) t0 (sym_eq T t0 (THead (Bind Void) u1 
-t1) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 
-H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead 
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+(THead (Bind Void) x2 x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: 
 (subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) 
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
-u0) t1 t3))))) x2 x3 H17 (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (\lambda (b: 
-B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head 
-(Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H14 x3 H19)))))))))) H16)) 
-(subst0_gen_head (Bind Void) u x0 x1 x i H15)))))))) H11)) (\lambda (H11: 
-(pr0 t1 (lift (S O) O t2))).(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).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: 
-T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (\lambda (b: B).(\lambda 
-(u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c 
-(CHead d (Bind Abbr) u) H8 u0) t1 (lift (S O) O t2) H11 (lift (S O) O x) 
-(subst0_lift_ge_S t2 x u i H10 O (le_O_n i))))))) (pr0_gen_void u1 t1 t2 
-H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Bind Void) u1 t1) H6))) 
-c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) 
-(refl_equal T (THead (Bind Void) u1 t1)) (refl_equal T x))))))).
+T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: 
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) 
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) t1 t3))))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (\lambda (b: 
+B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head 
+(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x3 H14)))))))))) H11)) 
+(subst0_gen_head (Bind Void) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0 
+t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: 
+T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: 
+T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) (\lambda (b: 
+B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head 
+(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) H6 (lift (S 
+O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) (pr0_gen_void u1 t1 
+t2 H5)))))))))))))) c y x H0))) H))))).
 
 theorem pr2_gen_lift:
  \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall 
@@ -3547,82 +3250,67 @@ theorem pr2_gen_lift:
 t2))))))))))
 \def
  \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda 
-(d: nat).(\lambda (H: (pr2 c (lift h d t1) x)).(\lambda (e: C).(\lambda (H0: 
-(drop h d c e)).(let H1 \def (match H in pr2 return (\lambda (c0: C).(\lambda 
-(t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t 
-(lift h d t1)) \to ((eq T t0 x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d 
-t2))) (\lambda (t2: T).(pr2 e t1 t2)))))))))) with [(pr2_free c0 t0 t2 H1) 
-\Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq T t0 (lift h d 
-t1))).(\lambda (H4: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (lift 
-h d t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t3: T).(eq T 
-x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))))) (\lambda (H5: (eq T t0 
-(lift h d t1))).(eq_ind T (lift h d t1) (\lambda (t: T).((eq T t2 x) \to 
-((pr0 t t2) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: 
-T).(pr2 e t1 t3)))))) (\lambda (H6: (eq T t2 x)).(eq_ind T x (\lambda (t: 
-T).((pr0 (lift h d t1) t) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) 
-(\lambda (t3: T).(pr2 e t1 t3))))) (\lambda (H7: (pr0 (lift h d t1) 
-x)).(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr0 
-t1 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 
-e t1 t3))) (\lambda (x0: T).(\lambda (H8: (eq T x (lift h d x0))).(\lambda 
-(H9: (pr0 t1 x0)).(eq_ind_r T (lift h d x0) (\lambda (t: T).(ex2 T (\lambda 
-(t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))) (ex_intro2 
-T (\lambda (t3: T).(eq T (lift h d x0) (lift h d t3))) (\lambda (t3: T).(pr2 
-e t1 t3)) x0 (refl_equal T (lift h d x0)) (pr2_free e t1 x0 H9)) x H8)))) 
-(pr0_gen_lift t1 x h d H7))) t2 (sym_eq T t2 x H6))) t0 (sym_eq T t0 (lift h 
-d t1) H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d0 u i H1 t0 
-t2 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t0 
-(lift h d t1))).(\lambda (H6: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T 
-t0 (lift h d t1)) \to ((eq T t x) \to ((getl i c1 (CHead d0 (Bind Abbr) u)) 
-\to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (ex2 T (\lambda (t3: T).(eq T x 
-(lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))))))) (\lambda (H7: (eq T t0 
-(lift h d t1))).(eq_ind T (lift h d t1) (\lambda (t3: T).((eq T t x) \to 
-((getl i c (CHead d0 (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) 
-\to (ex2 T (\lambda (t4: T).(eq T x (lift h d t4))) (\lambda (t4: T).(pr2 e 
-t1 t4)))))))) (\lambda (H8: (eq T t x)).(eq_ind T x (\lambda (t3: T).((getl i 
-c (CHead d0 (Bind Abbr) u)) \to ((pr0 (lift h d t1) t2) \to ((subst0 i u t2 
-t3) \to (ex2 T (\lambda (t4: T).(eq T x (lift h d t4))) (\lambda (t4: T).(pr2 
-e t1 t4))))))) (\lambda (H9: (getl i c (CHead d0 (Bind Abbr) u))).(\lambda 
-(H10: (pr0 (lift h d t1) t2)).(\lambda (H11: (subst0 i u t2 x)).(ex2_ind T 
+(d: nat).(\lambda (H: (pr2 c (lift h d t1) x)).(insert_eq T (lift h d t1) 
+(\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e) 
+\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e 
+t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: 
+C).(\lambda (t: T).(\lambda (t0: T).((eq T t (lift h d t1)) \to (\forall (e: 
+C).((drop h d c0 e) \to (ex2 T (\lambda (t2: T).(eq T t0 (lift h d t2))) 
+(\lambda (t2: T).(pr2 e t1 t2))))))))) (\lambda (c0: C).(\lambda (t0: 
+T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (lift h 
+d t1))).(\lambda (e: C).(\lambda (_: (drop h d c0 e)).(let H4 \def (eq_ind T 
+t0 (\lambda (t: T).(pr0 t t2)) H1 (lift h d t1) H2) in (ex2_ind T (\lambda 
+(t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 T 
+(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) 
+(\lambda (x0: T).(\lambda (H5: (eq T t2 (lift h d x0))).(\lambda (H6: (pr0 t1 
+x0)).(eq_ind_r T (lift h d x0) (\lambda (t: T).(ex2 T (\lambda (t3: T).(eq T 
+t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))) (ex_intro2 T (\lambda 
+(t3: T).(eq T (lift h d x0) (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) 
+x0 (refl_equal T (lift h d x0)) (pr2_free e t1 x0 H6)) t2 H5)))) 
+(pr0_gen_lift t1 t2 h d H4)))))))))) (\lambda (c0: C).(\lambda (d0: 
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d0 (Bind 
+Abbr) u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 
+t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0 
+(lift h d t1))).(\lambda (e: C).(\lambda (H5: (drop h d c0 e)).(let H6 \def 
+(eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (lift h d t1) H4) in (ex2_ind T 
 (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 
-T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) 
-(\lambda (x0: T).(\lambda (H12: (eq T t2 (lift h d x0))).(\lambda (H13: (pr0 
-t1 x0)).(let H14 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 x)) H11 
-(lift h d x0) H12) in (lt_le_e i d (ex2 T (\lambda (t3: T).(eq T x (lift h d 
-t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H15: (lt i d)).(let H16 \def 
-(eq_ind nat d (\lambda (n: nat).(drop h n c e)) H0 (S (plus i (minus d (S 
-i)))) (lt_plus_minus i d H15)) in (let H17 \def (eq_ind nat d (\lambda (n: 
-nat).(subst0 i u (lift h n x0) x)) H14 (S (plus i (minus d (S i)))) 
-(lt_plus_minus i d H15)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: 
+T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) 
+(\lambda (x0: T).(\lambda (H7: (eq T t2 (lift h d x0))).(\lambda (H8: (pr0 t1 
+x0)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (lift h 
+d x0) H7) in (lt_le_e i d (ex2 T (\lambda (t3: T).(eq T t (lift h d t3))) 
+(\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H10: (lt i d)).(let H11 \def 
+(eq_ind nat d (\lambda (n: nat).(subst0 i u (lift h n x0) t)) H9 (S (plus i 
+(minus d (S i)))) (lt_plus_minus i d H10)) in (let H12 \def (eq_ind nat d 
+(\lambda (n: nat).(drop h n c0 e)) H5 (S (plus i (minus d (S i)))) 
+(lt_plus_minus i d H10)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: 
 C).(eq T u (lift h (minus d (S i)) v)))) (\lambda (v: T).(\lambda (e0: 
 C).(getl i e (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: 
-C).(drop h (minus d (S i)) d0 e0))) (ex2 T (\lambda (t3: T).(eq T x (lift h d 
+C).(drop h (minus d (S i)) d0 e0))) (ex2 T (\lambda (t3: T).(eq T t (lift h d 
 t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x1: T).(\lambda (x2: 
-C).(\lambda (H18: (eq T u (lift h (minus d (S i)) x1))).(\lambda (H19: (getl 
+C).(\lambda (H13: (eq T u (lift h (minus d (S i)) x1))).(\lambda (H14: (getl 
 i e (CHead x2 (Bind Abbr) x1))).(\lambda (_: (drop h (minus d (S i)) d0 
-x2)).(let H21 \def (eq_ind T u (\lambda (t3: T).(subst0 i t3 (lift h (S (plus 
-i (minus d (S i)))) x0) x)) H17 (lift h (minus d (S i)) x1) H18) in (ex2_ind 
-T (\lambda (t3: T).(eq T x (lift h (S (plus i (minus d (S i)))) t3))) 
-(\lambda (t3: T).(subst0 i x1 x0 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h 
-d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x3: T).(\lambda (H22: (eq 
-T x (lift h (S (plus i (minus d (S i)))) x3))).(\lambda (H23: (subst0 i x1 x0 
-x3)).(let H24 \def (eq_ind_r nat (S (plus i (minus d (S i)))) (\lambda (n: 
-nat).(eq T x (lift h n x3))) H22 d (lt_plus_minus i d H15)) in (ex_intro2 T 
-(\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x3 
-H24 (pr2_delta e x2 x1 i H19 t1 x0 H13 x3 H23)))))) (subst0_gen_lift_lt x1 x0 
-x i h (minus d (S i)) H21)))))))) (getl_drop_conf_lt Abbr c d0 u i H9 e h 
-(minus d (S i)) H16))))) (\lambda (H15: (le d i)).(lt_le_e i (plus d h) (ex2 
-T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) 
-(\lambda (H16: (lt i (plus d h))).(subst0_gen_lift_false x0 u x h d i H15 H16 
-H14 (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e 
-t1 t3))))) (\lambda (H16: (le (plus d h) i)).(ex2_ind T (\lambda (t3: T).(eq 
-T x (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u x0 t3)) (ex2 T 
-(\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) 
-(\lambda (x1: T).(\lambda (H17: (eq T x (lift h d x1))).(\lambda (H18: 
-(subst0 (minus i h) u x0 x1)).(ex_intro2 T (\lambda (t3: T).(eq T x (lift h d 
-t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H17 (pr2_delta e d0 u (minus i h) 
-(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H9 e h d H0 H16) t1 x0 H13 x1 
-H18))))) (subst0_gen_lift_ge u x0 x i h d H14 H16)))))))))) (pr0_gen_lift t1 
-t2 h d H10))))) t (sym_eq T t x H8))) t0 (sym_eq T t0 (lift h d t1) H7))) c0 
-(sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c) (refl_equal T 
-(lift h d t1)) (refl_equal T x)))))))))).
+x2)).(let H16 \def (eq_ind T u (\lambda (t3: T).(subst0 i t3 (lift h (S (plus 
+i (minus d (S i)))) x0) t)) H11 (lift h (minus d (S i)) x1) H13) in (ex2_ind 
+T (\lambda (t3: T).(eq T t (lift h (S (plus i (minus d (S i)))) t3))) 
+(\lambda (t3: T).(subst0 i x1 x0 t3)) (ex2 T (\lambda (t3: T).(eq T t (lift h 
+d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x3: T).(\lambda (H17: (eq 
+T t (lift h (S (plus i (minus d (S i)))) x3))).(\lambda (H18: (subst0 i x1 x0 
+x3)).(let H19 \def (eq_ind_r nat (S (plus i (minus d (S i)))) (\lambda (n: 
+nat).(eq T t (lift h n x3))) H17 d (lt_plus_minus i d H10)) in (ex_intro2 T 
+(\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x3 
+H19 (pr2_delta e x2 x1 i H14 t1 x0 H8 x3 H18)))))) (subst0_gen_lift_lt x1 x0 
+t i h (minus d (S i)) H16)))))))) (getl_drop_conf_lt Abbr c0 d0 u i H1 e h 
+(minus d (S i)) H12))))) (\lambda (H10: (le d i)).(lt_le_e i (plus d h) (ex2 
+T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) 
+(\lambda (H11: (lt i (plus d h))).(subst0_gen_lift_false x0 u t h d i H10 H11 
+H9 (ex2 T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 
+t3))))) (\lambda (H11: (le (plus d h) i)).(ex2_ind T (\lambda (t3: T).(eq T t 
+(lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u x0 t3)) (ex2 T 
+(\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) 
+(\lambda (x1: T).(\lambda (H12: (eq T t (lift h d x1))).(\lambda (H13: 
+(subst0 (minus i h) u x0 x1)).(ex_intro2 T (\lambda (t3: T).(eq T t (lift h d 
+t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H12 (pr2_delta e d0 u (minus i h) 
+(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c0 H1 e h d H5 H11) t1 x0 H8 x1 
+H13))))) (subst0_gen_lift_ge u x0 t i h d H9 H11)))))))))) (pr0_gen_lift t1 
+t2 h d H6)))))))))))))))) c y x H0))) H)))))).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/pr2.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/pr2.ma
index ba7a54835..d9f206324 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/pr2.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/pr2.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr2/defs.ma".
+include "LambdaDelta-1/pr2/defs.ma".
 
-include "pr0/pr0.ma".
+include "LambdaDelta-1/pr0/pr0.ma".
 
-include "getl/props.ma".
+include "LambdaDelta-1/getl/props.ma".
 
 theorem pr2_confluence__pr2_free_free:
  \forall (c: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).((pr0 t0 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/props.ma
index 8b11644d3..74e2bf65e 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/props.ma
@@ -14,13 +14,13 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr2/defs.ma".
+include "LambdaDelta-1/pr2/defs.ma".
 
-include "pr0/props.ma".
+include "LambdaDelta-1/pr0/props.ma".
 
-include "getl/drop.ma".
+include "LambdaDelta-1/getl/drop.ma".
 
-include "getl/clear.ma".
+include "LambdaDelta-1/getl/clear.ma".
 
 theorem pr2_thin_dx:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
@@ -62,160 +62,120 @@ theorem pr2_head_2:
 t2)))))))
 \def
  \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(k: K).(K_ind (\lambda (k0: K).((pr2 (CHead c k0 u) t1 t2) \to (pr2 c (THead 
-k0 u t1) (THead k0 u t2)))) (\lambda (b: B).(\lambda (H: (pr2 (CHead c (Bind 
-b) u) t1 t2)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda 
-(t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 (CHead c (Bind 
-b) u)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (THead (Bind b) u t1) 
-(THead (Bind b) u t2))))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow 
-(\lambda (H1: (eq C c0 (CHead c (Bind b) u))).(\lambda (H2: (eq T t0 
-t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (_: 
-C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (THead (Bind 
-b) u t1) (THead (Bind b) u t2)))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1 
-(\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (THead (Bind b) u 
-t1) (THead (Bind b) u t2))))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 
-(\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) 
-u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead (Bind b) u t1) (THead 
-(Bind b) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6 (Bind b)))) t3 (sym_eq T 
-t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Bind b) u) H1) 
-H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t H2) \Rightarrow (\lambda 
-(H3: (eq C c0 (CHead c (Bind b) u))).(\lambda (H4: (eq T t0 t1)).(\lambda 
-(H5: (eq T t t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (c1: C).((eq T t0 
-t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u0)) \to ((pr0 t0 
-t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) 
-u t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T 
-t t2) \to ((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0 
-t4 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind 
-b) u t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4: 
-T).((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t3) 
-\to ((subst0 i u0 t3 t4) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u 
-t2)))))) (\lambda (H8: (getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) 
-u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u0 t3 t2)).(nat_ind 
-(\lambda (n: nat).((getl n (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to 
-((subst0 n u0 t3 t2) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u 
-t2))))) (\lambda (H11: (getl O (CHead c (Bind b) u) (CHead d (Bind Abbr) 
-u0))).(\lambda (H12: (subst0 O u0 t3 t2)).(let H13 \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) u0) (CHead c (Bind b) 
-u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind 
-b) u) (CHead d (Bind Abbr) u0) H11))) in ((let H14 \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 d 
+(k: K).(\lambda (H: (pr2 (CHead c k u) t1 t2)).(insert_eq C (CHead c k u) 
+(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead k u t1) (THead 
+k u t2))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: 
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c k u)) \to (pr2 c 
+(THead k u t) (THead k u t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda 
+(t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c k 
+u))).(pr2_free c (THead k u t3) (THead k u t4) (pr0_comp u u (pr0_refl u) t3 
+t4 H1 k))))))) (K_ind (\lambda (k0: K).(\forall (c0: C).(\forall (d: 
+C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Abbr) u0)) 
+\to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t: 
+T).((subst0 i u0 t4 t) \to ((eq C c0 (CHead c k0 u)) \to (pr2 c (THead k0 u 
+t3) (THead k0 u t)))))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u0: T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 
+(CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) 
+\to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Bind b) u)) 
+\to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u t)))))))))) (\lambda (H1: 
+(getl O c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 
+t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) u))).(let H5 \def (eq_ind C c0 
+(\lambda (c1: C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Bind b) 
+u) H4) in (let H6 \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) u0) (CHead c (Bind b) u) (clear_gen_bind b c 
+(CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind 
+Abbr) u0) 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 _ k0 _) 
+\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) 
+(CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u 
+(getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in ((let H8 
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) 
+with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d 
 (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) 
-u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H11))) in 
-((let H15 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: 
-C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t4) \Rightarrow t4])) 
-(CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d 
-(Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) 
-H11))) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H18 
-\def (eq_ind T u0 (\lambda (t4: T).(subst0 O t4 t3 t2)) H12 u H15) in (eq_ind 
-B Abbr (\lambda (b0: B).(pr2 c (THead (Bind b0) u t1) (THead (Bind b0) u 
-t2))) (pr2_free c (THead (Bind Abbr) u t1) (THead (Bind Abbr) u t2) 
-(pr0_delta u u (pr0_refl u) t1 t3 H9 t2 H18)) b H16))))) H14)) H13)))) 
-(\lambda (i0: nat).(\lambda (_: (((getl i0 (CHead c (Bind b) u) (CHead d 
-(Bind Abbr) u0)) \to ((subst0 i0 u0 t3 t2) \to (pr2 c (THead (Bind b) u t1) 
-(THead (Bind b) u t2)))))).(\lambda (H11: (getl (S i0) (CHead c (Bind b) u) 
-(CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 (S i0) u0 t3 
-t2)).(pr2_delta c d u0 (r (Bind b) i0) (getl_gen_S (Bind b) c (CHead d (Bind 
-Abbr) u0) u i0 H11) (THead (Bind b) u t1) (THead (Bind b) u t3) (pr0_comp u u 
-(pr0_refl u) t1 t3 H9 (Bind b)) (THead (Bind b) u t2) (subst0_snd (Bind b) u0 
-t2 t3 (r (Bind b) i0) H12 u)))))) i H8 H10)))) t (sym_eq T t t2 H7))) t0 
-(sym_eq T t0 t1 H6))) c0 (sym_eq C c0 (CHead c (Bind b) u) H3) H4 H5 H0 H1 
-H2))))]) in (H0 (refl_equal C (CHead c (Bind b) u)) (refl_equal T t1) 
-(refl_equal T t2))))) (\lambda (f: F).(\lambda (H: (pr2 (CHead c (Flat f) u) 
-t1 t2)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 (CHead c (Flat f) 
-u)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (THead (Flat f) u t1) 
-(THead (Flat f) u t2))))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow 
-(\lambda (H1: (eq C c0 (CHead c (Flat f) u))).(\lambda (H2: (eq T t0 
-t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Flat f) u) (\lambda (_: 
-C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (THead (Flat 
-f) u t1) (THead (Flat f) u t2)))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1 
-(\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (THead (Flat f) u 
-t1) (THead (Flat f) u t2))))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 
-(\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) 
-u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead (Flat f) u t1) (THead 
-(Flat f) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6 (Flat f)))) t3 (sym_eq T 
-t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Flat f) u) H1) 
-H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t H2) \Rightarrow (\lambda 
-(H3: (eq C c0 (CHead c (Flat f) u))).(\lambda (H4: (eq T t0 t1)).(\lambda 
-(H5: (eq T t t2)).(eq_ind C (CHead c (Flat f) u) (\lambda (c1: C).((eq T t0 
-t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u0)) \to ((pr0 t0 
-t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) 
-u t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T 
-t t2) \to ((getl i (CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to ((pr0 
-t4 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Flat f) u t1) (THead (Flat 
-f) u t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4: 
-T).((getl i (CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t3) 
-\to ((subst0 i u0 t3 t4) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u 
-t2)))))) (\lambda (H8: (getl i (CHead c (Flat f) u) (CHead d (Bind Abbr) 
-u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u0 t3 t2)).(nat_ind 
-(\lambda (n: nat).((getl n (CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to 
-((subst0 n u0 t3 t2) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u 
-t2))))) (\lambda (H11: (getl O (CHead c (Flat f) u) (CHead d (Bind Abbr) 
-u0))).(\lambda (H12: (subst0 O u0 t3 t2)).(pr2_delta c d u0 O (getl_intro O c 
-(CHead d (Bind Abbr) u0) c (drop_refl c) (clear_gen_flat f c (CHead d (Bind 
-Abbr) u0) u (getl_gen_O (CHead c (Flat f) u) (CHead d (Bind Abbr) u0) H11))) 
-(THead (Flat f) u t1) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t1 t3 
-H9 (Flat f)) (THead (Flat f) u t2) (subst0_snd (Flat f) u0 t2 t3 O H12 u)))) 
-(\lambda (i0: nat).(\lambda (_: (((getl i0 (CHead c (Flat f) u) (CHead d 
-(Bind Abbr) u0)) \to ((subst0 i0 u0 t3 t2) \to (pr2 c (THead (Flat f) u t1) 
-(THead (Flat f) u t2)))))).(\lambda (H11: (getl (S i0) (CHead c (Flat f) u) 
-(CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 (S i0) u0 t3 
-t2)).(pr2_delta c d u0 (r (Flat f) i0) (getl_gen_S (Flat f) c (CHead d (Bind 
-Abbr) u0) u i0 H11) (THead (Flat f) u t1) (THead (Flat f) u t3) (pr0_comp u u 
-(pr0_refl u) t1 t3 H9 (Flat f)) (THead (Flat f) u t2) (subst0_snd (Flat f) u0 
-t2 t3 (r (Flat f) i0) H12 u)))))) i H8 H10)))) t (sym_eq T t t2 H7))) t0 
-(sym_eq T t0 t1 H6))) c0 (sym_eq C c0 (CHead c (Flat f) u) H3) H4 H5 H0 H1 
-H2))))]) in (H0 (refl_equal C (CHead c (Flat f) u)) (refl_equal T t1) 
-(refl_equal T t2))))) k))))).
+u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in 
+(\lambda (H9: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H11 \def (eq_ind T 
+u0 (\lambda (t0: T).(subst0 O t0 t4 t)) H3 u H8) in (eq_ind B Abbr (\lambda 
+(b0: B).(pr2 c (THead (Bind b0) u t3) (THead (Bind b0) u t))) (pr2_free c 
+(THead (Bind Abbr) u t3) (THead (Bind Abbr) u t) (pr0_delta u u (pr0_refl u) 
+t3 t4 H2 t H11)) b H9))))) H7)) H6)))))))))) (\lambda (n: nat).(\lambda (H1: 
+(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: 
+T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead 
+c (Bind b) u)) \to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u 
+t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda 
+(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Bind b) 
+u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind 
+Abbr) u0))) H2 (CHead c (Bind b) u) H5) in (let H7 \def (eq_ind C c0 (\lambda 
+(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall 
+(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1 
+(CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u t5) (THead (Bind b) u 
+t0)))))))))) H1 (CHead c (Bind b) u) H5) in (pr2_delta c d u0 (r (Bind b) n) 
+(getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H6) (THead (Bind b) u t3) 
+(THead (Bind b) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Bind b)) (THead 
+(Bind b) u t) (subst0_snd (Bind b) u0 t t4 (r (Bind b) n) H4 u))))))))))))) 
+i)))))) (\lambda (f: F).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: 
+T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 (CHead d (Bind 
+Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall 
+(t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Flat f) u)) \to (pr2 c 
+(THead (Flat f) u t3) (THead (Flat f) u t)))))))))) (\lambda (H1: (getl O c0 
+(CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: 
+(pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 t)).(\lambda (H4: 
+(eq C c0 (CHead c (Flat f) u))).(let H5 \def (eq_ind C c0 (\lambda (c1: 
+C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Flat f) u) H4) in 
+(pr2_delta c d u0 O (getl_intro O c (CHead d (Bind Abbr) u0) c (drop_refl c) 
+(clear_gen_flat f c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Flat f) 
+u) (CHead d (Bind Abbr) u0) H5))) (THead (Flat f) u t3) (THead (Flat f) u t4) 
+(pr0_comp u u (pr0_refl u) t3 t4 H2 (Flat f)) (THead (Flat f) u t) 
+(subst0_snd (Flat f) u0 t t4 O H3 u)))))))))) (\lambda (n: nat).(\lambda (H1: 
+(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: 
+T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead 
+c (Flat f) u)) \to (pr2 c (THead (Flat f) u t3) (THead (Flat f) u 
+t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda 
+(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Flat f) 
+u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind 
+Abbr) u0))) H2 (CHead c (Flat f) u) H5) in (let H7 \def (eq_ind C c0 (\lambda 
+(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall 
+(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1 
+(CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u t5) (THead (Flat f) u 
+t0)))))))))) H1 (CHead c (Flat f) u) H5) in (pr2_delta c d u0 (r (Flat f) n) 
+(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H6) (THead (Flat f) u t3) 
+(THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Flat f)) (THead 
+(Flat f) u t) (subst0_snd (Flat f) u0 t t4 (r (Flat f) n) H4 u))))))))))))) 
+i)))))) k) y t1 t2 H0))) H)))))).
 
 theorem clear_pr2_trans:
  \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to 
 (\forall (c1: C).((clear c1 c2) \to (pr2 c1 t1 t2))))))
 \def
  \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c2 t1 
-t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def (match H in 
-pr2 return (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 
-c t t0)).((eq C c c2) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c1 t1 
-t2)))))))) with [(pr2_free c t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c 
-c2)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C c2 
-(\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c1 
-t1 t2))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 
-t2) \to ((pr0 t t3) \to (pr2 c1 t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind 
-T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c1 t1 t2))) (\lambda (H7: (pr0 t1 
-t2)).(pr2_free c1 t1 t2 H7)) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 
-H5))) c (sym_eq C c c2 H2) H3 H4 H1)))) | (pr2_delta c d u i H1 t0 t3 H2 t 
-H3) \Rightarrow (\lambda (H4: (eq C c c2)).(\lambda (H5: (eq T t0 
-t1)).(\lambda (H6: (eq T t t2)).(eq_ind C c2 (\lambda (c0: C).((eq T t0 t1) 
-\to ((eq T t t2) \to ((getl i c0 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) 
-\to ((subst0 i u t3 t) \to (pr2 c1 t1 t2))))))) (\lambda (H7: (eq T t0 
-t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i c2 (CHead d 
-(Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c1 t1 
-t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i c2 
-(CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c1 
-t1 t2))))) (\lambda (H9: (getl i c2 (CHead d (Bind Abbr) u))).(\lambda (H10: 
-(pr0 t1 t3)).(\lambda (H11: (subst0 i u t3 t2)).(pr2_delta c1 d u i 
-(clear_getl_trans i c2 (CHead d (Bind Abbr) u) H9 c1 H0) t1 t3 H10 t2 H11)))) 
-t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c (sym_eq C c c2 H4) H5 H6 H1 
-H2 H3))))]) in (H1 (refl_equal C c2) (refl_equal T t1) (refl_equal T 
-t2)))))))).
+t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\forall (c1: 
+C).((clear c1 c) \to (pr2 c1 t t0)))))) (\lambda (c: C).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c1: C).(\lambda (_: 
+(clear c1 c)).(pr2_free c1 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 (c1: 
+C).(\lambda (H3: (clear c1 c)).(pr2_delta c1 d u i (clear_getl_trans i c 
+(CHead d (Bind Abbr) u) H0 c1 H3) t3 t4 H1 t H2))))))))))))) c2 t1 t2 H)))).
 
 theorem pr2_cflat:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
 (f: F).(\forall (v: T).(pr2 (CHead c (Flat f) v) t1 t2))))))
 \def
  \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
-t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (f: 
-F).(\forall (v: T).(pr2 (CHead c0 (Flat f) v) t t0)))))) (\lambda (c0: 
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (f: 
-F).(\lambda (v: T).(pr2_free (CHead c0 (Flat f) v) t3 t4 H0))))))) (\lambda 
-(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl 
-i c0 (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 
-(f: F).(\lambda (v: T).(pr2_delta (CHead c0 (Flat f) v) d u i (getl_flat c0 
-(CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))))) c t1 t2 H)))).
+t2)).(\lambda (f: F).(\lambda (v: T).(pr2_ind (\lambda (c0: C).(\lambda (t: 
+T).(\lambda (t0: T).(pr2 (CHead c0 (Flat f) v) t t0)))) (\lambda (c0: 
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free 
+(CHead c0 (Flat f) v) t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda 
+(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (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)).(pr2_delta (CHead c0 (Flat f) v) d u 
+i (getl_flat c0 (CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))) c 
+t1 t2 H)))))).
 
 theorem pr2_ctail:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
@@ -288,50 +248,36 @@ h d t1) (lift h d t2)))))))))
 \def
  \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
 (H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 e t1 
-t2)).(let H1 \def (match H0 in pr2 return (\lambda (c0: C).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 e) \to ((eq T t t1) 
-\to ((eq T t0 t2) \to (pr2 c (lift h d t1) (lift h d t2))))))))) with 
-[(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0 e)).(\lambda (H3: 
-(eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C e (\lambda (_: C).((eq T 
-t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (lift h d t1) (lift h d 
-t2)))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 
-t2) \to ((pr0 t t3) \to (pr2 c (lift h d t1) (lift h d t2))))) (\lambda (H6: 
-(eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (lift h d 
-t1) (lift h d t2)))) (\lambda (H7: (pr0 t1 t2)).(pr2_free c (lift h d t1) 
-(lift h d t2) (pr0_lift t1 t2 H7 h d))) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T 
-t0 t1 H5))) c0 (sym_eq C c0 e H2) H3 H4 H1)))) | (pr2_delta c0 d0 u i H1 t0 
-t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 e)).(\lambda (H5: (eq T t0 
-t1)).(\lambda (H6: (eq T t t2)).(eq_ind C e (\lambda (c1: C).((eq T t0 t1) 
-\to ((eq T t t2) \to ((getl i c1 (CHead d0 (Bind Abbr) u)) \to ((pr0 t0 t3) 
-\to ((subst0 i u t3 t) \to (pr2 c (lift h d t1) (lift h d t2)))))))) (\lambda 
-(H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i e 
-(CHead d0 (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c 
-(lift h d t1) (lift h d t2))))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 
-(\lambda (t4: T).((getl i e (CHead d0 (Bind Abbr) u)) \to ((pr0 t1 t3) \to 
-((subst0 i u t3 t4) \to (pr2 c (lift h d t1) (lift h d t2)))))) (\lambda (H9: 
-(getl i e (CHead d0 (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda 
-(H11: (subst0 i u t3 t2)).(lt_le_e i d (pr2 c (lift h d t1) (lift h d t2)) 
-(\lambda (H12: (lt i d)).(let H13 \def (drop_getl_trans_le i d (le_S_n i d 
-(le_S (S i) d H12)) c e h H (CHead d0 (Bind Abbr) u) H9) in (ex3_2_ind C C 
-(\lambda (e0: C).(\lambda (_: C).(drop i O c e0))) (\lambda (e0: C).(\lambda 
-(e1: C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear 
-e1 (CHead d0 (Bind Abbr) u)))) (pr2 c (lift h d t1) (lift h d t2)) (\lambda 
-(x0: C).(\lambda (x1: C).(\lambda (H14: (drop i O c x0)).(\lambda (H15: (drop 
-h (minus d i) x0 x1)).(\lambda (H16: (clear x1 (CHead d0 (Bind Abbr) 
-u))).(let H17 \def (eq_ind nat (minus d i) (\lambda (n: nat).(drop h n x0 
-x1)) H15 (S (minus d (S i))) (minus_x_Sy d i H12)) in (let H18 \def 
-(drop_clear_S x1 x0 h (minus d (S i)) H17 Abbr d0 u H16) in (ex2_ind C 
-(\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h (minus d (S i)) 
-u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c (lift h d t1) 
-(lift h d t2)) (\lambda (x: C).(\lambda (H19: (clear x0 (CHead x (Bind Abbr) 
-(lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S i)) x 
-d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead x 
-(Bind Abbr) (lift h (minus d (S i)) u)) x0 H14 H19) (lift h d t1) (lift h d 
-t3) (pr0_lift t1 t3 H10 h d) (lift h d t2) (subst0_lift_lt t3 t2 u i H11 d 
-H12 h))))) H18)))))))) H13))) (\lambda (H12: (le d i)).(pr2_delta c d0 u 
-(plus i h) (drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H9 H12) 
-(lift h d t1) (lift h d t3) (pr0_lift t1 t3 H10 h d) (lift h d t2) 
-(subst0_lift_ge t3 t2 u i h H11 d H12))))))) t (sym_eq T t t2 H8))) t0 
-(sym_eq T t0 t1 H7))) c0 (sym_eq C c0 e H4) H5 H6 H1 H2 H3))))]) in (H1 
-(refl_equal C e) (refl_equal T t1) (refl_equal T t2)))))))))).
+t2)).(insert_eq C e (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c 
+(lift h d t1) (lift h d t2))) (\lambda (y: C).(\lambda (H1: (pr2 y t1 
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 e) 
+\to (pr2 c (lift h d t) (lift h d t0)))))) (\lambda (c0: C).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 
+e)).(pr2_free c (lift h d t3) (lift h d t4) (pr0_lift t3 t4 H2 h d))))))) 
+(\lambda (c0: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H2: (getl i c0 (CHead d0 (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 e)).(let H6 \def (eq_ind C c0 (\lambda (c1: 
+C).(getl i c1 (CHead d0 (Bind Abbr) u))) H2 e H5) in (lt_le_e i d (pr2 c 
+(lift h d t3) (lift h d t)) (\lambda (H7: (lt i d)).(let H8 \def 
+(drop_getl_trans_le i d (le_S_n i d (le_S (S i) d H7)) c e h H (CHead d0 
+(Bind Abbr) u) H6) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i 
+O c e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) 
+(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d0 (Bind Abbr) u)))) (pr2 c 
+(lift h d t3) (lift h d t)) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H9: 
+(drop i O c x0)).(\lambda (H10: (drop h (minus d i) x0 x1)).(\lambda (H11: 
+(clear x1 (CHead d0 (Bind Abbr) u))).(let H12 \def (eq_ind nat (minus d i) 
+(\lambda (n: nat).(drop h n x0 x1)) H10 (S (minus d (S i))) (minus_x_Sy d i 
+H7)) in (let H13 \def (drop_clear_S x1 x0 h (minus d (S i)) H12 Abbr d0 u 
+H11) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h 
+(minus d (S i)) u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c 
+(lift h d t3) (lift h d t)) (\lambda (x: C).(\lambda (H14: (clear x0 (CHead x 
+(Bind Abbr) (lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S 
+i)) x d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead 
+x (Bind Abbr) (lift h (minus d (S i)) u)) x0 H9 H14) (lift h d t3) (lift h d 
+t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_lt t4 t u i H4 d H7 
+h))))) H13)))))))) H8))) (\lambda (H7: (le d i)).(pr2_delta c d0 u (plus i h) 
+(drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H6 H7) (lift h d t3) 
+(lift h d t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_ge t4 t u i h 
+H4 d H7)))))))))))))))) y t1 t2 H1))) H0)))))))).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/subst1.ma
index 9d230c6b1..936e655f1 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/subst1.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/subst1.ma
@@ -14,19 +14,19 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr2/defs.ma".
+include "LambdaDelta-1/pr2/defs.ma".
 
-include "pr0/subst1.ma".
+include "LambdaDelta-1/pr0/subst1.ma".
 
-include "pr0/fwd.ma".
+include "LambdaDelta-1/pr0/fwd.ma".
 
-include "csubst1/getl.ma".
+include "LambdaDelta-1/csubst1/getl.ma".
 
-include "csubst1/fwd.ma".
+include "LambdaDelta-1/csubst1/fwd.ma".
 
-include "subst1/subst1.ma".
+include "LambdaDelta-1/subst1/subst1.ma".
 
-include "getl/drop.ma".
+include "LambdaDelta-1/getl/drop.ma".
 
 theorem pr2_delta1:
  \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c 
@@ -48,80 +48,64 @@ w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))))
 \def
  \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda 
 (H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (H0: (pr2 c t1 t2)).(let H1 \def (match H0 in pr2 return (\lambda 
-(c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C 
-c0 c) \to ((eq T t t1) \to ((eq T t0 t2) \to (\forall (w1: T).((subst1 i v t1 
-w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v 
-t2 w2)))))))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq 
-C c0 c)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C c 
-(\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (\forall 
-(w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) 
-(\lambda (w2: T).(subst1 i v t2 w2))))))))) (\lambda (H5: (eq T t0 
-t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (\forall 
-(w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) 
-(\lambda (w2: T).(subst1 i v t2 w2)))))))) (\lambda (H6: (eq T t3 
-t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (\forall (w1: T).((subst1 i 
-v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 
-i v t2 w2))))))) (\lambda (H7: (pr0 t1 t2)).(\lambda (w1: T).(\lambda (H8: 
-(subst1 i v t1 w1)).(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst1 i v t2 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: 
-T).(subst1 i v t2 w2))) (\lambda (x: T).(\lambda (H9: (pr0 w1 x)).(\lambda 
-(H10: (subst1 i v t2 x)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) 
-(\lambda (w2: T).(subst1 i v t2 w2)) x (pr2_free c w1 x H9) H10)))) 
-(pr0_subst1 t1 t2 H7 v w1 i H8 v (pr0_refl v)))))) t3 (sym_eq T t3 t2 H6))) 
-t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d 
-u i0 H1 t0 t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: 
-(eq T t0 t1)).(\lambda (H6: (eq T t t2)).(eq_ind C c (\lambda (c1: C).((eq T 
-t0 t1) \to ((eq T t t2) \to ((getl i0 c1 (CHead d (Bind Abbr) u)) \to ((pr0 
-t0 t3) \to ((subst0 i0 u t3 t) \to (\forall (w1: T).((subst1 i v t1 w1) \to 
-(ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 
-w2))))))))))) (\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq 
-T t t2) \to ((getl i0 c (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to 
-((subst0 i0 u t3 t) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T 
-(\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))))) 
-(\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i0 c (CHead d 
-(Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i0 u t3 t4) \to (\forall (w1: 
-T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda 
-(w2: T).(subst1 i v t2 w2))))))))) (\lambda (H9: (getl i0 c (CHead d (Bind 
-Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda (H11: (subst0 i0 u t3 
-t2)).(\lambda (w1: T).(\lambda (H12: (subst1 i v t1 w1)).(ex2_ind T (\lambda 
-(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T (\lambda 
-(w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x: 
-T).(\lambda (H13: (pr0 w1 x)).(\lambda (H14: (subst1 i v t3 x)).(neq_eq_e i 
-i0 (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 
-w2))) (\lambda (H15: (not (eq nat i i0))).(ex2_ind T (\lambda (t4: T).(subst1 
-i v t2 t4)) (\lambda (t4: T).(subst1 i0 u x t4)) (ex2 T (\lambda (w2: T).(pr2 
-c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x0: T).(\lambda 
-(H16: (subst1 i v t2 x0)).(\lambda (H17: (subst1 i0 u x x0)).(ex_intro2 T 
-(\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)) x0 
-(pr2_delta1 c d u i0 H9 w1 x H13 x0 H17) H16)))) (subst1_confluence_neq t3 t2 
-u i0 (subst1_single i0 u t3 t2 H11) x v i H14 (sym_not_eq nat i i0 H15)))) 
-(\lambda (H15: (eq nat i i0)).(let H16 \def (eq_ind_r nat i0 (\lambda (n: 
-nat).(subst0 n u t3 t2)) H11 i H15) in (let H17 \def (eq_ind_r nat i0 
-(\lambda (n: nat).(getl n c (CHead d (Bind Abbr) u))) H9 i H15) in (let H18 
-\def (eq_ind C (CHead e (Bind Abbr) v) (\lambda (c1: C).(getl i c c1)) H 
+T).(\lambda (H0: (pr2 c t1 t2)).(insert_eq C c (\lambda (c0: C).(pr2 c0 t1 
+t2)) (\lambda (c0: C).(\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T 
+(\lambda (w2: T).(pr2 c0 w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))) 
+(\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: 
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to (\forall (w1: 
+T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr2 c0 w1 w2)) (\lambda 
+(w2: T).(subst1 i v t0 w2))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda 
+(t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3: (eq C c0 c)).(\lambda (w1: 
+T).(\lambda (H4: (subst1 i v t3 w1)).(eq_ind_r C c (\lambda (c1: C).(ex2 T 
+(\lambda (w2: T).(pr2 c1 w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2)))) 
+(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2)) 
+(ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2))) 
+(\lambda (x: T).(\lambda (H5: (pr0 w1 x)).(\lambda (H6: (subst1 i v t4 
+x)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v 
+t4 w2)) x (pr2_free c w1 x H5) H6)))) (pr0_subst1 t3 t4 H2 v w1 i H4 v 
+(pr0_refl v))) c0 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (i0: nat).(\lambda (H2: (getl i0 c0 (CHead d (Bind Abbr) 
+u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda 
+(t: T).(\lambda (H4: (subst0 i0 u t4 t)).(\lambda (H5: (eq C c0 c)).(\lambda 
+(w1: T).(\lambda (H6: (subst1 i v t3 w1)).(let H7 \def (eq_ind C c0 (\lambda 
+(c1: C).(getl i0 c1 (CHead d (Bind Abbr) u))) H2 c H5) in (eq_ind_r C c 
+(\lambda (c1: C).(ex2 T (\lambda (w2: T).(pr2 c1 w1 w2)) (\lambda (w2: 
+T).(subst1 i v t w2)))) (ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda 
+(w2: T).(subst1 i v t4 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda 
+(w2: T).(subst1 i v t w2))) (\lambda (x: T).(\lambda (H8: (pr0 w1 
+x)).(\lambda (H9: (subst1 i v t4 x)).(neq_eq_e i i0 (ex2 T (\lambda (w2: 
+T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t w2))) (\lambda (H10: (not 
+(eq nat i i0))).(ex2_ind T (\lambda (t0: T).(subst1 i v t t0)) (\lambda (t0: 
+T).(subst1 i0 u x t0)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: 
+T).(subst1 i v t w2))) (\lambda (x0: T).(\lambda (H11: (subst1 i v t 
+x0)).(\lambda (H12: (subst1 i0 u x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c 
+w1 w2)) (\lambda (w2: T).(subst1 i v t w2)) x0 (pr2_delta1 c d u i0 H7 w1 x 
+H8 x0 H12) H11)))) (subst1_confluence_neq t4 t u i0 (subst1_single i0 u t4 t 
+H4) x v i H9 (sym_not_eq nat i i0 H10)))) (\lambda (H10: (eq nat i i0)).(let 
+H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u t4 t)) H4 i H10) in 
+(let H12 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d (Bind 
+Abbr) u))) H7 i H10) in (let H13 \def (eq_ind C (CHead e (Bind Abbr) v) 
+(\lambda (c1: C).(getl i c c1)) H (CHead d (Bind Abbr) u) (getl_mono c (CHead 
+e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in (let H14 \def (f_equal 
+C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) 
+\Rightarrow e | (CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind Abbr) v) 
 (CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d 
-(Bind Abbr) u) H17)) in (let H19 \def (f_equal C C (\lambda (e0: C).(match e0 
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c1 _ _) 
-\Rightarrow c1])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u) (getl_mono 
-c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H17)) in ((let H20 \def 
-(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow v | (CHead _ _ t4) \Rightarrow t4])) (CHead e (Bind 
-Abbr) v) (CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H 
-(CHead d (Bind Abbr) u) H17)) in (\lambda (H21: (eq C e d)).(let H22 \def 
-(eq_ind_r T u (\lambda (t4: T).(getl i c (CHead d (Bind Abbr) t4))) H18 v 
-H20) in (let H23 \def (eq_ind_r T u (\lambda (t4: T).(subst0 i t4 t3 t2)) H16 
-v H20) in (let H24 \def (eq_ind_r C d (\lambda (c1: C).(getl i c (CHead c1 
-(Bind Abbr) v))) H22 e H21) in (ex2_ind T (\lambda (t4: T).(subst1 i v t2 
-t4)) (\lambda (t4: T).(subst1 i v x t4)) (ex2 T (\lambda (w2: T).(pr2 c w1 
-w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x0: T).(\lambda (H25: 
-(subst1 i v t2 x0)).(\lambda (H26: (subst1 i v x x0)).(ex_intro2 T (\lambda 
-(w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)) x0 (pr2_delta1 c 
-e v i H24 w1 x H13 x0 H26) H25)))) (subst1_confluence_eq t3 t2 v i 
-(subst1_single i v t3 t2 H23) x H14))))))) H19)))))))))) (pr0_subst1 t1 t3 
-H10 v w1 i H12 v (pr0_refl v)))))))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 
-t1 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c) 
-(refl_equal T t1) (refl_equal T t2)))))))))).
+(Bind Abbr) u) H12)) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match 
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ 
+t0) \Rightarrow t0])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u) 
+(getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in 
+(\lambda (H16: (eq C e d)).(let H17 \def (eq_ind_r T u (\lambda (t0: T).(getl 
+i c (CHead d (Bind Abbr) t0))) H13 v H15) in (let H18 \def (eq_ind_r T u 
+(\lambda (t0: T).(subst0 i t0 t4 t)) H11 v H15) in (let H19 \def (eq_ind_r C 
+d (\lambda (c1: C).(getl i c (CHead c1 (Bind Abbr) v))) H17 e H16) in 
+(ex2_ind T (\lambda (t0: T).(subst1 i v t t0)) (\lambda (t0: T).(subst1 i v x 
+t0)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t 
+w2))) (\lambda (x0: T).(\lambda (H20: (subst1 i v t x0)).(\lambda (H21: 
+(subst1 i v x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: 
+T).(subst1 i v t w2)) x0 (pr2_delta1 c e v i H19 w1 x H8 x0 H21) H20)))) 
+(subst1_confluence_eq t4 t v i (subst1_single i v t4 t H18) x H9))))))) 
+H14)))))))))) (pr0_subst1 t3 t4 H3 v w1 i H6 v (pr0_refl v))) c0 
+H5))))))))))))))) y t1 t2 H1))) H0)))))))).
 
 theorem pr2_gen_cabbr:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/defs.ma
index cd26300e7..94eb8ea82 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr2/defs.ma".
+include "LambdaDelta-1/pr2/defs.ma".
 
 inductive pr3 (c: C): T \to (T \to Prop) \def
 | pr3_refl: \forall (t: T).(pr3 c t t)
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/fwd.ma
index 690ccfce3..258df5e4c 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/fwd.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr3/props.ma".
+include "LambdaDelta-1/pr3/props.ma".
 
-include "pr2/fwd.ma".
+include "LambdaDelta-1/pr2/fwd.ma".
 
 theorem pr3_gen_sort:
  \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TSort n) x) \to 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/iso.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/iso.ma
index 9e3d88402..51e8083d9 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/iso.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/iso.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr3/fwd.ma".
+include "LambdaDelta-1/pr3/fwd.ma".
 
-include "iso/props.ma".
+include "LambdaDelta-1/iso/props.ma".
 
-include "tlist/props.ma".
+include "LambdaDelta-1/tlist/props.ma".
 
 theorem pr3_iso_appls_abbr:
  \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/pr1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/pr1.ma
index 872444dc8..f75383c61 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/pr1.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/pr1.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr3/defs.ma".
+include "LambdaDelta-1/pr3/defs.ma".
 
-include "pr1/defs.ma".
+include "LambdaDelta-1/pr1/defs.ma".
 
 theorem pr3_pr1:
  \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (c: C).(pr3 c t1 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/pr3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/pr3.ma
index 62e34c4ee..24ccd52d0 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/pr3.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/pr3.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr3/props.ma".
+include "LambdaDelta-1/pr3/props.ma".
 
-include "pr2/pr2.ma".
+include "LambdaDelta-1/pr2/pr2.ma".
 
 theorem pr3_strip:
  \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/props.ma
index 67607a1e2..3569888eb 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/props.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr3/pr1.ma".
+include "LambdaDelta-1/pr3/pr1.ma".
 
-include "pr2/props.ma".
+include "LambdaDelta-1/pr2/props.ma".
 
-include "pr1/props.ma".
+include "LambdaDelta-1/pr1/props.ma".
 
 theorem clear_pr3_trans:
  \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr3 c2 t1 t2) \to 
@@ -142,84 +142,69 @@ theorem pr3_pr0_pr2_t:
 \def
  \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (c: 
 C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2 
-(CHead c k u2) t1 t2)).(let H1 \def (match H0 in pr2 return (\lambda (c0: 
-C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 
-(CHead c k u2)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr3 (CHead c k u1) t1 
-t2)))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0 
-(CHead c k u2))).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 
-t2)).(eq_ind C (CHead c k u2) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) 
-\to ((pr0 t0 t3) \to (pr3 (CHead c k u1) t1 t2))))) (\lambda (H5: (eq T t0 
-t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr3 
-(CHead c k u1) t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda 
-(t: T).((pr0 t1 t) \to (pr3 (CHead c k u1) t1 t2))) (\lambda (H7: (pr0 t1 
-t2)).(pr3_pr2 (CHead c k u1) t1 t2 (pr2_free (CHead c k u1) t1 t2 H7))) t3 
-(sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 (CHead c k u2) 
-H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t0 t3 H2 t H3) \Rightarrow (\lambda 
-(H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq 
-T t t2)).(eq_ind C (CHead c k u2) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t 
-t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i 
-u t3 t) \to (pr3 (CHead c k u1) t1 t2))))))) (\lambda (H7: (eq T t0 
-t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c k u2) 
-(CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr3 
-(CHead c k u1) t1 t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda 
-(t4: T).((getl i (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to 
-((subst0 i u t3 t4) \to (pr3 (CHead c k u1) t1 t2))))) (\lambda (H9: (getl i 
-(CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda 
-(H11: (subst0 i u t3 t2)).(nat_ind (\lambda (n: nat).((getl n (CHead c k u2) 
-(CHead d (Bind Abbr) u)) \to ((subst0 n u t3 t2) \to (pr3 (CHead c k u1) t1 
-t2)))) (\lambda (H12: (getl O (CHead c k u2) (CHead d (Bind Abbr) 
-u))).(\lambda (H13: (subst0 O u t3 t2)).(K_ind (\lambda (k0: K).((getl O 
-(CHead c k0 u2) (CHead d (Bind Abbr) u)) \to (pr3 (CHead c k0 u1) t1 t2))) 
-(\lambda (b: B).(\lambda (H14: (getl O (CHead c (Bind b) u2) (CHead d (Bind 
-Abbr) u))).(let H15 \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) u2) (clear_gen_bind b c 
-(CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind 
-Abbr) u) H14))) in ((let H16 \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 d (Bind Abbr) u) 
-(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 
-(getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H14))) in ((let H17 
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) 
-with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d 
-(Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) 
-u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H14))) in 
-(\lambda (H18: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H20 \def (eq_ind 
-T u (\lambda (t4: T).(subst0 O t4 t3 t2)) H13 u2 H17) in (eq_ind B Abbr 
-(\lambda (b0: B).(pr3 (CHead c (Bind b0) u1) t1 t2)) (ex2_ind T (\lambda (t4: 
-T).(subst0 O u1 t3 t4)) (\lambda (t4: T).(pr0 t4 t2)) (pr3 (CHead c (Bind 
-Abbr) u1) t1 t2) (\lambda (x: T).(\lambda (H21: (subst0 O u1 t3 x)).(\lambda 
-(H22: (pr0 x t2)).(pr3_sing (CHead c (Bind Abbr) u1) x t1 (pr2_delta (CHead c 
-(Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 t3 H10 x H21) t2 (pr3_pr2 
-(CHead c (Bind Abbr) u1) x t2 (pr2_free (CHead c (Bind Abbr) u1) x t2 
-H22)))))) (pr0_subst0_back u2 t3 t2 O H20 u1 H)) b H18))))) H16)) H15)))) 
-(\lambda (f: F).(\lambda (H14: (getl O (CHead c (Flat f) u2) (CHead d (Bind 
-Abbr) u))).(pr3_pr2 (CHead c (Flat f) u1) t1 t2 (pr2_cflat c t1 t2 (pr2_delta 
-c d u O (getl_intro O c (CHead d (Bind Abbr) u) c (drop_refl c) 
-(clear_gen_flat f c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Flat f) 
-u2) (CHead d (Bind Abbr) u) H14))) t1 t3 H10 t2 H13) f u1)))) k H12))) 
-(\lambda (i0: nat).(\lambda (IHi: (((getl i0 (CHead c k u2) (CHead d (Bind 
-Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c k u1) t1 
-t2))))).(\lambda (H12: (getl (S i0) (CHead c k u2) (CHead d (Bind Abbr) 
-u))).(\lambda (H13: (subst0 (S i0) u t3 t2)).(K_ind (\lambda (k0: K).((getl 
-(S i0) (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((((getl i0 (CHead c k0 
-u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c k0 
-u1) t1 t2)))) \to (pr3 (CHead c k0 u1) t1 t2)))) (\lambda (b: B).(\lambda 
-(H14: (getl (S i0) (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(\lambda 
-(_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 
-u t3 t2) \to (pr3 (CHead c (Bind b) u1) t1 t2))))).(pr3_pr2 (CHead c (Bind b) 
-u1) t1 t2 (pr2_delta (CHead c (Bind b) u1) 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) u2 
-i0 H14) u1) t1 t3 H10 t2 H13))))) (\lambda (f: F).(\lambda (H14: (getl (S i0) 
-(CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(\lambda (_: (((getl i0 
-(CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to 
-(pr3 (CHead c (Flat f) u1) t1 t2))))).(pr3_pr2 (CHead c (Flat f) u1) t1 t2 
-(pr2_cflat c t1 t2 (pr2_delta c d u (r (Flat f) i0) (getl_gen_S (Flat f) c 
-(CHead d (Bind Abbr) u) u2 i0 H14) t1 t3 H10 t2 H13) f u1))))) k H12 IHi))))) 
-i H9 H11)))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0 
-(CHead c k u2) H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C (CHead c k u2)) 
-(refl_equal T t1) (refl_equal T t2)))))))))).
+(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0 
+t1 t2)) (\lambda (_: C).(pr3 (CHead c k u1) 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 k u2)) \to (pr3 (CHead c k u1) t t0))))) (\lambda (c0: 
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: 
+(eq C c0 (CHead c k u2))).(pr3_pr2 (CHead c k u1) t3 t4 (pr2_free (CHead c k 
+u1) 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 k u2))).(let H6 \def 
+(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead 
+c k u2) H5) in (nat_ind (\lambda (n: nat).((getl n (CHead c k u2) (CHead d 
+(Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pr3 (CHead c k u1) t3 t)))) 
+(\lambda (H7: (getl O (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H8: 
+(subst0 O u t4 t)).(K_ind (\lambda (k0: K).((getl O (CHead c k0 u2) (CHead d 
+(Bind Abbr) u)) \to (pr3 (CHead c k0 u1) t3 t))) (\lambda (b: B).(\lambda 
+(H9: (getl O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(let H10 \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) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 
+(getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H9))) in ((let H11 
+\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 d (Bind Abbr) u) (CHead c (Bind b) u2) 
+(clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) 
+u2) (CHead d (Bind Abbr) u) H9))) in ((let H12 \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) 
+u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind 
+b) u2) (CHead d (Bind Abbr) u) H9))) in (\lambda (H13: (eq B Abbr 
+b)).(\lambda (_: (eq C d c)).(let H15 \def (eq_ind T u (\lambda (t0: 
+T).(subst0 O t0 t4 t)) H8 u2 H12) in (eq_ind B Abbr (\lambda (b0: B).(pr3 
+(CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 
+t0)) (\lambda (t0: T).(pr0 t0 t)) (pr3 (CHead c (Bind Abbr) u1) t3 t) 
+(\lambda (x: T).(\lambda (H16: (subst0 O u1 t4 x)).(\lambda (H17: (pr0 x 
+t)).(pr3_sing (CHead c (Bind Abbr) u1) x t3 (pr2_delta (CHead c (Bind Abbr) 
+u1) c u1 O (getl_refl Abbr c u1) t3 t4 H3 x H16) t (pr3_pr2 (CHead c (Bind 
+Abbr) u1) x t (pr2_free (CHead c (Bind Abbr) u1) x t H17)))))) 
+(pr0_subst0_back u2 t4 t O H15 u1 H)) b H13))))) H11)) H10)))) (\lambda (f: 
+F).(\lambda (H9: (getl O (CHead c (Flat f) u2) (CHead d (Bind Abbr) 
+u))).(pr3_pr2 (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u O 
+(getl_intro O c (CHead d (Bind Abbr) u) c (drop_refl c) (clear_gen_flat f c 
+(CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Flat f) u2) (CHead d (Bind 
+Abbr) u) H9))) t3 t4 H3 t H8) f u1)))) k H7))) (\lambda (i0: nat).(\lambda 
+(IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 
+t) \to (pr3 (CHead c k u1) t3 t))))).(\lambda (H7: (getl (S i0) (CHead c k 
+u2) (CHead d (Bind Abbr) u))).(\lambda (H8: (subst0 (S i0) u t4 t)).(K_ind 
+(\lambda (k0: K).((getl (S i0) (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to 
+((((getl i0 (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 t) 
+\to (pr3 (CHead c k0 u1) t3 t)))) \to (pr3 (CHead c k0 u1) t3 t)))) (\lambda 
+(b: B).(\lambda (H9: (getl (S i0) (CHead c (Bind b) u2) (CHead d (Bind Abbr) 
+u))).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) 
+\to ((subst0 i0 u t4 t) \to (pr3 (CHead c (Bind b) u1) t3 t))))).(pr3_pr2 
+(CHead c (Bind b) u1) t3 t (pr2_delta (CHead c (Bind b) u1) 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) u2 i0 H9) u1) t3 t4 H3 t H8))))) (\lambda (f: 
+F).(\lambda (H9: (getl (S i0) (CHead c (Flat f) u2) (CHead d (Bind Abbr) 
+u))).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) 
+\to ((subst0 i0 u t4 t) \to (pr3 (CHead c (Flat f) u1) t3 t))))).(pr3_pr2 
+(CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0) 
+(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u) u2 i0 H9) t3 t4 H3 t H8) f 
+u1))))) k H7 IHi))))) i H6 H4))))))))))))) y t1 t2 H1))) H0)))))))).
 
 theorem pr3_pr2_pr2_t:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall 
@@ -227,119 +212,80 @@ theorem pr3_pr2_pr2_t:
 (CHead c k u1) t1 t2))))))))
 \def
  \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1 
-u2)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t u1) 
-\to ((eq T t0 u2) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 
-(CHead c k u2) t1 t2) \to (pr3 (CHead c k u1) t1 t2)))))))))))) with 
-[(pr2_free c0 t1 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: 
-(eq T t1 u1)).(\lambda (H3: (eq T t2 u2)).(eq_ind C c (\lambda (_: C).((eq T 
-t1 u1) \to ((eq T t2 u2) \to ((pr0 t1 t2) \to (\forall (t3: T).(\forall (t4: 
-T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 
-t4))))))))) (\lambda (H4: (eq T t1 u1)).(eq_ind T u1 (\lambda (t: T).((eq T 
-t2 u2) \to ((pr0 t t2) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: 
-K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4)))))))) (\lambda 
-(H5: (eq T t2 u2)).(eq_ind T u2 (\lambda (t: T).((pr0 u1 t) \to (\forall (t3: 
-T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 
-(CHead c k u1) t3 t4))))))) (\lambda (H6: (pr0 u1 u2)).(\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H7: (pr2 (CHead c k u2) t3 
-t4)).(pr3_pr0_pr2_t u1 u2 H6 c t3 t4 k H7)))))) t2 (sym_eq T t2 u2 H5))) t1 
-(sym_eq T t1 u1 H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u 
-i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq 
-T t1 u1)).(\lambda (H5: (eq T t u2)).(eq_ind C c (\lambda (c1: C).((eq T t1 
-u1) \to ((eq T t u2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1 
-t2) \to ((subst0 i u t2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: 
-K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4))))))))))) 
-(\lambda (H6: (eq T t1 u1)).(eq_ind T u1 (\lambda (t0: T).((eq T t u2) \to 
-((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) 
-\to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 
-t4) \to (pr3 (CHead c k u1) t3 t4)))))))))) (\lambda (H7: (eq T t 
-u2)).(eq_ind T u2 (\lambda (t0: T).((getl i c (CHead d (Bind Abbr) u)) \to 
-((pr0 u1 t2) \to ((subst0 i u t2 t0) \to (\forall (t3: T).(\forall (t4: 
-T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 
-t4))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: 
-(pr0 u1 t2)).(\lambda (H10: (subst0 i u t2 u2)).(\lambda (t3: T).(\lambda 
-(t0: T).(\lambda (k: K).(\lambda (H11: (pr2 (CHead c k u2) t3 t0)).(let H12 
-\def (match H11 in pr2 return (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: 
-T).(\lambda (_: (pr2 c1 t4 t5)).((eq C c1 (CHead c k u2)) \to ((eq T t4 t3) 
-\to ((eq T t5 t0) \to (pr3 (CHead c k u1) t3 t0)))))))) with [(pr2_free c1 t4 
-t5 H12) \Rightarrow (\lambda (H13: (eq C c1 (CHead c k u2))).(\lambda (H14: 
-(eq T t4 t3)).(\lambda (H15: (eq T t5 t0)).(eq_ind C (CHead c k u2) (\lambda 
-(_: C).((eq T t4 t3) \to ((eq T t5 t0) \to ((pr0 t4 t5) \to (pr3 (CHead c k 
-u1) t3 t0))))) (\lambda (H16: (eq T t4 t3)).(eq_ind T t3 (\lambda (t6: 
-T).((eq T t5 t0) \to ((pr0 t6 t5) \to (pr3 (CHead c k u1) t3 t0)))) (\lambda 
-(H17: (eq T t5 t0)).(eq_ind T t0 (\lambda (t6: T).((pr0 t3 t6) \to (pr3 
-(CHead c k u1) t3 t0))) (\lambda (H18: (pr0 t3 t0)).(pr3_pr2 (CHead c k u1) 
-t3 t0 (pr2_free (CHead c k u1) t3 t0 H18))) t5 (sym_eq T t5 t0 H17))) t4 
-(sym_eq T t4 t3 H16))) c1 (sym_eq C c1 (CHead c k u2) H13) H14 H15 H12)))) | 
-(pr2_delta c1 d0 u0 i0 H12 t4 t5 H13 t6 H14) \Rightarrow (\lambda (H15: (eq C 
-c1 (CHead c k u2))).(\lambda (H16: (eq T t4 t3)).(\lambda (H17: (eq T t6 
-t0)).(eq_ind C (CHead c k u2) (\lambda (c2: C).((eq T t4 t3) \to ((eq T t6 
-t0) \to ((getl i0 c2 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t4 t5) \to ((subst0 
-i0 u0 t5 t6) \to (pr3 (CHead c k u1) t3 t0))))))) (\lambda (H18: (eq T t4 
-t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t6 t0) \to ((getl i0 (CHead c k u2) 
-(CHead d0 (Bind Abbr) u0)) \to ((pr0 t7 t5) \to ((subst0 i0 u0 t5 t6) \to 
-(pr3 (CHead c k u1) t3 t0)))))) (\lambda (H19: (eq T t6 t0)).(eq_ind T t0 
-(\lambda (t7: T).((getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to 
-((pr0 t3 t5) \to ((subst0 i0 u0 t5 t7) \to (pr3 (CHead c k u1) t3 t0))))) 
-(\lambda (H20: (getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda 
-(H21: (pr0 t3 t5)).(\lambda (H22: (subst0 i0 u0 t5 t0)).(nat_ind (\lambda (n: 
-nat).((getl n (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5 
-t0) \to (pr3 (CHead c k u1) t3 t0)))) (\lambda (H23: (getl O (CHead c k u2) 
-(CHead d0 (Bind Abbr) u0))).(\lambda (H24: (subst0 O u0 t5 t0)).(K_ind 
-(\lambda (k0: K).((clear (CHead c k0 u2) (CHead d0 (Bind Abbr) u0)) \to (pr3 
-(CHead c k0 u1) t3 t0))) (\lambda (b: B).(\lambda (H25: (clear (CHead c (Bind 
-b) u2) (CHead d0 (Bind Abbr) u0))).(let H26 \def (f_equal C C (\lambda (e: 
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 | 
-(CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) 
-u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H25)) in ((let H27 \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 
+u2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1: 
+T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t0) t1 t2) \to (pr3 
+(CHead c0 k t) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: 
+K).(\lambda (H1: (pr2 (CHead c0 k t2) t0 t3)).(pr3_pr0_pr2_t t1 t2 H0 c0 t0 
+t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
+(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2: 
+(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda 
+(H3: (pr2 (CHead c0 k t) t0 t3)).(insert_eq C (CHead c0 k t) (\lambda (c1: 
+C).(pr2 c1 t0 t3)) (\lambda (_: C).(pr3 (CHead c0 k t1) t0 t3)) (\lambda (y: 
+C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4: 
+T).(\lambda (t5: T).((eq C c1 (CHead c0 k t)) \to (pr3 (CHead c0 k t1) t4 
+t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0 
+t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t))).(pr3_pr2 (CHead c0 k t1) t4 t5 
+(pr2_free (CHead c0 k t1) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0: 
+C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0 
+(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4 
+t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C 
+c1 (CHead c0 k t))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2 
+(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t) H8) in (nat_ind (\lambda (n: 
+nat).((getl n (CHead c0 k t) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5 
+t6) \to (pr3 (CHead c0 k t1) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t) 
+(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind 
+(\lambda (k0: K).((clear (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 
+(CHead c0 k0 t1) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0 
+(Bind b) t) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda 
+(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 
+| (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind 
+b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H14 
+\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 d0 (Bind Abbr) u0) (CHead c (Bind b) u2) 
-(clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H25)) in ((let H28 \def 
+\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) 
+(clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H15 \def 
 (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
 [(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind 
-Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) 
-u2 H25)) in (\lambda (H29: (eq B Abbr b)).(\lambda (_: (eq C d0 c)).(let H31 
-\def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t0)) H24 u2 H28) in 
-(eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c (Bind b0) u1) t3 t0)) (ex2_ind 
+Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) 
+u0) t H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let 
+H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) H11 t H15) in 
+(eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c0 (Bind b0) t1) t4 t6)) (ex2_ind 
 T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(subst0 (S (plus i 
-O)) u t7 t0)) (pr3 (CHead c (Bind Abbr) u1) t3 t0) (\lambda (x: T).(\lambda 
-(H32: (subst0 O t2 t5 x)).(\lambda (H33: (subst0 (S (plus i O)) u x t0)).(let 
-H34 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O 
-i))) in (let H35 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n 
-u x t0)) H33 (S i) H34) in (ex2_ind T (\lambda (t7: T).(subst0 O u1 t5 t7)) 
-(\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c (Bind Abbr) u1) t3 t0) (\lambda 
-(x0: T).(\lambda (H36: (subst0 O u1 t5 x0)).(\lambda (H37: (pr0 x0 
-x)).(pr3_sing (CHead c (Bind Abbr) u1) x0 t3 (pr2_delta (CHead c (Bind Abbr) 
-u1) c u1 O (getl_refl Abbr c u1) t3 t5 H21 x0 H36) t0 (pr3_pr2 (CHead c (Bind 
-Abbr) u1) x0 t0 (pr2_delta (CHead c (Bind Abbr) u1) d u (S i) 
-(getl_clear_bind Abbr (CHead c (Bind Abbr) u1) c u1 (clear_bind Abbr c u1) 
-(CHead d (Bind Abbr) u) i H8) x0 x H37 t0 H35)))))) (pr0_subst0_back t2 t5 x 
-O H32 u1 H9))))))) (subst0_subst0 t5 t0 u2 O H31 t2 u i H10)) b H29))))) 
-H27)) H26)))) (\lambda (f: F).(\lambda (H25: (clear (CHead c (Flat f) u2) 
-(CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c (Flat f) u1) t3 t0 (pr2_cflat c 
-t3 t0 (pr2_delta c d0 u0 O (getl_intro O c (CHead d0 (Bind Abbr) u0) c 
-(drop_refl c) (clear_gen_flat f c (CHead d0 (Bind Abbr) u0) u2 H25)) t3 t5 
-H21 t0 H24) f u1)))) k (getl_gen_O (CHead c k u2) (CHead d0 (Bind Abbr) u0) 
-H23)))) (\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c k u2) (CHead d0 
-(Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t0) \to (pr3 (CHead c k u1) t3 
-t0))))).(\lambda (H23: (getl (S i1) (CHead c k u2) (CHead d0 (Bind Abbr) 
-u0))).(\lambda (H24: (subst0 (S i1) u0 t5 t0)).(K_ind (\lambda (k0: K).((getl 
-(S i1) (CHead c k0 u2) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c k0 u1) t3 
-t0))) (\lambda (b: B).(\lambda (H25: (getl (S i1) (CHead c (Bind b) u2) 
-(CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c (Bind b) u1) t3 t0 (pr2_delta 
-(CHead c (Bind b) u1) d0 u0 (S i1) (getl_head (Bind b) i1 c (CHead d0 (Bind 
-Abbr) u0) (getl_gen_S (Bind b) c (CHead d0 (Bind Abbr) u0) u2 i1 H25) u1) t3 
-t5 H21 t0 H24)))) (\lambda (f: F).(\lambda (H25: (getl (S i1) (CHead c (Flat 
-f) u2) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c (Flat f) u1) t3 t0 
-(pr2_cflat c t3 t0 (pr2_delta c d0 u0 (r (Flat f) i1) (getl_gen_S (Flat f) c 
-(CHead d0 (Bind Abbr) u0) u2 i1 H25) t3 t5 H21 t0 H24) f u1)))) k H23))))) i0 
-H20 H22)))) t6 (sym_eq T t6 t0 H19))) t4 (sym_eq T t4 t3 H18))) c1 (sym_eq C 
-c1 (CHead c k u2) H15) H16 H17 H12 H13 H14))))]) in (H12 (refl_equal C (CHead 
-c k u2)) (refl_equal T t3) (refl_equal T t0)))))))))) t (sym_eq T t u2 H7))) 
-t1 (sym_eq T t1 u1 H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 
-(refl_equal C c) (refl_equal T u1) (refl_equal T u2)))))).
+O)) u t7 t6)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda (x: T).(\lambda 
+(H19: (subst0 O t2 t5 x)).(\lambda (H20: (subst0 (S (plus i O)) u x t6)).(let 
+H21 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O 
+i))) in (let H22 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n 
+u x t6)) H20 (S i) H21) in (ex2_ind T (\lambda (t7: T).(subst0 O t1 t5 t7)) 
+(\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda 
+(x0: T).(\lambda (H23: (subst0 O t1 t5 x0)).(\lambda (H24: (pr0 x0 
+x)).(pr3_sing (CHead c0 (Bind Abbr) t1) x0 t4 (pr2_delta (CHead c0 (Bind 
+Abbr) t1) c0 t1 O (getl_refl Abbr c0 t1) t4 t5 H6 x0 H23) t6 (pr3_pr2 (CHead 
+c0 (Bind Abbr) t1) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t1) d u (S i) 
+(getl_clear_bind Abbr (CHead c0 (Bind Abbr) t1) c0 t1 (clear_bind Abbr c0 t1) 
+(CHead d (Bind Abbr) u) i H0) x0 x H24 t6 H22)))))) (pr0_subst0_back t2 t5 x 
+O H19 t1 H1))))))) (subst0_subst0 t5 t6 t O H18 t2 u i H2)) b H16))))) H14)) 
+H13)))) (\lambda (f: F).(\lambda (H12: (clear (CHead c0 (Flat f) t) (CHead d0 
+(Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6 
+(pr2_delta c0 d0 u0 O (getl_intro O c0 (CHead d0 (Bind Abbr) u0) c0 
+(drop_refl c0) (clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t H12)) t4 t5 
+H6 t6 H11) f t1)))) k (getl_gen_O (CHead c0 k t) (CHead d0 (Bind Abbr) u0) 
+H10)))) (\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c0 k t) (CHead d0 
+(Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to (pr3 (CHead c0 k t1) t4 
+t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t) (CHead d0 (Bind Abbr) 
+u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind (\lambda (k0: K).((getl 
+(S i1) (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c0 k0 t1) 
+t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (S i1) (CHead c0 (Bind b) t) 
+(CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Bind b) t1) t4 t6 (pr2_delta 
+(CHead c0 (Bind b) t1) d0 u0 (S i1) (getl_head (Bind b) i1 c0 (CHead d0 (Bind 
+Abbr) u0) (getl_gen_S (Bind b) c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t1) t4 
+t5 H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (S i1) (CHead c0 (Flat 
+f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6 
+(pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) (getl_gen_S (Flat f) 
+c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t4 t5 H6 t6 H11) f t1)))) k H10))))) 
+i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u1 u2 H)))).
 
 theorem pr3_pr2_pr3_t:
  \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/subst1.ma
index 83c8cb75c..5ec290cd8 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/subst1.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/subst1.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr3/defs.ma".
+include "LambdaDelta-1/pr3/defs.ma".
 
-include "pr2/subst1.ma".
+include "LambdaDelta-1/pr2/subst1.ma".
 
 theorem pr3_subst1:
  \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/wcpr0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/wcpr0.ma
index df905f39c..1469dbf9f 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/wcpr0.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr3/wcpr0.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr3/props.ma".
+include "LambdaDelta-1/pr3/props.ma".
 
-include "wcpr0/getl.ma".
+include "LambdaDelta-1/wcpr0/getl.ma".
 
 theorem pr3_wcpr0_t:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (t1: 
@@ -34,44 +34,30 @@ T).(\lambda (H3: (pr3 (CHead c3 k u2) t1 t2)).(pr3_ind (CHead c3 k u1)
 T).(pr3_refl (CHead c0 k u1) t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda 
 (H4: (pr2 (CHead c3 k u1) t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead 
 c3 k u1) t0 t4)).(\lambda (H6: (pr3 (CHead c0 k u1) t0 t4)).(pr3_t t0 t3 
-(CHead c0 k u1) (let H7 \def (match H4 in pr2 return (\lambda (c: C).(\lambda 
-(t: T).(\lambda (t5: T).(\lambda (_: (pr2 c t t5)).((eq C c (CHead c3 k u1)) 
-\to ((eq T t t3) \to ((eq T t5 t0) \to (pr3 (CHead c0 k u1) t3 t0)))))))) 
-with [(pr2_free c t5 t6 H7) \Rightarrow (\lambda (H8: (eq C c (CHead c3 k 
-u1))).(\lambda (H9: (eq T t5 t3)).(\lambda (H10: (eq T t6 t0)).(eq_ind C 
-(CHead c3 k u1) (\lambda (_: C).((eq T t5 t3) \to ((eq T t6 t0) \to ((pr0 t5 
-t6) \to (pr3 (CHead c0 k u1) t3 t0))))) (\lambda (H11: (eq T t5 t3)).(eq_ind 
-T t3 (\lambda (t: T).((eq T t6 t0) \to ((pr0 t t6) \to (pr3 (CHead c0 k u1) 
-t3 t0)))) (\lambda (H12: (eq T t6 t0)).(eq_ind T t0 (\lambda (t: T).((pr0 t3 
-t) \to (pr3 (CHead c0 k u1) t3 t0))) (\lambda (H13: (pr0 t3 t0)).(pr3_pr2 
-(CHead c0 k u1) t3 t0 (pr2_free (CHead c0 k u1) t3 t0 H13))) t6 (sym_eq T t6 
-t0 H12))) t5 (sym_eq T t5 t3 H11))) c (sym_eq C c (CHead c3 k u1) H8) H9 H10 
-H7)))) | (pr2_delta c d u i H7 t5 t6 H8 t H9) \Rightarrow (\lambda (H10: (eq 
-C c (CHead c3 k u1))).(\lambda (H11: (eq T t5 t3)).(\lambda (H12: (eq T t 
-t0)).(eq_ind C (CHead c3 k u1) (\lambda (c4: C).((eq T t5 t3) \to ((eq T t 
-t0) \to ((getl i c4 (CHead d (Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i 
-u t6 t) \to (pr3 (CHead c0 k u1) t3 t0))))))) (\lambda (H13: (eq T t5 
-t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t t0) \to ((getl i (CHead c3 k u1) 
-(CHead d (Bind Abbr) u)) \to ((pr0 t7 t6) \to ((subst0 i u t6 t) \to (pr3 
-(CHead c0 k u1) t3 t0)))))) (\lambda (H14: (eq T t t0)).(eq_ind T t0 (\lambda 
-(t7: T).((getl i (CHead c3 k u1) (CHead d (Bind Abbr) u)) \to ((pr0 t3 t6) 
-\to ((subst0 i u t6 t7) \to (pr3 (CHead c0 k u1) t3 t0))))) (\lambda (H15: 
-(getl i (CHead c3 k u1) (CHead d (Bind Abbr) u))).(\lambda (H16: (pr0 t3 
-t6)).(\lambda (H17: (subst0 i u t6 t0)).(ex3_2_ind C T (\lambda (e2: 
+(CHead c0 k u1) (insert_eq C (CHead c3 k u1) (\lambda (c: C).(pr2 c t3 t0)) 
+(\lambda (_: C).(pr3 (CHead c0 k u1) t3 t0)) (\lambda (y: C).(\lambda (H7: 
+(pr2 y t3 t0)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t5: T).((eq 
+C c (CHead c3 k u1)) \to (pr3 (CHead c0 k u1) t t5))))) (\lambda (c: 
+C).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H8: (pr0 t5 t6)).(\lambda (_: 
+(eq C c (CHead c3 k u1))).(pr3_pr2 (CHead c0 k u1) t5 t6 (pr2_free (CHead c0 
+k u1) t5 t6 H8))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (i: nat).(\lambda (H8: (getl i c (CHead d (Bind Abbr) 
+u))).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H9: (pr0 t5 t6)).(\lambda 
+(t: T).(\lambda (H10: (subst0 i u t6 t)).(\lambda (H11: (eq C c (CHead c3 k 
+u1))).(let H12 \def (eq_ind C c (\lambda (c4: C).(getl i c4 (CHead d (Bind 
+Abbr) u))) H8 (CHead c3 k u1) H11) in (ex3_2_ind C T (\lambda (e2: 
 C).(\lambda (u3: T).(getl i (CHead c0 k u1) (CHead e2 (Bind Abbr) u3)))) 
 (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 d))) (\lambda (_: C).(\lambda (u3: 
-T).(pr0 u3 u))) (pr3 (CHead c0 k u1) t3 t0) (\lambda (x0: C).(\lambda (x1: 
-T).(\lambda (H18: (getl i (CHead c0 k u1) (CHead x0 (Bind Abbr) 
-x1))).(\lambda (_: (wcpr0 x0 d)).(\lambda (H20: (pr0 x1 u)).(ex2_ind T 
-(\lambda (t7: T).(subst0 i x1 t6 t7)) (\lambda (t7: T).(pr0 t7 t0)) (pr3 
-(CHead c0 k u1) t3 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x1 t6 
-x)).(\lambda (H22: (pr0 x t0)).(pr3_sing (CHead c0 k u1) x t3 (pr2_delta 
-(CHead c0 k u1) x0 x1 i H18 t3 t6 H16 x H21) t0 (pr3_pr2 (CHead c0 k u1) x t0 
-(pr2_free (CHead c0 k u1) x t0 H22)))))) (pr0_subst0_back u t6 t0 i H17 x1 
-H20))))))) (wcpr0_getl_back (CHead c3 k u1) (CHead c0 k u1) (wcpr0_comp c0 c3 
-H0 u1 u1 (pr0_refl u1) k) i d u (Bind Abbr) H15))))) t (sym_eq T t t0 H14))) 
-t5 (sym_eq T t5 t3 H13))) c (sym_eq C c (CHead c3 k u1) H10) H11 H12 H7 H8 
-H9))))]) in (H7 (refl_equal C (CHead c3 k u1)) (refl_equal T t3) (refl_equal 
-T t0))) t4 H6))))))) t1 t2 (pr3_pr2_pr3_t c3 u2 t1 t2 k H3 u1 (pr2_free c3 u1 
-u2 H2)))))))))))))) c2 c1 H))).
+T).(pr0 u3 u))) (pr3 (CHead c0 k u1) t5 t) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (H13: (getl i (CHead c0 k u1) (CHead x0 (Bind Abbr) 
+x1))).(\lambda (_: (wcpr0 x0 d)).(\lambda (H15: (pr0 x1 u)).(ex2_ind T 
+(\lambda (t7: T).(subst0 i x1 t6 t7)) (\lambda (t7: T).(pr0 t7 t)) (pr3 
+(CHead c0 k u1) t5 t) (\lambda (x: T).(\lambda (H16: (subst0 i x1 t6 
+x)).(\lambda (H17: (pr0 x t)).(pr3_sing (CHead c0 k u1) x t5 (pr2_delta 
+(CHead c0 k u1) x0 x1 i H13 t5 t6 H9 x H16) t (pr3_pr2 (CHead c0 k u1) x t 
+(pr2_free (CHead c0 k u1) x t H17)))))) (pr0_subst0_back u t6 t i H10 x1 
+H15))))))) (wcpr0_getl_back (CHead c3 k u1) (CHead c0 k u1) (wcpr0_comp c0 c3 
+H0 u1 u1 (pr0_refl u1) k) i d u (Bind Abbr) H12)))))))))))))) y t3 t0 H7))) 
+H4) t4 H6))))))) t1 t2 (pr3_pr2_pr3_t c3 u2 t1 t2 k H3 u1 (pr2_free c3 u1 u2 
+H2)))))))))))))) c2 c1 H))).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/preamble.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/preamble.ma
index 5abecfd80..e493455eb 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/preamble.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/preamble.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "../Base-1/theory.ma".
+include "Base-1/theory.ma".
 
 alias id "and_ind" = "cic:/Coq/Init/Logic/and_ind.con".
 alias id "bool_ind" = "cic:/Coq/Init/Datatypes/bool_ind.con".
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/r/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/r/defs.ma
index 83ae2b2d2..a4b960202 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/r/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/r/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "T/defs.ma".
+include "LambdaDelta-1/T/defs.ma".
 
 definition r:
  K \to (nat \to nat)
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/r/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/r/props.ma
index 20ae7a5cf..a768ab93c 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/r/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/r/props.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "r/defs.ma".
+include "LambdaDelta-1/r/defs.ma".
 
-include "s/defs.ma".
+include "LambdaDelta-1/s/defs.ma".
 
 theorem r_S:
  \forall (k: K).(\forall (i: nat).(eq nat (r k (S i)) (S (r k i))))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/root b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/root
deleted file mode 100644
index 73f353af7..000000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/root
+++ /dev/null
@@ -1 +0,0 @@
-baseuri=cic:/matita/LAMBDA-TYPES/LambdaDelta-1
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/s/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/s/defs.ma
index dbbf0665c..d0e871e0b 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/s/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/s/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "T/defs.ma".
+include "LambdaDelta-1/T/defs.ma".
 
 definition s:
  K \to (nat \to nat)
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/s/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/s/props.ma
index b4c556f30..5d6b682a1 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/s/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/s/props.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "s/defs.ma".
+include "LambdaDelta-1/s/defs.ma".
 
 theorem s_S:
  \forall (k: K).(\forall (i: nat).(eq nat (s k (S i)) (S (s k i))))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/arity.ma
index 7258d8d1a..16393b04b 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/arity.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/arity.ma
@@ -14,13 +14,13 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubc/arity.ma".
+include "LambdaDelta-1/csubc/arity.ma".
 
-include "csubc/getl.ma".
+include "LambdaDelta-1/csubc/getl.ma".
 
-include "csubc/drop1.ma".
+include "LambdaDelta-1/csubc/drop1.ma".
 
-include "csubc/props.ma".
+include "LambdaDelta-1/csubc/props.ma".
 
 theorem sc3_arity_csubc:
  \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 
@@ -53,247 +53,213 @@ 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 
+x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1 (ptrans is i) u) (Bind 
+Abbr) H10) in (let H11 \def H_x1 in (or_ind (ex2 C (\lambda (c3: C).(eq C x0 
+(CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x 
+c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K 
+(Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: 
+A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w))))) (sc3 g a0 c2 (lift1 
+is (TLRef i))) (\lambda (H12: (ex2 C (\lambda (c3: C).(eq C x0 (CHead c3 
+(Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x 
+c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abbr) (lift1 
+(ptrans is i) u)))) (\lambda (c3: C).(csubc g x c3)) (sc3 g a0 c2 (lift1 is 
+(TLRef i))) (\lambda (x1: C).(\lambda (H13: (eq C x0 (CHead x1 (Bind Abbr) 
+(lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x x1)).(let H15 \def (eq_ind 
+C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H9 (CHead x1 (Bind Abbr) 
+(lift1 (ptrans is i) u)) H13) 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 
+(trans is i) x1 (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) 
+is i)) O (lift1 (ptrans is i) u)) (lift1_free is i u)) H15) (lift1 is (TLRef 
+i)) (lift1_lref is i))))))) H12)) (\lambda (H12: (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abst))))) (\lambda 
+(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans 
+is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 
+w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq 
+K (Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: 
+A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w)))) (sc3 g a0 c2 (lift1 is 
+(TLRef i))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H13: 
+(eq K (Bind Abbr) (Bind Abst))).(\lambda (H14: (eq C x0 (CHead x1 (Bind Abbr) 
+x2))).(\lambda (_: (csubc g x x1)).(\lambda (_: (sc3 g (asucc g x3) x (lift1 
+(ptrans is i) u))).(\lambda (_: (sc3 g x3 x1 x2)).(let H18 \def (eq_ind C x0 
+(\lambda (c0: C).(getl (trans is i) c2 c0)) H9 (CHead x1 (Bind Abbr) x2) H14) 
+in (let H19 \def (eq_ind K (Bind Abbr) (\lambda (ee: K).(match ee 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 (Bind Abst) H13) 
+in (False_ind (sc3 g a0 c2 (lift1 is (TLRef i))) H19))))))))))) H12)) 
+H11)))))) 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 (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))))).
+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 H_x1 \def (csubc_gen_head_l g x x0 (lift1 (ptrans is i) u) (Bind 
+Abst) H11) in (let H12 \def H_x1 in (or_ind (ex2 C (\lambda (c3: C).(eq C x0 
+(CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x 
+c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K 
+(Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: 
+A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w))))) (sc3 g a0 c2 (lift1 
+is (TLRef i))) (\lambda (H13: (ex2 C (\lambda (c3: C).(eq C x0 (CHead c3 
+(Bind Abst) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x 
+c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 
+(ptrans is i) u)))) (\lambda (c3: C).(csubc g x c3)) (sc3 g a0 c2 (lift1 is 
+(TLRef i))) (\lambda (x1: C).(\lambda (H14: (eq C x0 (CHead x1 (Bind Abst) 
+(lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x x1)).(let H16 \def (eq_ind 
+C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H10 (CHead x1 (Bind Abst) 
+(lift1 (ptrans is i) u)) H14) 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 x1 (lift1 (ptrans is i) u) (trans is i) 
+H16) I) (lift1 is (TLRef i)) (lift1_lref is i))))))) H13)) (\lambda (H13: 
+(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind 
+Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C 
+x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
+A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g 
+(asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: C).(\lambda (w: 
+T).(\lambda (a1: A).(sc3 g a1 c3 w)))))).(ex5_3_ind C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda 
+(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans 
+is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 
+w)))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: C).(\lambda (x2: 
+T).(\lambda (x3: A).(\lambda (_: (eq K (Bind Abst) (Bind Abst))).(\lambda 
+(H15: (eq C x0 (CHead x1 (Bind Abbr) x2))).(\lambda (_: (csubc g x 
+x1)).(\lambda (H17: (sc3 g (asucc g x3) x (lift1 (ptrans is i) u))).(\lambda 
+(H18: (sc3 g x3 x1 x2)).(let H19 \def (eq_ind C x0 (\lambda (c0: C).(getl 
+(trans is i) c2 c0)) H10 (CHead x1 (Bind Abbr) x2) H15) 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) x1 x2 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 x3) H17) in (sc3_repl g x3 c2 (lift (S 
+(trans is i)) O x2) (sc3_lift g x3 x1 x2 H18 c2 (S (trans is i)) O (getl_drop 
+Abbr c2 x1 x2 (trans is i) H19)) a0 (asucc_inj g x3 a0 (arity_mono g x (lift1 
+(ptrans is i) u) (asucc g x3) H_y1 (asucc g a0) H_y0))))) H19) (lift1 is 
+(TLRef i)) (lift1_lref is i)))))))))))) H13)) H12)))))) 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 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/defs.ma
index be8cda450..5947d1bbd 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/defs.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "sn3/defs.ma".
+include "LambdaDelta-1/sn3/defs.ma".
 
-include "arity/defs.ma".
+include "LambdaDelta-1/arity/defs.ma".
 
-include "drop1/defs.ma".
+include "LambdaDelta-1/drop1/defs.ma".
 
 definition sc3:
  G \to (A \to (C \to (T \to Prop)))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/props.ma
index 79645a1fd..8cf36c0b4 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/props.ma
@@ -14,25 +14,25 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "sc3/defs.ma".
+include "LambdaDelta-1/sc3/defs.ma".
 
-include "sn3/lift1.ma".
+include "LambdaDelta-1/sn3/lift1.ma".
 
-include "nf2/lift1.ma".
+include "LambdaDelta-1/nf2/lift1.ma".
 
-include "csuba/arity.ma".
+include "LambdaDelta-1/csuba/arity.ma".
 
-include "arity/lift1.ma".
+include "LambdaDelta-1/arity/lift1.ma".
 
-include "arity/aprem.ma".
+include "LambdaDelta-1/arity/aprem.ma".
 
-include "llt/props.ma".
+include "LambdaDelta-1/llt/props.ma".
 
-include "drop1/getl.ma".
+include "LambdaDelta-1/drop1/getl.ma".
 
-include "drop1/props.ma".
+include "LambdaDelta-1/drop1/props.ma".
 
-include "lift1/props.ma".
+include "LambdaDelta-1/lift1/props.ma".
 
 theorem sc3_arity_gen:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((sc3 g a c 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/defs.ma
index ddf14abc2..9aa57868f 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr3/defs.ma".
+include "LambdaDelta-1/pr3/defs.ma".
 
 inductive sn3 (c: C): T \to Prop \def
 | sn3_sing: \forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to (\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/fwd.ma
index c9db131a1..c55628d8a 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/fwd.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "sn3/defs.ma".
+include "LambdaDelta-1/sn3/defs.ma".
 
-include "pr3/props.ma".
+include "LambdaDelta-1/pr3/props.ma".
 
 theorem sn3_gen_bind:
  \forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/lift1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/lift1.ma
index 44fb2ca8e..fec70b8f9 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/lift1.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/lift1.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "sn3/props.ma".
+include "LambdaDelta-1/sn3/props.ma".
 
-include "drop1/defs.ma".
+include "LambdaDelta-1/drop1/defs.ma".
 
-include "lift1/fwd.ma".
+include "LambdaDelta-1/lift1/fwd.ma".
 
 theorem sns3_lifts1:
  \forall (e: C).(\forall (hds: PList).(\forall (c: C).((drop1 hds c e) \to 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/nf2.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/nf2.ma
index 152088179..b979db1a5 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/nf2.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/nf2.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "sn3/defs.ma".
+include "LambdaDelta-1/sn3/defs.ma".
 
-include "nf2/dec.ma".
+include "LambdaDelta-1/nf2/dec.ma".
 
-include "nf2/pr3.ma".
+include "LambdaDelta-1/nf2/pr3.ma".
 
 theorem sn3_nf2:
  \forall (c: C).(\forall (t: T).((nf2 c t) \to (sn3 c t)))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/props.ma
index afd3fdbc9..254589a27 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/props.ma
@@ -14,15 +14,13 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "sn3/nf2.ma".
+include "LambdaDelta-1/sn3/nf2.ma".
 
-include "sn3/fwd.ma".
+include "LambdaDelta-1/sn3/fwd.ma".
 
-include "nf2/iso.ma".
+include "LambdaDelta-1/nf2/iso.ma".
 
-include "pr3/iso.ma".
-
-include "iso/props.ma".
+include "LambdaDelta-1/pr3/iso.ma".
 
 theorem sn3_pr3_trans:
  \forall (c: C).(\forall (t1: T).((sn3 c t1) \to (\forall (t2: T).((pr3 c t1 
@@ -181,6 +179,71 @@ 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_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)))))).
+
 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) 
@@ -307,11 +370,9 @@ 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)))).
+(Bind b) t1) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3) (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 
@@ -2431,68 +2492,3 @@ 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/LambdaDelta-1/spare.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/spare.ma
index c79beed0b..6022d3fa9 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/spare.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/spare.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "theory.ma".
+include "LambdaDelta-1/theory.ma".
 
 definition cbk:
  C \to nat
@@ -141,50 +141,24 @@ TNil u)))
 | tys3_cons: \forall (t: T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts: 
 TList).((tys3 g c ts u) \to (tys3 g c (TCons t ts) u))))).
 
-theorem tys3_inv_coq:
- \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (u: T).(\forall 
-(P: ((G \to (C \to (TList \to (T \to Prop)))))).((((tys3 g c ts u) \to 
-(\forall (u0: T).(\forall (u1: T).((eq TList TNil ts) \to ((eq T u0 u) \to 
-((ty3 g c u0 u1) \to (P g c ts u)))))))) \to ((((tys3 g c ts u) \to (\forall 
-(t: T).(\forall (u0: T).(\forall (ts0: TList).((eq TList (TCons t ts0) ts) 
-\to ((eq T u0 u) \to ((ty3 g c t u0) \to ((tys3 g c ts0 u0) \to (P g c ts 
-u)))))))))) \to ((tys3 g c ts u) \to (P g c ts u))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (u: T).(\lambda 
-(P: ((G \to (C \to (TList \to (T \to Prop)))))).(\lambda (H: (((tys3 g c ts 
-u) \to (\forall (u0: T).(\forall (u1: T).((eq TList TNil ts) \to ((eq T u0 u) 
-\to ((ty3 g c u0 u1) \to (P g c ts u))))))))).(\lambda (H0: (((tys3 g c ts u) 
-\to (\forall (t: T).(\forall (u0: T).(\forall (ts0: TList).((eq TList (TCons 
-t ts0) ts) \to ((eq T u0 u) \to ((ty3 g c t u0) \to ((tys3 g c ts0 u0) \to (P 
-g c ts u))))))))))).(\lambda (H1: (tys3 g c ts u)).(let H2 \def (match H1 in 
-tys3 return (\lambda (t: TList).(\lambda (t0: T).(\lambda (_: (tys3 ? ? t 
-t0)).((eq TList t ts) \to ((eq T t0 u) \to (P g c ts u)))))) with [(tys3_nil 
-u0 u1 H2) \Rightarrow (\lambda (H3: (eq TList TNil ts)).(\lambda (H4: (eq T 
-u0 u)).(H H1 u0 u1 H3 H4 H2))) | (tys3_cons t u0 H2 ts0 H3) \Rightarrow 
-(\lambda (H4: (eq TList (TCons t ts0) ts)).(\lambda (H5: (eq T u0 u)).(H0 H1 
-t u0 ts0 H4 H5 H2 H3)))]) in (H2 (refl_equal TList ts) (refl_equal T 
-u)))))))))).
-
 theorem tys3_gen_nil:
  \forall (g: G).(\forall (c: C).(\forall (u: T).((tys3 g c TNil u) \to (ex T 
 (\lambda (u0: T).(ty3 g c u u0))))))
 \def
  \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (tys3 g c TNil 
-u)).(tys3_inv_coq g c TNil u (\lambda (g0: G).(\lambda (c0: C).(\lambda (_: 
-TList).(\lambda (t0: T).(ex T (\lambda (u0: T).(ty3 g0 c0 t0 u0))))))) 
-(\lambda (_: (tys3 g c TNil u)).(\lambda (u0: T).(\lambda (u1: T).(\lambda 
-(_: (eq TList TNil TNil)).(\lambda (H2: (eq T u0 u)).(\lambda (H3: (ty3 g c 
-u0 u1)).(let H4 \def (eq_ind T u0 (\lambda (t: T).(ty3 g c t u1)) H3 u H2) in 
-(ex_intro T (\lambda (u2: T).(ty3 g c u u2)) u1 H4)))))))) (\lambda (_: (tys3 
-g c TNil u)).(\lambda (t: T).(\lambda (u0: T).(\lambda (ts0: TList).(\lambda 
-(H2: (eq TList (TCons t ts0) TNil)).(\lambda (H3: (eq T u0 u)).(\lambda (H1: 
-(ty3 g c t u0)).(\lambda (H4: (tys3 g c ts0 u0)).(let H5 \def (eq_ind T u0 
-(\lambda (t0: T).(tys3 g c ts0 t0)) H4 u H3) in (let H6 \def (eq_ind T u0 
-(\lambda (t0: T).(ty3 g c t t0)) H1 u H3) in (let H7 \def (eq_ind TList 
-(TCons t ts0) (\lambda (ee: TList).(match ee in TList return (\lambda (_: 
-TList).Prop) with [TNil \Rightarrow False | (TCons _ _) \Rightarrow True])) I 
-TNil H2) in (False_ind (ex T (\lambda (u1: T).(ty3 g c u u1))) H7)))))))))))) 
-H)))).
+u)).(insert_eq TList TNil (\lambda (t: TList).(tys3 g c t u)) (\lambda (_: 
+TList).(ex T (\lambda (u0: T).(ty3 g c u u0)))) (\lambda (y: TList).(\lambda 
+(H0: (tys3 g c y u)).(tys3_ind g c (\lambda (t: TList).(\lambda (t0: T).((eq 
+TList t TNil) \to (ex T (\lambda (u0: T).(ty3 g c t0 u0)))))) (\lambda (u0: 
+T).(\lambda (u1: T).(\lambda (H1: (ty3 g c u0 u1)).(\lambda (_: (eq TList 
+TNil TNil)).(ex_intro T (\lambda (u2: T).(ty3 g c u0 u2)) u1 H1))))) (\lambda 
+(t: T).(\lambda (u0: T).(\lambda (_: (ty3 g c t u0)).(\lambda (ts: 
+TList).(\lambda (_: (tys3 g c ts u0)).(\lambda (_: (((eq TList ts TNil) \to 
+(ex T (\lambda (u1: T).(ty3 g c u0 u1)))))).(\lambda (H4: (eq TList (TCons t 
+ts) TNil)).(let H5 \def (eq_ind TList (TCons t ts) (\lambda (ee: 
+TList).(match ee in TList return (\lambda (_: TList).Prop) with [TNil 
+\Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H4) in (False_ind 
+(ex T (\lambda (u1: T).(ty3 g c u0 u1))) H5))))))))) y u H0))) H)))).
 
 theorem tys3_gen_cons:
  \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (t: T).(\forall 
@@ -192,30 +166,30 @@ theorem tys3_gen_cons:
 u)))))))
 \def
  \lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (t: T).(\lambda 
-(u: T).(\lambda (H: (tys3 g c (TCons t ts) u)).(tys3_inv_coq g c (TCons t ts) 
-u (\lambda (g0: G).(\lambda (c0: C).(\lambda (_: TList).(\lambda (t1: 
-T).(land (ty3 g0 c0 t t1) (tys3 g0 c0 ts t1)))))) (\lambda (_: (tys3 g c 
-(TCons t ts) u)).(\lambda (u0: T).(\lambda (u1: T).(\lambda (H1: (eq TList 
-TNil (TCons t ts))).(\lambda (H2: (eq T u0 u)).(\lambda (H3: (ty3 g c u0 
-u1)).(let H4 \def (eq_ind T u0 (\lambda (t0: T).(ty3 g c t0 u1)) H3 u H2) in 
-(let H5 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList 
+(u: T).(\lambda (H: (tys3 g c (TCons t ts) u)).(insert_eq TList (TCons t ts) 
+(\lambda (t0: TList).(tys3 g c t0 u)) (\lambda (_: TList).(land (ty3 g c t u) 
+(tys3 g c ts u))) (\lambda (y: TList).(\lambda (H0: (tys3 g c y u)).(tys3_ind 
+g c (\lambda (t0: TList).(\lambda (t1: T).((eq TList t0 (TCons t ts)) \to 
+(land (ty3 g c t t1) (tys3 g c ts t1))))) (\lambda (u0: T).(\lambda (u1: 
+T).(\lambda (_: (ty3 g c u0 u1)).(\lambda (H2: (eq TList TNil (TCons t 
+ts))).(let H3 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList 
 return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _) 
-\Rightarrow False])) I (TCons t ts) H1) in (False_ind (land (ty3 g c t u) 
-(tys3 g c ts u)) H5))))))))) (\lambda (_: (tys3 g c (TCons t ts) u)).(\lambda 
-(t0: T).(\lambda (u0: T).(\lambda (ts0: TList).(\lambda (H2: (eq TList (TCons 
-t0 ts0) (TCons t ts))).(\lambda (H3: (eq T u0 u)).(\lambda (H1: (ty3 g c t0 
-u0)).(\lambda (H4: (tys3 g c ts0 u0)).(let H5 \def (eq_ind T u0 (\lambda (t1: 
-T).(tys3 g c ts0 t1)) H4 u H3) in (let H6 \def (eq_ind T u0 (\lambda (t1: 
-T).(ty3 g c t0 t1)) H1 u H3) in (let H7 \def (f_equal TList T (\lambda (e: 
-TList).(match e in TList return (\lambda (_: TList).T) with [TNil \Rightarrow 
-t0 | (TCons t1 _) \Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H2) in ((let 
-H8 \def (f_equal TList TList (\lambda (e: TList).(match e in TList return 
-(\lambda (_: TList).TList) with [TNil \Rightarrow ts0 | (TCons _ t1) 
-\Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H2) in (\lambda (H9: (eq T t0 
-t)).(let H10 \def (eq_ind TList ts0 (\lambda (t1: TList).(tys3 g c t1 u)) H5 
-ts H8) in (let H11 \def (eq_ind T t0 (\lambda (t1: T).(ty3 g c t1 u)) H6 t 
-H9) in (conj (ty3 g c t u) (tys3 g c ts u) H11 H10))))) H7)))))))))))) 
-H)))))).
+\Rightarrow False])) I (TCons t ts) H2) in (False_ind (land (ty3 g c t u0) 
+(tys3 g c ts u0)) H3)))))) (\lambda (t0: T).(\lambda (u0: T).(\lambda (H1: 
+(ty3 g c t0 u0)).(\lambda (ts0: TList).(\lambda (H2: (tys3 g c ts0 
+u0)).(\lambda (H3: (((eq TList ts0 (TCons t ts)) \to (land (ty3 g c t u0) 
+(tys3 g c ts u0))))).(\lambda (H4: (eq TList (TCons t0 ts0) (TCons t 
+ts))).(let H5 \def (f_equal TList T (\lambda (e: TList).(match e in TList 
+return (\lambda (_: TList).T) with [TNil \Rightarrow t0 | (TCons t1 _) 
+\Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H4) in ((let H6 \def (f_equal 
+TList TList (\lambda (e: TList).(match e in TList return (\lambda (_: 
+TList).TList) with [TNil \Rightarrow ts0 | (TCons _ t1) \Rightarrow t1])) 
+(TCons t0 ts0) (TCons t ts) H4) in (\lambda (H7: (eq T t0 t)).(let H8 \def 
+(eq_ind TList ts0 (\lambda (t1: TList).((eq TList t1 (TCons t ts)) \to (land 
+(ty3 g c t u0) (tys3 g c ts u0)))) H3 ts H6) in (let H9 \def (eq_ind TList 
+ts0 (\lambda (t1: TList).(tys3 g c t1 u0)) H2 ts H6) in (let H10 \def (eq_ind 
+T t0 (\lambda (t1: T).(ty3 g c t1 u0)) H1 t H7) in (conj (ty3 g c t u0) (tys3 
+g c ts u0) H10 H9)))))) H5))))))))) y u H0))) H)))))).
 
 theorem ty3_getl_subst0:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/dec.ma
index bbdc36832..d93394c4a 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/dec.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/dec.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "subst0/defs.ma".
+include "LambdaDelta-1/subst0/defs.ma".
 
-include "lift/props.ma".
+include "LambdaDelta-1/lift/props.ma".
 
 theorem dnf_dec2:
  \forall (t: T).(\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/defs.ma
index 63eececf3..ef65debe3 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "lift/defs.ma".
+include "LambdaDelta-1/lift/defs.ma".
 
 inductive subst0: nat \to (T \to (T \to (T \to Prop))) \def
 | subst0_lref: \forall (v: T).(\forall (i: nat).(subst0 i v (TLRef i) (lift 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/fwd.ma
index 612a3261a..43d9e4df3 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/fwd.ma
@@ -14,198 +14,88 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "subst0/defs.ma".
+include "LambdaDelta-1/subst0/defs.ma".
 
-include "lift/props.ma".
-
-theorem subst0_inv_coq:
- \forall (i: nat).(\forall (v: T).(\forall (t1: T).(\forall (t2: T).(\forall 
-(P: ((nat \to (T \to (T \to (T \to Prop)))))).((((subst0 i v t1 t2) \to 
-(\forall (v0: T).(\forall (i0: nat).((eq nat i0 i) \to ((eq T v0 v) \to ((eq 
-T (TLRef i0) t1) \to ((eq T (lift (S i0) O v0) t2) \to (P i v t1 t2))))))))) 
-\to ((((subst0 i v t1 t2) \to (\forall (v0: T).(\forall (u2: T).(\forall (u1: 
-T).(\forall (i0: nat).(\forall (t: T).(\forall (k: K).((eq nat i0 i) \to ((eq 
-T v0 v) \to ((eq T (THead k u1 t) t1) \to ((eq T (THead k u2 t) t2) \to 
-((subst0 i0 v0 u1 u2) \to (P i v t1 t2)))))))))))))) \to ((((subst0 i v t1 
-t2) \to (\forall (k: K).(\forall (v0: T).(\forall (t0: T).(\forall (t3: 
-T).(\forall (i0: nat).(\forall (u: T).((eq nat i0 i) \to ((eq T v0 v) \to 
-((eq T (THead k u t3) t1) \to ((eq T (THead k u t0) t2) \to ((subst0 (s k i0) 
-v0 t3 t0) \to (P i v t1 t2)))))))))))))) \to ((((subst0 i v t1 t2) \to 
-(\forall (v0: T).(\forall (u1: T).(\forall (u2: T).(\forall (i0: 
-nat).(\forall (k: K).(\forall (t0: T).(\forall (t3: T).((eq nat i0 i) \to 
-((eq T v0 v) \to ((eq T (THead k u1 t0) t1) \to ((eq T (THead k u2 t3) t2) 
-\to ((subst0 i0 v0 u1 u2) \to ((subst0 (s k i0) v0 t0 t3) \to (P i v t1 
-t2)))))))))))))))) \to ((subst0 i v t1 t2) \to (P i v t1 t2))))))))))
-\def
- \lambda (i: nat).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(P: ((nat \to (T \to (T \to (T \to Prop)))))).(\lambda (H: (((subst0 i v t1 
-t2) \to (\forall (v0: T).(\forall (i0: nat).((eq nat i0 i) \to ((eq T v0 v) 
-\to ((eq T (TLRef i0) t1) \to ((eq T (lift (S i0) O v0) t2) \to (P i v t1 
-t2)))))))))).(\lambda (H0: (((subst0 i v t1 t2) \to (\forall (v0: T).(\forall 
-(u2: T).(\forall (u1: T).(\forall (i0: nat).(\forall (t: T).(\forall (k: 
-K).((eq nat i0 i) \to ((eq T v0 v) \to ((eq T (THead k u1 t) t1) \to ((eq T 
-(THead k u2 t) t2) \to ((subst0 i0 v0 u1 u2) \to (P i v t1 
-t2))))))))))))))).(\lambda (H1: (((subst0 i v t1 t2) \to (\forall (k: 
-K).(\forall (v0: T).(\forall (t0: T).(\forall (t3: T).(\forall (i0: 
-nat).(\forall (u: T).((eq nat i0 i) \to ((eq T v0 v) \to ((eq T (THead k u 
-t3) t1) \to ((eq T (THead k u t0) t2) \to ((subst0 (s k i0) v0 t3 t0) \to (P 
-i v t1 t2))))))))))))))).(\lambda (H2: (((subst0 i v t1 t2) \to (\forall (v0: 
-T).(\forall (u1: T).(\forall (u2: T).(\forall (i0: nat).(\forall (k: 
-K).(\forall (t0: T).(\forall (t3: T).((eq nat i0 i) \to ((eq T v0 v) \to ((eq 
-T (THead k u1 t0) t1) \to ((eq T (THead k u2 t3) t2) \to ((subst0 i0 v0 u1 
-u2) \to ((subst0 (s k i0) v0 t0 t3) \to (P i v t1 
-t2))))))))))))))))).(\lambda (H3: (subst0 i v t1 t2)).(let H4 \def (match H3 
-in subst0 return (\lambda (n: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda 
-(t3: T).(\lambda (_: (subst0 n t t0 t3)).((eq nat n i) \to ((eq T t v) \to 
-((eq T t0 t1) \to ((eq T t3 t2) \to (P i v t1 t2)))))))))) with [(subst0_lref 
-v0 i0) \Rightarrow (\lambda (H4: (eq nat i0 i)).(\lambda (H5: (eq T v0 
-v)).(\lambda (H6: (eq T (TLRef i0) t1)).(\lambda (H7: (eq T (lift (S i0) O 
-v0) t2)).(H H3 v0 i0 H4 H5 H6 H7))))) | (subst0_fst v0 u2 u1 i0 H4 t k) 
-\Rightarrow (\lambda (H5: (eq nat i0 i)).(\lambda (H6: (eq T v0 v)).(\lambda 
-(H7: (eq T (THead k u1 t) t1)).(\lambda (H8: (eq T (THead k u2 t) t2)).(H0 H3 
-v0 u2 u1 i0 t k H5 H6 H7 H8 H4))))) | (subst0_snd k v0 t0 t3 i0 H4 u) 
-\Rightarrow (\lambda (H5: (eq nat i0 i)).(\lambda (H6: (eq T v0 v)).(\lambda 
-(H7: (eq T (THead k u t3) t1)).(\lambda (H8: (eq T (THead k u t0) t2)).(H1 H3 
-k v0 t0 t3 i0 u H5 H6 H7 H8 H4))))) | (subst0_both v0 u1 u2 i0 H4 k t0 t3 H5) 
-\Rightarrow (\lambda (H6: (eq nat i0 i)).(\lambda (H7: (eq T v0 v)).(\lambda 
-(H8: (eq T (THead k u1 t0) t1)).(\lambda (H9: (eq T (THead k u2 t3) t2)).(H2 
-H3 v0 u1 u2 i0 k t0 t3 H6 H7 H8 H9 H4 H5)))))]) in (H4 (refl_equal nat i) 
-(refl_equal T v) (refl_equal T t1) (refl_equal T t2)))))))))))).
+include "LambdaDelta-1/lift/props.ma".
 
 theorem 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).(subst0_inv_coq i v (TSort 
-n) x (\lambda (_: nat).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).P)))) 
-(\lambda (H0: (subst0 i v (TSort n) x)).(\lambda (v0: T).(\lambda (i0: 
-nat).(\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: 
-(eq T (TLRef i0) (TSort n))).(\lambda (H4: (eq T (lift (S i0) O v0) x)).(let 
-H5 \def (eq_ind nat i0 (\lambda (n0: nat).(eq T (lift (S n0) O v0) x)) H4 i 
-H1) in (let H6 \def (eq_ind nat i0 (\lambda (n0: nat).(eq T (TLRef n0) (TSort 
-n))) H3 i H1) in (let H7 \def (eq_ind T v0 (\lambda (t: T).(eq T (lift (S i) 
-O t) x)) H5 v H2) in (let H8 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v 
-(TSort n) t)) H0 (lift (S i) O v) H7) in (let H9 \def (eq_ind_r T x (\lambda 
-(t: T).(subst0 i v (TSort n) t)) H (lift (S i) O v) H7) in (let H10 \def 
-(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: 
+(H: (subst0 i v (TSort n) x)).(\lambda (P: Prop).(insert_eq T (TSort n) 
+(\lambda (t: T).(subst0 i v t x)) (\lambda (_: T).P) (\lambda (y: T).(\lambda 
+(H0: (subst0 i v y x)).(subst0_ind (\lambda (_: nat).(\lambda (_: T).(\lambda 
+(t0: T).(\lambda (_: T).((eq T t0 (TSort n)) \to P))))) (\lambda (_: 
+T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TSort n))).(let H2 \def 
+(eq_ind T (TLRef i0) (\lambda (ee: T).(match ee in T return (\lambda (_: 
 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (TSort n) H6) in (False_ind P 
-H10)))))))))))))) (\lambda (H0: (subst0 i v (TSort n) x)).(\lambda (v0: 
-T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (t: 
-T).(\lambda (k: K).(\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)).(\lambda (H5: (subst0 i0 v0 u1 u2)).(let H6 \def (eq_ind nat i0 
-(\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H5 i H1) in (let H7 \def (eq_ind T 
-v0 (\lambda (t0: T).(subst0 i t0 u1 u2)) H6 v H2) in (let H8 \def (eq_ind_r T 
-x (\lambda (t0: T).(subst0 i v (TSort n) t0)) H0 (THead k u2 t) H4) in (let 
-H9 \def (eq_ind_r T x (\lambda (t0: T).(subst0 i v (TSort n) t0)) H (THead k 
-u2 t) H4) in (let H10 \def (eq_ind T (THead k u1 t) (\lambda (ee: T).(match 
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) 
-H3) in (False_ind P H10)))))))))))))))))) (\lambda (H0: (subst0 i v (TSort n) 
-x)).(\lambda (k: K).(\lambda (v0: T).(\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (i0: nat).(\lambda (u: T).(\lambda (H1: (eq nat i0 i)).(\lambda 
-(H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u t3) (TSort n))).(\lambda 
-(H4: (eq T (THead k u t0) x)).(\lambda (H5: (subst0 (s k i0) v0 t3 t0)).(let 
-H6 \def (eq_ind nat i0 (\lambda (n0: nat).(subst0 (s k n0) v0 t3 t0)) H5 i 
-H1) in (let H7 \def (eq_ind T v0 (\lambda (t: T).(subst0 (s k i) t t3 t0)) H6 
-v H2) in (let H8 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (TSort n) t)) 
-H0 (THead k u t0) H4) in (let H9 \def (eq_ind_r T x (\lambda (t: T).(subst0 i 
-v (TSort n) t)) H (THead k u t0) H4) in (let H10 \def (eq_ind T (THead k u 
-t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TSort n) H3) in (False_ind P H10)))))))))))))))))) 
-(\lambda (H0: (subst0 i v (TSort n) x)).(\lambda (v0: T).(\lambda (u1: 
-T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (k: K).(\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (H2: (eq nat i0 i)).(\lambda (H3: (eq T v0 
-v)).(\lambda (H4: (eq T (THead k u1 t0) (TSort n))).(\lambda (H5: (eq T 
-(THead k u2 t3) x)).(\lambda (H1: (subst0 i0 v0 u1 u2)).(\lambda (H6: (subst0 
-(s k i0) v0 t0 t3)).(let H7 \def (eq_ind nat i0 (\lambda (n0: nat).(subst0 (s 
-k n0) v0 t0 t3)) H6 i H2) in (let H8 \def (eq_ind nat i0 (\lambda (n0: 
-nat).(subst0 n0 v0 u1 u2)) H1 i H2) in (let H9 \def (eq_ind T v0 (\lambda (t: 
-T).(subst0 (s k i) t t0 t3)) H7 v H3) in (let H10 \def (eq_ind T v0 (\lambda 
-(t: T).(subst0 i t u1 u2)) H8 v H3) in (let H11 \def (eq_ind_r T x (\lambda 
-(t: T).(subst0 i v (TSort n) t)) H0 (THead k u2 t3) H5) in (let H12 \def 
-(eq_ind_r T x (\lambda (t: T).(subst0 i v (TSort n) t)) H (THead k u2 t3) H5) 
-in (let H13 \def (eq_ind T (THead k u1 t0) (\lambda (ee: T).(match ee in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H4) in 
-(False_ind P H13)))))))))))))))))))))) H)))))).
+(THead _ _ _) \Rightarrow False])) I (TSort n) H1) in (False_ind P H2))))) 
+(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: 
+nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) 
+\to P))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T (THead k u1 t) 
+(TSort n))).(let H4 \def (eq_ind T (THead k u1 t) (\lambda (ee: T).(match ee 
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef 
+_) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in 
+(False_ind P H4))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: 
+T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v0 t1 
+t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (u: T).(\lambda 
+(H3: (eq T (THead k u t1) (TSort n))).(let H4 \def (eq_ind T (THead k u t1) 
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TSort n) H3) in (False_ind P H4))))))))))) (\lambda (v0: 
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0 
+i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to P))).(\lambda (k: 
+K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (s k i0) v0 t1 
+t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (H5: (eq T (THead k 
+u1 t1) (TSort n))).(let H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: 
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
+(TSort n) H5) in (False_ind P H6)))))))))))))) i v y x H0))) H)))))).
 
 theorem 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)).(subst0_inv_coq i v (TLRef n) x (\lambda (n0: 
-nat).(\lambda (t: T).(\lambda (_: T).(\lambda (t1: T).(land (eq nat n n0) (eq 
-T t1 (lift (S n) O t))))))) (\lambda (H0: (subst0 i v (TLRef n) x)).(\lambda 
-(v0: T).(\lambda (i0: nat).(\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T 
-v0 v)).(\lambda (H3: (eq T (TLRef i0) (TLRef n))).(\lambda (H4: (eq T (lift 
-(S i0) O v0) x)).(let H5 \def (eq_ind nat i0 (\lambda (n0: nat).(eq T (lift 
-(S n0) O v0) x)) H4 i H1) in (let H6 \def (eq_ind nat i0 (\lambda (n0: 
-nat).(eq T (TLRef n0) (TLRef n))) H3 i H1) in (let H7 \def (eq_ind T v0 
-(\lambda (t: T).(eq T (lift (S i) O t) x)) H5 v H2) in (let H8 \def (eq_ind_r 
-T x (\lambda (t: T).(subst0 i v (TLRef n) t)) H0 (lift (S i) O v) H7) in (let 
-H9 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (TLRef n) t)) H (lift (S i) 
-O v) H7) in (eq_ind T (lift (S i) O v) (\lambda (t: T).(land (eq nat n i) (eq 
-T t (lift (S n) O v)))) (let H10 \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) H6) in 
-(let H11 \def (eq_ind_r nat n (\lambda (n0: nat).(subst0 i v (TLRef n0) (lift 
-(S i) O v))) H8 i H10) in (let H12 \def (eq_ind_r nat n (\lambda (n0: 
-nat).(subst0 i v (TLRef n0) (lift (S i) O v))) H9 i H10) in (eq_ind nat i 
-(\lambda (n0: nat).(land (eq nat n0 i) (eq T (lift (S i) O v) (lift (S n0) O 
-v)))) (conj (eq nat i i) (eq T (lift (S i) O v) (lift (S i) O v)) (refl_equal 
-nat i) (refl_equal T (lift (S i) O v))) n H10)))) x H7))))))))))))) (\lambda 
-(H0: (subst0 i v (TLRef n) x)).(\lambda (v0: T).(\lambda (u2: T).(\lambda 
-(u1: T).(\lambda (i0: nat).(\lambda (t: T).(\lambda (k: K).(\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)).(\lambda (H5: (subst0 i0 
-v0 u1 u2)).(let H6 \def (eq_ind nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1 
-u2)) H5 i H1) in (let H7 \def (eq_ind T v0 (\lambda (t0: T).(subst0 i t0 u1 
-u2)) H6 v H2) in (let H8 \def (eq_ind_r T x (\lambda (t0: T).(subst0 i v 
-(TLRef n) t0)) H0 (THead k u2 t) H4) in (let H9 \def (eq_ind_r T x (\lambda 
-(t0: T).(subst0 i v (TLRef n) t0)) H (THead k u2 t) H4) in (eq_ind T (THead k 
-u2 t) (\lambda (t0: T).(land (eq nat n i) (eq T t0 (lift (S n) O v)))) (let 
-H10 \def (eq_ind T (THead k u1 t) (\lambda (ee: T).(match ee in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H3) in 
-(False_ind (land (eq nat n i) (eq T (THead k u2 t) (lift (S n) O v))) H10)) x 
-H4))))))))))))))))) (\lambda (H0: (subst0 i v (TLRef n) x)).(\lambda (k: 
-K).(\lambda (v0: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: 
-nat).(\lambda (u: T).(\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 
-v)).(\lambda (H3: (eq T (THead k u t3) (TLRef n))).(\lambda (H4: (eq T (THead 
-k u t0) x)).(\lambda (H5: (subst0 (s k i0) v0 t3 t0)).(let H6 \def (eq_ind 
-nat i0 (\lambda (n0: nat).(subst0 (s k n0) v0 t3 t0)) H5 i H1) in (let H7 
-\def (eq_ind T v0 (\lambda (t: T).(subst0 (s k i) t t3 t0)) H6 v H2) in (let 
-H8 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (TLRef n) t)) H0 (THead k u 
-t0) H4) in (let H9 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (TLRef n) 
-t)) H (THead k u t0) H4) in (eq_ind T (THead k u t0) (\lambda (t: T).(land 
-(eq nat n i) (eq T t (lift (S n) O v)))) (let H10 \def (eq_ind T (THead k u 
-t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TLRef n) H3) in (False_ind (land (eq nat n i) (eq T 
-(THead k u t0) (lift (S n) O v))) H10)) x H4))))))))))))))))) (\lambda (H0: 
-(subst0 i v (TLRef n) x)).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (i0: nat).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (H2: (eq nat i0 i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq 
-T (THead k u1 t0) (TLRef n))).(\lambda (H5: (eq T (THead k u2 t3) 
-x)).(\lambda (H1: (subst0 i0 v0 u1 u2)).(\lambda (H6: (subst0 (s k i0) v0 t0 
-t3)).(let H7 \def (eq_ind nat i0 (\lambda (n0: nat).(subst0 (s k n0) v0 t0 
-t3)) H6 i H2) in (let H8 \def (eq_ind nat i0 (\lambda (n0: nat).(subst0 n0 v0 
-u1 u2)) H1 i H2) in (let H9 \def (eq_ind T v0 (\lambda (t: T).(subst0 (s k i) 
-t t0 t3)) H7 v H3) in (let H10 \def (eq_ind T v0 (\lambda (t: T).(subst0 i t 
-u1 u2)) H8 v H3) in (let H11 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v 
-(TLRef n) t)) H0 (THead k u2 t3) H5) in (let H12 \def (eq_ind_r T x (\lambda 
-(t: T).(subst0 i v (TLRef n) t)) H (THead k u2 t3) H5) in (eq_ind T (THead k 
-u2 t3) (\lambda (t: T).(land (eq nat n i) (eq T t (lift (S n) O v)))) (let 
-H13 \def (eq_ind T (THead k u1 t0) (\lambda (ee: T).(match ee in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H4) in 
-(False_ind (land (eq nat n i) (eq T (THead k u2 t3) (lift (S n) O v))) H13)) 
-x H5))))))))))))))))))))) H))))).
+(H: (subst0 i v (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(subst0 
+i v t x)) (\lambda (_: T).(land (eq nat n i) (eq T x (lift (S n) O v)))) 
+(\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda (n0: 
+nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).((eq T t0 (TLRef n)) 
+\to (land (eq nat n n0) (eq T t1 (lift (S n) O t)))))))) (\lambda (v0: 
+T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TLRef n))).(let H2 \def 
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with 
+[(TSort _) \Rightarrow i0 | (TLRef n0) \Rightarrow n0 | (THead _ _ _) 
+\Rightarrow i0])) (TLRef i0) (TLRef n) H1) in (eq_ind_r nat n (\lambda (n0: 
+nat).(land (eq nat n n0) (eq T (lift (S n0) O v0) (lift (S n) O v0)))) (conj 
+(eq nat n n) (eq T (lift (S n) O v0) (lift (S n) O v0)) (refl_equal nat n) 
+(refl_equal T (lift (S n) O v0))) i0 H2))))) (\lambda (v0: T).(\lambda (u2: 
+T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1 
+u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2 
+(lift (S n) O v0)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T 
+(THead k u1 t) (TLRef n))).(let H4 \def (eq_ind T (THead k u1 t) (\lambda 
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u2 
+t) (lift (S n) O v0))) H4))))))))))) (\lambda (k: K).(\lambda (v0: 
+T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 
+(s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (land (eq nat n (s 
+k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda (u: T).(\lambda (H3: (eq T 
+(THead k u t1) (TLRef n))).(let H4 \def (eq_ind T (THead k u t1) (\lambda 
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u 
+t2) (lift (S n) O v0))) H4))))))))))) (\lambda (v0: T).(\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1 
+u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2 
+(lift (S n) O v0)))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef 
+n)) \to (land (eq nat n (s k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda 
+(H5: (eq T (THead k u1 t1) (TLRef n))).(let H6 \def (eq_ind T (THead k u1 t1) 
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TLRef n) H5) in (False_ind (land (eq nat n i0) (eq T (THead k u2 
+t2) (lift (S n) O v0))) H6)))))))))))))) i v y x H0))) H))))).
 
 theorem subst0_gen_head:
  \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall 
@@ -218,190 +108,195 @@ 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)).(subst0_inv_coq i v (THead k u1 t1) x (\lambda (n: nat).(\lambda (t: 
-T).(\lambda (_: T).(\lambda (t2: T).(or3 (ex2 T (\lambda (u2: T).(eq T t2 
-(THead k u2 t1))) (\lambda (u2: T).(subst0 n t u1 u2))) (ex2 T (\lambda (t3: 
-T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k n) t t1 t3))) 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 n t u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s k n) t t1 t3))))))))) (\lambda (H0: (subst0 i 
-v (THead k u1 t1) x)).(\lambda (v0: T).(\lambda (i0: nat).(\lambda (H1: (eq 
-nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (TLRef i0) (THead k 
-u1 t1))).(\lambda (H4: (eq T (lift (S i0) O v0) x)).(let H5 \def (eq_ind nat 
-i0 (\lambda (n: nat).(eq T (lift (S n) O v0) x)) H4 i H1) in (let H6 \def 
-(eq_ind nat i0 (\lambda (n: nat).(eq T (TLRef n) (THead k u1 t1))) H3 i H1) 
-in (let H7 \def (eq_ind T v0 (\lambda (t: T).(eq T (lift (S i) O t) x)) H5 v 
-H2) in (let H8 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (THead k u1 t1) 
-t)) H0 (lift (S i) O v) H7) in (let H9 \def (eq_ind_r T x (\lambda (t: 
-T).(subst0 i v (THead k u1 t1) t)) H (lift (S i) O v) H7) in (eq_ind T (lift 
-(S i) O v) (\lambda (t: T).(or3 (ex2 T (\lambda (u2: T).(eq T t (THead k u2 
-t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T t 
-(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 t (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)))))) (let H10 \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 k u1 t1) H6) in (False_ind (or3 (ex2 T (\lambda (u2: T).(eq 
-T (lift (S i) O v) (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) 
-(ex2 T (\lambda (t2: T).(eq T (lift (S i) O v) (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 (lift (S i) O v) (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))))) H10)) x H7))))))))))))) (\lambda (H0: (subst0 i v (THead k u1 t1) 
-x)).(\lambda (v0: T).(\lambda (u2: T).(\lambda (u0: T).(\lambda (i0: 
-nat).(\lambda (t: T).(\lambda (k0: K).(\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)).(\lambda (H5: (subst0 i0 v0 u0 
-u2)).(let H6 \def (eq_ind nat i0 (\lambda (n: nat).(subst0 n v0 u0 u2)) H5 i 
-H1) in (let H7 \def (eq_ind T v0 (\lambda (t0: T).(subst0 i t0 u0 u2)) H6 v 
-H2) in (let H8 \def (eq_ind_r T x (\lambda (t0: T).(subst0 i v (THead k u1 
-t1) t0)) H0 (THead k0 u2 t) H4) in (let H9 \def (eq_ind_r T x (\lambda (t0: 
-T).(subst0 i v (THead k u1 t1) t0)) H (THead k0 u2 t) H4) in (eq_ind T (THead 
-k0 u2 t) (\lambda (t0: T).(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)))))) (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 t) 
-(THead k u1 t1) H3) in ((let H11 \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) H3) in ((let H12 \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) H3) in 
-(\lambda (H13: (eq T u0 u1)).(\lambda (H14: (eq K k0 k)).(let H15 \def 
-(eq_ind K k0 (\lambda (k1: K).(subst0 i v (THead k u1 t1) (THead k1 u2 t))) 
-H9 k H14) in (let H16 \def (eq_ind K k0 (\lambda (k1: K).(subst0 i v (THead k 
-u1 t1) (THead k1 u2 t))) H8 k H14) in (eq_ind_r K k (\lambda (k1: K).(or3 
-(ex2 T (\lambda (u3: T).(eq T (THead k1 u2 t) (THead k u3 t1))) (\lambda (u3: 
-T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k1 u2 t) (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 k1 u2 t) (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)))))) (let H17 \def (eq_ind T t (\lambda (t0: 
-T).(subst0 i v (THead k u1 t1) (THead k u2 t0))) H15 t1 H12) in (let H18 \def 
-(eq_ind T t (\lambda (t0: T).(subst0 i v (THead k u1 t1) (THead k u2 t0))) 
-H16 t1 H12) in (eq_ind_r T t1 (\lambda (t0: T).(or3 (ex2 T (\lambda (u3: 
-T).(eq T (THead k u2 t0) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 
-u3))) (ex2 T (\lambda (t2: T).(eq T (THead k u2 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 (THead k u2 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)))))) (let H19 \def (eq_ind T u0 (\lambda (t0: 
-T).(subst0 i v t0 u2)) H7 u1 H13) in (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)) H19))) 
-t H12))) k0 H14)))))) H11)) H10)) x H4))))))))))))))))) (\lambda (H0: (subst0 
-i v (THead k u1 t1) x)).(\lambda (k0: K).(\lambda (v0: T).(\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (u: T).(\lambda (H1: (eq nat 
-i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k0 u t3) (THead 
-k u1 t1))).(\lambda (H4: (eq T (THead k0 u t0) x)).(\lambda (H5: (subst0 (s 
-k0 i0) v0 t3 t0)).(let H6 \def (eq_ind nat i0 (\lambda (n: nat).(subst0 (s k0 
-n) v0 t3 t0)) H5 i H1) in (let H7 \def (eq_ind T v0 (\lambda (t: T).(subst0 
-(s k0 i) t t3 t0)) H6 v H2) in (let H8 \def (eq_ind_r T x (\lambda (t: 
-T).(subst0 i v (THead k u1 t1) t)) H0 (THead k0 u t0) H4) in (let H9 \def 
-(eq_ind_r T x (\lambda (t: T).(subst0 i v (THead k u1 t1) t)) H (THead k0 u 
-t0) H4) in (eq_ind T (THead k0 u t0) (\lambda (t: T).(or3 (ex2 T (\lambda 
-(u2: T).(eq T t (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 
-T (\lambda (t2: T).(eq T t (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 t (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)))))) (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 u t3) (THead k u1 t1) H3) in ((let H11 \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 t3) 
-(THead k u1 t1) H3) in ((let H12 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) 
-\Rightarrow t3 | (THead _ _ t) \Rightarrow t])) (THead k0 u t3) (THead k u1 
-t1) H3) in (\lambda (H13: (eq T u u1)).(\lambda (H14: (eq K k0 k)).(let H15 
-\def (eq_ind T u (\lambda (t: T).(subst0 i v (THead k u1 t1) (THead k0 t 
-t0))) H9 u1 H13) in (let H16 \def (eq_ind T u (\lambda (t: T).(subst0 i v 
-(THead k u1 t1) (THead k0 t t0))) H8 u1 H13) in (eq_ind_r T u1 (\lambda (t: 
-T).(or3 (ex2 T (\lambda (u2: T).(eq T (THead k0 t t0) (THead k u2 t1))) 
-(\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k0 
-t t0) (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 (THead k0 t t0) (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)))))) (let H17 \def (eq_ind T t3 
-(\lambda (t: T).(subst0 (s k0 i) v t t0)) H7 t1 H12) in (let H18 \def (eq_ind 
-K k0 (\lambda (k1: K).(subst0 i v (THead k u1 t1) (THead k1 u1 t0))) H15 k 
-H14) in (let H19 \def (eq_ind K k0 (\lambda (k1: K).(subst0 i v (THead k u1 
-t1) (THead k1 u1 t0))) H16 k H14) in (let H20 \def (eq_ind K k0 (\lambda (k1: 
-K).(subst0 (s k1 i) v t1 t0)) H17 k H14) in (eq_ind_r K k (\lambda (k1: 
-K).(or3 (ex2 T (\lambda (u2: T).(eq T (THead k1 u1 t0) (THead k u2 t1))) 
-(\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k1 
-u1 t0) (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 (THead k1 u1 t0) (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)))))) (or3_intro1 (ex2 T (\lambda 
-(u2: T).(eq T (THead k u1 t0) (THead k u2 t1))) (\lambda (u2: T).(subst0 i v 
-u1 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k u1 t0) (THead k u1 t2))) 
+x)).(insert_eq T (THead k u1 t1) (\lambda (t: T).(subst0 i v t x)) (\lambda 
+(_: T).(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 (THead k u1 t0) (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)))) (ex_intro2 T (\lambda (t2: T).(eq T (THead k 
-u1 t0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2)) t0 
-(refl_equal T (THead k u1 t0)) H20)) k0 H14))))) u H13)))))) H11)) H10)) x 
-H4))))))))))))))))) (\lambda (H0: (subst0 i v (THead k u1 t1) x)).(\lambda 
-(v0: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (k0: 
-K).(\lambda (t0: T).(\lambda (t3: T).(\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 t3) x)).(\lambda (H1: (subst0 i0 v0 u0 
-u2)).(\lambda (H6: (subst0 (s k0 i0) v0 t0 t3)).(let H7 \def (eq_ind nat i0 
-(\lambda (n: nat).(subst0 (s k0 n) v0 t0 t3)) H6 i H2) in (let H8 \def 
-(eq_ind nat i0 (\lambda (n: nat).(subst0 n v0 u0 u2)) H1 i H2) in (let H9 
-\def (eq_ind T v0 (\lambda (t: T).(subst0 (s k0 i) t t0 t3)) H7 v H3) in (let 
-H10 \def (eq_ind T v0 (\lambda (t: T).(subst0 i t u0 u2)) H8 v H3) in (let 
-H11 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (THead k u1 t1) t)) H0 
-(THead k0 u2 t3) H5) in (let H12 \def (eq_ind_r T x (\lambda (t: T).(subst0 i 
-v (THead k u1 t1) t)) H (THead k0 u2 t3) H5) in (eq_ind T (THead k0 u2 t3) 
-(\lambda (t: T).(or3 (ex2 T (\lambda (u3: T).(eq T t (THead k u3 t1))) 
-(\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T t (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 t (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)))))) (let H13 \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) H4) in 
-((let H14 \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) H4) in ((let H15 \def 
+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 (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda 
+(n: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t2: T).((eq T t0 (THead k 
+u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda 
+(u2: T).(subst0 n t u1 u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 
+t3))) (\lambda (t3: T).(subst0 (s k n) t t1 t3))) (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 n t u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k n) t t1 
+t3)))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef 
+i0) (THead k u1 t1))).(let H2 \def (eq_ind T (TLRef i0) (\lambda (ee: 
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
+(THead k u1 t1) H1) in (False_ind (or3 (ex2 T (\lambda (u2: T).(eq T (lift (S 
+i0) O v0) (THead k u2 t1))) (\lambda (u2: T).(subst0 i0 v0 u1 u2))) (ex2 T 
+(\lambda (t2: T).(eq T (lift (S i0) O v0) (THead k u1 t2))) (\lambda (t2: 
+T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T (lift (S i0) O v0) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i0 v0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) 
+v0 t1 t2))))) H2))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u0: 
+T).(\lambda (i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq 
+T u0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 
+t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T 
+u2 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (t: T).(\lambda (k0: 
+K).(\lambda (H3: (eq T (THead k0 u0 t) (THead k u1 t1))).(let H4 \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) H3) 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 _ t0 _) 
+\Rightarrow t0])) (THead k0 u0 t) (THead k 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 t | (TLRef _) \Rightarrow t | (THead _ _ t0) 
+\Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in (\lambda (H7: (eq T 
+u0 u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T 
+(\lambda (u3: T).(eq T (THead k1 u2 t) (THead k u3 t1))) (\lambda (u3: 
+T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k1 u2 t) 
+(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T (THead k1 u2 t) (THead k u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: 
+T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (eq_ind_r T t1 (\lambda 
+(t0: T).(or3 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t0) (THead k u3 t1))) 
+(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead 
+k u2 t0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t0) (THead k 
+u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda 
+(_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (let H9 \def (eq_ind 
+T u0 (\lambda (t0: T).((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda 
+(u3: T).(eq T u2 (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) 
+(ex2 T (\lambda (t2: T).(eq T u2 (THead k u1 t2))) (\lambda (t2: T).(subst0 
+(s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 
+(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) 
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))))))) H2 u1 H7) 
+in (let H10 \def (eq_ind T u0 (\lambda (t0: T).(subst0 i0 v0 t0 u2)) H1 u1 
+H7) in (or3_intro0 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 
+t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T 
+(THead k u2 t1) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 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 i0 v0 u1 u3))) 
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))) (ex_intro2 T 
+(\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3: 
+T).(subst0 i0 v0 u1 u3)) u2 (refl_equal T (THead k u2 t1)) H10)))) t H6) k0 
+H8)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (v0: T).(\lambda (t2: 
+T).(\lambda (t0: T).(\lambda (i0: nat).(\lambda (H1: (subst0 (s k0 i0) v0 t0 
+t2)).(\lambda (H2: (((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: 
+T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2))) 
+(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 
+(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
+T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0 
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 
+t3)))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead k0 u t0) (THead k u1 
+t1))).(let H4 \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) H3) 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 k0 u t0) (THead k 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 t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) 
+\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in (\lambda (H7: (eq T u 
+u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex2 T 
+(\lambda (u2: T).(eq T (THead k0 t t2) (THead k u2 t1))) (\lambda (u2: 
+T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k0 t t2) 
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k0 t t2) (THead k u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (let H9 \def (eq_ind T t0 
+(\lambda (t: T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: 
+T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2))) 
+(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 
+(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
+T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0 
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 
+t3))))))) H2 t1 H6) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s 
+k0 i0) v0 t t2)) H1 t1 H6) in (let H11 \def (eq_ind K k0 (\lambda (k1: 
+K).((eq T t1 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 
+(THead k u2 t1))) (\lambda (u2: T).(subst0 (s k1 i0) v0 u1 u2))) (ex2 T 
+(\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s 
+k1 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
+(THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 
+t3))))))) H9 k H8) in (let H12 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s 
+k1 i0) v0 t1 t2)) H10 k H8) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T 
+(\lambda (u2: T).(eq T (THead k1 u1 t2) (THead k u2 t1))) (\lambda (u2: 
+T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k1 u1 t2) 
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k1 u1 t2) (THead k u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (or3_intro1 (ex2 T 
+(\lambda (u2: T).(eq T (THead k u1 t2) (THead k u2 t1))) (\lambda (u2: 
+T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k u1 t2) 
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 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 i0 v0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))) (ex_intro2 T (\lambda (t3: 
+T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) 
+v0 t1 t3)) t2 (refl_equal T (THead k u1 t2)) H12)) k0 H8))))) u H7)))) H5)) 
+H4))))))))))) (\lambda (v0: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda 
+(i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq T u0 (THead 
+k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1))) 
+(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T u2 
+(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (k0: K).(\lambda (t0: 
+T).(\lambda (t2: T).(\lambda (H3: (subst0 (s k0 i0) v0 t0 t2)).(\lambda (H4: 
+(((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead 
+k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda 
+(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0)) 
+v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3))) 
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 
+t3)))))))).(\lambda (H5: (eq T (THead k0 u0 t0) (THead k u1 t1))).(let H6 
+\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) H5) in ((let H7 \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) H5) in ((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) H4) in (\lambda (H16: (eq T 
-u0 u1)).(\lambda (H17: (eq K k0 k)).(let H18 \def (eq_ind T t0 (\lambda (t: 
-T).(subst0 (s k0 i) v t t3)) H9 t1 H15) in (let H19 \def (eq_ind K k0 
-(\lambda (k1: K).(subst0 i v (THead k u1 t1) (THead k1 u2 t3))) H12 k H17) in 
-(let H20 \def (eq_ind K k0 (\lambda (k1: K).(subst0 i v (THead k u1 t1) 
-(THead k1 u2 t3))) H11 k H17) in (let H21 \def (eq_ind K k0 (\lambda (k1: 
-K).(subst0 (s k1 i) v t1 t3)) H18 k H17) in (eq_ind_r K k (\lambda (k1: 
-K).(or3 (ex2 T (\lambda (u3: T).(eq T (THead k1 u2 t3) (THead k u3 t1))) 
-(\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k1 
-u2 t3) (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 k1 u2 t3) (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)))))) (let H22 \def (eq_ind T u0 
-(\lambda (t: T).(subst0 i v t u2)) H10 u1 H16) in (or3_intro2 (ex2 T (\lambda 
-(u3: T).(eq T (THead k u2 t3) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v 
-u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k u2 t3) (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 t3) (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)))) (ex3_2_intro T T (\lambda (u3: T).(\lambda 
-(t2: T).(eq T (THead k u2 t3) (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))) u2 t3 (refl_equal T (THead k u2 t3)) H22 H21))) k0 H17)))))))) 
-H14)) H13)) x H5))))))))))))))))))))) H))))))).
+\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in (\lambda (H9: (eq T 
+u0 u1)).(\lambda (H10: (eq K k0 k)).(let H11 \def (eq_ind T t0 (\lambda (t: 
+T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead 
+k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda 
+(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0)) 
+v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3))) 
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 t3))))))) H4 
+t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s k0 i0) v0 t 
+t2)) H3 t1 H8) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq T t1 
+(THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3 t1))) 
+(\lambda (u3: T).(subst0 (s k1 i0) v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T 
+t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))) 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))))))) H11 k H10) in 
+(let H14 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s k1 i0) v0 t1 t2)) H12 
+k H10) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T (\lambda (u3: T).(eq T 
+(THead k1 u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) 
+(ex2 T (\lambda (t3: T).(eq T (THead k1 u2 t2) (THead k u1 t3))) (\lambda 
+(t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda 
+(t3: T).(eq T (THead k1 u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda 
+(_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k 
+i0) v0 t1 t3)))))) (let H15 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead 
+k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1))) 
+(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T u2 
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead k u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s k i0) v0 t1 t3))))))) H2 u1 H9) in (let H16 \def (eq_ind T u0 
+(\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H9) in (or3_intro2 (ex2 T (\lambda 
+(u3: T).(eq T (THead k u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 
+v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead k u2 t2) (THead k u1 t3))) 
+(\lambda (t3: T).(subst0 (s k i0) v0 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 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s k i0) v0 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 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k 
+i0) v0 t1 t3))) u2 t2 (refl_equal T (THead k u2 t2)) H16 H14)))) k0 
+H10)))))))) H7)) H6)))))))))))))) i v y x H0))) H))))))).
 
 theorem subst0_gen_lift_lt:
  \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/props.ma
index c36cd597e..a9ded88cf 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/props.ma
@@ -14,9 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "subst0/fwd.ma".
-
-include "lift/props.ma".
+include "LambdaDelta-1/subst0/fwd.ma".
 
 theorem subst0_refl:
  \forall (u: T).(\forall (t: T).(\forall (d: nat).((subst0 d u t t) \to 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/subst0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/subst0.ma
index c8a35b3a3..b234990ee 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/subst0.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/subst0.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "subst0/props.ma".
+include "LambdaDelta-1/subst0/props.ma".
 
 theorem subst0_subst0:
  \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma
index 87c1fec52..3cad5ab9c 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "subst0/defs.ma".
+include "LambdaDelta-1/subst0/defs.ma".
 
-include "lift/props.ma".
+include "LambdaDelta-1/lift/props.ma".
 
-include "lift/tlt.ma".
+include "LambdaDelta-1/lift/tlt.ma".
 
 theorem subst0_weight_le:
  \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/defs.ma
index 62cfc2e48..8e874231a 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "subst0/defs.ma".
+include "LambdaDelta-1/subst0/defs.ma".
 
 inductive subst1 (i: nat) (v: T) (t1: T): T \to Prop \def
 | subst1_refl: subst1 i v t1 t1
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/fwd.ma
index c7eef274d..8610452f1 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/fwd.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "subst1/defs.ma".
+include "LambdaDelta-1/subst1/defs.ma".
 
-include "subst0/props.ma".
+include "LambdaDelta-1/subst0/props.ma".
 
 theorem subst1_gen_sort:
  \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/props.ma
index f16462a86..5a4059bfc 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/props.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "subst1/defs.ma".
+include "LambdaDelta-1/subst1/defs.ma".
 
-include "subst0/props.ma".
+include "LambdaDelta-1/subst0/props.ma".
 
 theorem subst1_head:
  \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: nat).((subst1 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/subst1.ma
index 797c1f1db..c683a4940 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/subst1.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/subst1.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "subst1/fwd.ma".
+include "LambdaDelta-1/subst1/fwd.ma".
 
-include "subst0/subst0.ma".
+include "LambdaDelta-1/subst0/subst0.ma".
 
 theorem subst1_subst1:
  \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau0/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau0/defs.ma
index 598876804..58ff7e171 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau0/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau0/defs.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "G/defs.ma".
+include "LambdaDelta-1/G/defs.ma".
 
-include "getl/defs.ma".
+include "LambdaDelta-1/getl/defs.ma".
 
 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 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau0/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau0/props.ma
index ddba3d947..36c004667 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau0/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau0/props.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "tau0/defs.ma".
+include "LambdaDelta-1/tau0/defs.ma".
 
-include "getl/drop.ma".
+include "LambdaDelta-1/getl/drop.ma".
 
 theorem tau0_lift:
  \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((tau0 g e 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau1/cnt.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau1/cnt.ma
index 4419db104..571001db0 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau1/cnt.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau1/cnt.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "tau1/props.ma".
+include "LambdaDelta-1/tau1/props.ma".
 
-include "cnt/props.ma".
+include "LambdaDelta-1/cnt/props.ma".
 
 theorem tau1_cnt:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau0 g c 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau1/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau1/defs.ma
index aeeea9566..559bf5da7 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau1/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau1/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "tau0/defs.ma".
+include "LambdaDelta-1/tau0/defs.ma".
 
 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))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau1/props.ma
index 434611c57..d7fc28bbb 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau1/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau1/props.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "tau1/defs.ma".
+include "LambdaDelta-1/tau1/defs.ma".
 
-include "tau0/props.ma".
+include "LambdaDelta-1/tau0/props.ma".
 
 theorem tau1_trans:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau1 g c 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/theory.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/theory.ma
index 72517f559..04950ea01 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/theory.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/theory.ma
@@ -14,23 +14,23 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "subst0/tlt.ma".
+include "LambdaDelta-1/subst0/tlt.ma".
 
-include "tau1/cnt.ma".
+include "LambdaDelta-1/tau1/cnt.ma".
 
-include "ex0/props.ma".
+include "LambdaDelta-1/ex0/props.ma".
 
-include "wcpr0/fwd.ma".
+include "LambdaDelta-1/wcpr0/fwd.ma".
 
-include "pr3/wcpr0.ma".
+include "LambdaDelta-1/pr3/wcpr0.ma".
 
-include "ex2/props.ma".
+include "LambdaDelta-1/ex2/props.ma".
 
-include "ex1/props.ma".
+include "LambdaDelta-1/ex1/props.ma".
 
-include "ty3/tau0.ma".
+include "LambdaDelta-1/ty3/tau0.ma".
 
-include "ty3/nf2.ma".
+include "LambdaDelta-1/ty3/nf2.ma".
 
-include "ty3/dec.ma".
+include "LambdaDelta-1/ty3/dec.ma".
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlist/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlist/defs.ma
index 51437021c..638aa68b4 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlist/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlist/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "T/defs.ma".
+include "LambdaDelta-1/T/defs.ma".
 
 inductive TList: Set \def
 | TNil: TList
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlist/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlist/props.ma
index 518fc312a..2a56bb13a 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlist/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlist/props.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "tlist/defs.ma".
+include "LambdaDelta-1/tlist/defs.ma".
 
 theorem tslt_wf__q_ind:
  \forall (P: ((TList \to Prop))).(((\forall (n: nat).((\lambda (P0: ((TList 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlt/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlt/defs.ma
index 3e24d5adf..c6a41c86c 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlt/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlt/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "T/defs.ma".
+include "LambdaDelta-1/T/defs.ma".
 
 definition wadd:
  ((nat \to nat)) \to (nat \to (nat \to nat))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlt/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlt/props.ma
index 56f46e4ea..bd6794ad0 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlt/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlt/props.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "tlt/defs.ma".
+include "LambdaDelta-1/tlt/defs.ma".
 
 theorem wadd_le:
  \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/arity.ma
index f285aa538..5f63defc6 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/arity.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/arity.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "ty3/pr3_props.ma".
+include "LambdaDelta-1/ty3/pr3_props.ma".
 
-include "arity/pr3.ma".
+include "LambdaDelta-1/arity/pr3.ma".
 
-include "asucc/fwd.ma".
+include "LambdaDelta-1/asucc/fwd.ma".
 
 theorem ty3_arity:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/arity_props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/arity_props.ma
index 2249354ab..5357e58e2 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/arity_props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/arity_props.ma
@@ -14,11 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "ty3/arity.ma".
+include "LambdaDelta-1/ty3/arity.ma".
 
-include "ty3/fwd.ma".
-
-include "sc3/arity.ma".
+include "LambdaDelta-1/sc3/arity.ma".
 
 theorem ty3_predicative:
  \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (u: 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/dec.ma
index 4121b6018..1936dd061 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/dec.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/dec.ma
@@ -14,13 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "ty3/pr3_props.ma".
+include "LambdaDelta-1/pc3/dec.ma".
 
-include "pc3/dec.ma".
+include "LambdaDelta-1/getl/flt.ma".
 
-include "getl/flt.ma".
-
-include "getl/dec.ma".
+include "LambdaDelta-1/getl/dec.ma".
 
 theorem ty3_inference:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(or (ex T (\lambda (t2: 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/defs.ma
index 33628e22d..d30e8bdd2 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/defs.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "G/defs.ma".
+include "LambdaDelta-1/G/defs.ma".
 
-include "pc3/defs.ma".
+include "LambdaDelta-1/pc3/defs.ma".
 
 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) 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fsubst0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fsubst0.ma
index 8a7506ceb..ee064039c 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fsubst0.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fsubst0.ma
@@ -14,13 +14,13 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "ty3/props.ma".
+include "LambdaDelta-1/ty3/props.ma".
 
-include "pc3/fsubst0.ma".
+include "LambdaDelta-1/pc3/fsubst0.ma".
 
-include "csubst0/props.ma".
+include "LambdaDelta-1/csubst0/props.ma".
 
-include "getl/getl.ma".
+include "LambdaDelta-1/getl/getl.ma".
 
 theorem ty3_fsubst0:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fwd.ma
index 60d2b11ee..5a88b676b 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fwd.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "ty3/defs.ma".
+include "LambdaDelta-1/ty3/defs.ma".
 
-include "pc3/props.ma".
+include "LambdaDelta-1/pc3/props.ma".
 
 theorem ty3_gen_sort:
  \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/nf2.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/nf2.ma
index 1d5a2f7ef..f76e49289 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/nf2.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/nf2.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "ty3/arity.ma".
+include "LambdaDelta-1/ty3/arity.ma".
 
-include "pc3/nf2.ma".
+include "LambdaDelta-1/pc3/nf2.ma".
 
-include "nf2/arity.ma".
+include "LambdaDelta-1/nf2/arity.ma".
 
 definition ty3_nf2_inv_abst_premise:
  C \to (T \to (T \to Prop))
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3.ma
index 46d1784f6..b689bf35c 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3.ma
@@ -14,17 +14,17 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "csubt/ty3.ma".
+include "LambdaDelta-1/csubt/ty3.ma".
 
-include "ty3/subst1.ma".
+include "LambdaDelta-1/ty3/subst1.ma".
 
-include "ty3/fsubst0.ma".
+include "LambdaDelta-1/ty3/fsubst0.ma".
 
-include "pc3/pc1.ma".
+include "LambdaDelta-1/pc3/pc1.ma".
 
-include "pc3/wcpr0.ma".
+include "LambdaDelta-1/pc3/wcpr0.ma".
 
-include "pc1/props.ma".
+include "LambdaDelta-1/pc1/props.ma".
 
 theorem ty3_sred_wcpr0_pr0:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3_props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3_props.ma
index 8e3dfb9e6..14ce7c088 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3_props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3_props.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "ty3/pr3.ma".
+include "LambdaDelta-1/ty3/pr3.ma".
 
 theorem ty3_cred_pr2:
  \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr2 c v1 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/props.ma
index a2c71de45..4ffc591ad 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/props.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "ty3/fwd.ma".
+include "LambdaDelta-1/ty3/fwd.ma".
 
-include "pc3/fwd.ma".
+include "LambdaDelta-1/pc3/fwd.ma".
 
 theorem ty3_lift:
  \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((ty3 g e 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma
index 3ae10ea9f..6dfdb48e4 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma
@@ -14,17 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "ty3/props.ma".
+include "LambdaDelta-1/ty3/props.ma".
 
-include "pc3/subst1.ma".
+include "LambdaDelta-1/pc3/subst1.ma".
 
-include "pc3/fwd.ma".
-
-include "csubst1/getl.ma".
-
-include "csubst1/fwd.ma".
-
-include "getl/getl.ma".
+include "LambdaDelta-1/getl/getl.ma".
 
 theorem ty3_gen_cabbr:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/tau0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/tau0.ma
index f7221d81b..ceaa58478 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/tau0.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/tau0.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "ty3/pr3_props.ma".
+include "LambdaDelta-1/ty3/pr3_props.ma".
 
-include "tau0/defs.ma".
+include "LambdaDelta-1/tau0/defs.ma".
 
 theorem ty3_tau0:
  \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/wcpr0/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/wcpr0/defs.ma
index cd6638c54..2685a4995 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/wcpr0/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/wcpr0/defs.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "pr0/defs.ma".
+include "LambdaDelta-1/pr0/defs.ma".
 
-include "C/defs.ma".
+include "LambdaDelta-1/C/defs.ma".
 
 inductive wcpr0: C \to (C \to Prop) \def
 | wcpr0_refl: \forall (c: C).(wcpr0 c c)
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/wcpr0/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/wcpr0/fwd.ma
index 0b4723fcb..7005751af 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/wcpr0/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/wcpr0/fwd.ma
@@ -14,27 +14,26 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "wcpr0/defs.ma".
+include "LambdaDelta-1/wcpr0/defs.ma".
 
 theorem wcpr0_gen_sort:
  \forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort 
 n))))
 \def
- \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n) x)).(let H0 
-\def (match H in wcpr0 return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: 
-(wcpr0 c c0)).((eq C c (CSort n)) \to ((eq C c0 x) \to (eq C x (CSort 
-n))))))) with [(wcpr0_refl c) \Rightarrow (\lambda (H0: (eq C c (CSort 
-n))).(\lambda (H1: (eq C c x)).(eq_ind C (CSort n) (\lambda (c0: C).((eq C c0 
-x) \to (eq C x (CSort n)))) (\lambda (H2: (eq C (CSort n) x)).(eq_ind C 
-(CSort n) (\lambda (c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x 
-H2)) c (sym_eq C c (CSort n) H0) H1))) | (wcpr0_comp c1 c2 H0 u1 u2 H1 k) 
-\Rightarrow (\lambda (H2: (eq C (CHead c1 k u1) (CSort n))).(\lambda (H3: (eq 
-C (CHead c2 k u2) x)).((let H4 \def (eq_ind C (CHead c1 k u1) (\lambda (e: 
-C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow 
-False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H2) in (False_ind ((eq 
-C (CHead c2 k u2) x) \to ((wcpr0 c1 c2) \to ((pr0 u1 u2) \to (eq C x (CSort 
-n))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CSort n)) (refl_equal C 
-x))))).
+ \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n) 
+x)).(insert_eq C (CSort n) (\lambda (c: C).(wcpr0 c x)) (\lambda (c: C).(eq C 
+x c)) (\lambda (y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: 
+C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) (\lambda (c: 
+C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e: 
+C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(eq C c0 c0)) 
+(refl_equal C (CSort n)) c H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda 
+(_: (wcpr0 c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2 
+c1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(k: K).(\lambda (H4: (eq C (CHead c1 k u1) (CSort n))).(let H5 \def (eq_ind C 
+(CHead c1 k u1) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) 
+with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I 
+(CSort n) H4) in (False_ind (eq C (CHead c2 k u2) (CHead c1 k u1)) 
+H5))))))))))) y x H0))) H))).
 
 theorem wcpr0_gen_head:
  \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0 
@@ -43,58 +42,58 @@ C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
 T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))
 \def
  \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda 
-(H: (wcpr0 (CHead c1 k u1) x)).(let H0 \def (match H in wcpr0 return (\lambda 
-(c: C).(\lambda (c0: C).(\lambda (_: (wcpr0 c c0)).((eq C c (CHead c1 k u1)) 
-\to ((eq C c0 x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: 
+(H: (wcpr0 (CHead c1 k u1) x)).(insert_eq C (CHead c1 k u1) (\lambda (c: 
+C).(wcpr0 c x)) (\lambda (c: C).(or (eq C x c) (ex3_2 C T (\lambda (c2: 
 C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: 
-T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))) with 
-[(wcpr0_refl c) \Rightarrow (\lambda (H0: (eq C c (CHead c1 k u1))).(\lambda 
-(H1: (eq C c x)).(eq_ind C (CHead c1 k u1) (\lambda (c0: C).((eq C c0 x) \to 
-(or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq 
-C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) 
-(\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))) (\lambda (H2: (eq C (CHead 
-c1 k u1) x)).(eq_ind C (CHead c1 k u1) (\lambda (c0: C).(or (eq C c0 (CHead 
-c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq C c0 (CHead c2 k 
-u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda (_: 
-C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C (CHead c1 k u1) (CHead 
-c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq C (CHead c1 k u1) 
-(CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda 
-(_: C).(\lambda (u2: T).(pr0 u1 u2)))) (refl_equal C (CHead c1 k u1))) x H2)) 
-c (sym_eq C c (CHead c1 k u1) H0) H1))) | (wcpr0_comp c0 c2 H0 u0 u2 H1 k0) 
-\Rightarrow (\lambda (H2: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(\lambda 
-(H3: (eq C (CHead c2 k0 u2) x)).((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 c0 k0 u0) (CHead c1 k u1) H2) in ((let 
-H5 \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 u0) (CHead c1 k u1) H2) 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 k0 u0) (CHead c1 k u1) H2) in 
-(eq_ind C c1 (\lambda (c: C).((eq K k0 k) \to ((eq T u0 u1) \to ((eq C (CHead 
-c2 k0 u2) x) \to ((wcpr0 c c2) \to ((pr0 u0 u2) \to (or (eq C x (CHead c1 k 
-u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C x (CHead c3 k u3)))) 
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (\lambda 
+(y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: C).(\lambda (c0: 
+C).((eq C c (CHead c1 k u1)) \to (or (eq C c0 c) (ex3_2 C T (\lambda (c2: 
+C).(\lambda (u2: T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: 
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))) 
+(\lambda (c: C).(\lambda (H1: (eq C c (CHead c1 k u1))).(let H2 \def (f_equal 
+C C (\lambda (e: C).e) c (CHead c1 k u1) H1) in (eq_ind_r C (CHead c1 k u1) 
+(\lambda (c0: C).(or (eq C c0 c0) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: 
+T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 
+c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C 
+(CHead c1 k u1) (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: 
+T).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: 
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))) 
+(refl_equal C (CHead c1 k u1))) c H2)))) (\lambda (c0: C).(\lambda (c2: 
+C).(\lambda (H1: (wcpr0 c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to 
+(or (eq C c2 c0) (ex3_2 C T (\lambda (c3: C).(\lambda (u2: T).(eq C c2 (CHead 
+c3 k u2)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: 
+C).(\lambda (u2: T).(pr0 u1 u2)))))))).(\lambda (u0: T).(\lambda (u2: 
+T).(\lambda (H3: (pr0 u0 u2)).(\lambda (k0: K).(\lambda (H4: (eq C (CHead c0 
+k0 u0) (CHead c1 k u1))).(let H5 \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 k0 u0) (CHead c1 k u1) H4) in ((let H6 \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 u0) 
+(CHead c1 k u1) H4) in ((let H7 \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 k0 u0) (CHead c1 k u1) H4) in (\lambda (H8: (eq K 
+k0 k)).(\lambda (H9: (eq C c0 c1)).(eq_ind_r K k (\lambda (k1: K).(or (eq C 
+(CHead c2 k1 u2) (CHead c0 k1 u0)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: 
+T).(eq C (CHead c2 k1 u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: 
+T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))) (let H10 
+\def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H3 u1 H7) in (eq_ind_r T u1 
+(\lambda (t: T).(or (eq C (CHead c2 k u2) (CHead c0 k t)) (ex3_2 C T (\lambda 
+(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda 
+(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 
+u1 u3)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k 
+u1)) \to (or (eq C c2 c) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C 
+c2 (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) 
+(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))))) H2 c1 H9) in (let H12 \def 
+(eq_ind C c0 (\lambda (c: C).(wcpr0 c c2)) H1 c1 H9) in (eq_ind_r C c1 
+(\lambda (c: C).(or (eq C (CHead c2 k u2) (CHead c k u1)) (ex3_2 C T (\lambda 
+(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda 
+(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 
+u1 u3)))))) (or_intror (eq C (CHead c2 k u2) (CHead c1 k u1)) (ex3_2 C T 
+(\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) 
 (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda 
-(u3: T).(pr0 u1 u3))))))))))) (\lambda (H7: (eq K k0 k)).(eq_ind K k (\lambda 
-(k1: K).((eq T u0 u1) \to ((eq C (CHead c2 k1 u2) x) \to ((wcpr0 c1 c2) \to 
-((pr0 u0 u2) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: 
-C).(\lambda (u3: T).(eq C x (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: 
-T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))))))) 
-(\lambda (H8: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 k 
-u2) x) \to ((wcpr0 c1 c2) \to ((pr0 t u2) \to (or (eq C x (CHead c1 k u1)) 
-(ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C x (CHead c3 k u3)))) 
-(\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda 
-(u3: T).(pr0 u1 u3))))))))) (\lambda (H9: (eq C (CHead c2 k u2) x)).(eq_ind C 
-(CHead c2 k u2) (\lambda (c: C).((wcpr0 c1 c2) \to ((pr0 u1 u2) \to (or (eq C 
-c (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C c 
-(CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda 
-(_: C).(\lambda (u3: T).(pr0 u1 u3)))))))) (\lambda (H10: (wcpr0 c1 
-c2)).(\lambda (H11: (pr0 u1 u2)).(or_intror (eq C (CHead c2 k u2) (CHead c1 k 
-u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) 
-(CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda 
-(_: C).(\lambda (u3: T).(pr0 u1 u3)))) (ex3_2_intro C T (\lambda (c3: 
-C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: 
-C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 
-u3))) c2 u2 (refl_equal C (CHead c2 k u2)) H10 H11)))) x H9)) u0 (sym_eq T u0 
-u1 H8))) k0 (sym_eq K k0 k H7))) c0 (sym_eq C c0 c1 H6))) H5)) H4)) H3 H0 
-H1)))]) in (H0 (refl_equal C (CHead c1 k u1)) (refl_equal C x))))))).
+(u3: T).(pr0 u1 u3)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (u3: T).(eq 
+C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 
+c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c2 u2 (refl_equal C 
+(CHead c2 k u2)) H12 H10)) c0 H9))) u0 H7)) k0 H8)))) H6)) H5))))))))))) y x 
+H0))) H))))).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/wcpr0/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/wcpr0/getl.ma
index 41e0cb34a..e66c1c7a6 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/wcpr0/getl.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/wcpr0/getl.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "wcpr0/defs.ma".
+include "LambdaDelta-1/wcpr0/defs.ma".
 
-include "getl/props.ma".
+include "LambdaDelta-1/getl/props.ma".
 
 theorem wcpr0_drop:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: 
@@ -35,94 +35,87 @@ u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
 u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead 
 e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
 C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl 
-u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 
-c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: 
-T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda 
-(e2: C).(\lambda (u2: T).(drop h O c3 (CHead e2 k u2)))) (\lambda (e2: 
+u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 
+c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: 
+T).(\forall (k: K).((drop h O c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda 
+(e2: C).(\lambda (u2: T).(drop h O c4 (CHead e2 k u2)))) (\lambda (e2: 
 C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 
 u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 
 u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall 
-(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c0 k u1) (CHead 
+(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 k u1) (CHead 
 e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead 
-c3 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
+c4 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
 (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: 
-C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c0 k u1) 
-(CHead e1 k0 u0))).(let H4 \def (match (drop_gen_refl (CHead c0 k u1) (CHead 
-e1 k0 u0) H3) in eq return (\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c 
-(CHead e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O 
-(CHead c3 k u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
-e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))))) with [refl_equal 
-\Rightarrow (\lambda (H4: (eq C (CHead c0 k u1) (CHead e1 k0 u0))).(let H5 
-\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 c0 k 
-u1) (CHead e1 k0 u0) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match 
-e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k1 
-_) \Rightarrow k1])) (CHead c0 k u1) (CHead e1 k0 u0) 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 k u1) 
-(CHead e1 k0 u0) H4) in (eq_ind C e1 (\lambda (_: C).((eq K k k0) \to ((eq T 
-u1 u0) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k 
-u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
-(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))))) (\lambda (H8: (eq K k 
-k0)).(eq_ind K k0 (\lambda (k1: K).((eq T u1 u0) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u3: T).(drop O O (CHead c3 k1 u2) (CHead e2 k0 u3)))) (\lambda 
-(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 
-u0 u3)))))) (\lambda (H9: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).(ex3_2 C 
-T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k0 u2) (CHead e2 k0 
-u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
-C).(\lambda (u3: T).(pr0 u0 u3))))) (let H10 \def (eq_ind T u1 (\lambda (t: 
-T).(pr0 t u2)) H2 u0 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(wcpr0 
-c c3)) H0 e1 H7) in (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop 
-O O (CHead c3 k0 u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: 
-T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) c3 u2 
-(drop_refl (CHead c3 k0 u2)) H11 H10))) u1 (sym_eq T u1 u0 H9))) k (sym_eq K 
-k k0 H8))) c0 (sym_eq C c0 e1 H7))) H6)) H5)))]) in (H4 (refl_equal C (CHead 
-e1 k0 u0)))))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: 
-C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c0 k0 u1) (CHead e1 k1 
-u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 k0 
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u1) 
+(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C 
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow c3 | (CHead c _ _) 
+\Rightarrow c])) (CHead c3 k u1) (CHead e1 k0 u0) (drop_gen_refl (CHead c3 k 
+u1) (CHead e1 k0 u0) 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 _ k1 _) \Rightarrow k1])) (CHead c3 k u1) (CHead e1 k0 u0) 
+(drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in ((let H6 \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 c3 k u1) 
+(CHead e1 k0 u0) (drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in 
+(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c3 e1)).(eq_ind K k (\lambda 
+(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k 
+u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
+(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind T u1 (\lambda (t: 
+T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2) 
+(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda 
+(_: C).(\lambda (u3: T).(pr0 t u3))))) (eq_ind C c3 (\lambda (c: C).(ex3_2 C 
+T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k 
+u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u1 u3))))) (ex3_2_intro C T (\lambda (e2: 
+C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k u3)))) (\lambda 
+(e2: C).(\lambda (_: T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 
+u1 u3))) c4 u2 (drop_refl (CHead c4 k u2)) H0 H2) e1 H8) u0 H6) k0 H7)))) 
+H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: 
+C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c3 k0 u1) (CHead e1 k1 
+u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 k0 
 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
 (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) \to (\forall (e1: 
-C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c0 k0 u1) (CHead 
+C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c3 k0 u1) (CHead 
 e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O 
-(CHead c3 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
+(CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b: 
 B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall 
-(k0: K).((drop n O (CHead c0 (Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T 
-(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Bind b) u2) (CHead e2 
+(k0: K).((drop n O (CHead c3 (Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T 
+(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Bind b) u2) (CHead e2 
 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
 C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: 
-T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c0 (Bind b) u1) (CHead 
-e1 k0 u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n O c3 
-(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda 
-(_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda 
-(u3: T).(drop (S n) O (CHead c3 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda 
-(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 
-u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop n O c3 (CHead 
-x0 k0 x1))).(\lambda (H6: (wcpr0 e1 x0)).(\lambda (H7: (pr0 u0 
+T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Bind b) u1) (CHead 
+e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c3 (CHead e1 
+k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n 
+O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
+(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Bind b) u2) (CHead e2 k0 u3)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda 
+(u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop 
+n O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 
 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead 
-c3 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
+c4 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop 
-(Bind b) n c3 (CHead x0 k0 x1) H5 u2) H6 H7)))))) (H1 n e1 u0 k0 
-(drop_gen_drop (Bind b) c0 (CHead e1 k0 u0) u1 n H4)))))))))) (\lambda (f: 
+(Bind b) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) (\lambda (f: 
 F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall 
-(k0: K).((drop n O (CHead c0 (Flat f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T 
-(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Flat f) u2) (CHead e2 
+(k0: K).((drop n O (CHead c3 (Flat f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T 
+(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Flat f) u2) (CHead e2 
 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
 C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: 
-T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c0 (Flat f) u1) (CHead 
-e1 k0 u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O c3 
-(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda 
-(_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda 
-(u3: T).(drop (S n) O (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda 
-(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 
-u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop (S n) O c3 
-(CHead x0 k0 x1))).(\lambda (H6: (wcpr0 e1 x0)).(\lambda (H7: (pr0 u0 
-x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead 
-c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
-e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop 
-(Flat f) n c3 (CHead x0 k0 x1) H5 u2) H6 H7)))))) (H1 (S n) e1 u0 k0 
-(drop_gen_drop (Flat f) c0 (CHead e1 k0 u0) u1 n H4)))))))))) k) h)))))))))) 
-c1 c2 H))).
+T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Flat f) u1) (CHead 
+e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c3 (CHead 
+e1 k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop 
+(S n) O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 
+e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda 
+(u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop 
+(S n) O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 
+u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O 
+(CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: 
+T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 
+(drop_drop (Flat f) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k) 
+h)))))))))) c1 c2 H))).
 
 theorem wcpr0_drop_back:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: 
@@ -141,94 +134,87 @@ u2 u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
 u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead 
 e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
 C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl 
-u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 
-c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: 
-T).(\forall (k: K).((drop h O c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda 
-(e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2: 
+u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 
+c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: 
+T).(\forall (k: K).((drop h O c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda 
+(e2: C).(\lambda (u2: T).(drop h O c3 (CHead e2 k u2)))) (\lambda (e2: 
 C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 
 u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 
 u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall 
-(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 k u2) (CHead 
+(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c4 k u2) (CHead 
 e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead 
-c0 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
+c3 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
 (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: 
-C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u2) 
-(CHead e1 k0 u0))).(let H4 \def (match (drop_gen_refl (CHead c3 k u2) (CHead 
-e1 k0 u0) H3) in eq return (\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c 
-(CHead e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O 
-(CHead c0 k u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
-e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))))) with [refl_equal 
-\Rightarrow (\lambda (H4: (eq C (CHead c3 k u2) (CHead e1 k0 u0))).(let H5 
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) 
-with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c3 k 
-u2) (CHead e1 k0 u0) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match 
-e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k1 
-_) \Rightarrow k1])) (CHead c3 k u2) (CHead e1 k0 u0) H4) in ((let H7 \def 
-(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 k u2) 
-(CHead e1 k0 u0) H4) in (eq_ind C e1 (\lambda (_: C).((eq K k k0) \to ((eq T 
-u2 u0) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c0 k 
-u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
-(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))))) (\lambda (H8: (eq K k 
-k0)).(eq_ind K k0 (\lambda (k1: K).((eq T u2 u0) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u3: T).(drop O O (CHead c0 k1 u1) (CHead e2 k0 u3)))) (\lambda 
-(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 
-u3 u0)))))) (\lambda (H9: (eq T u2 u0)).(eq_ind T u0 (\lambda (_: T).(ex3_2 C 
-T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c0 k0 u1) (CHead e2 k0 
-u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
-C).(\lambda (u3: T).(pr0 u3 u0))))) (let H10 \def (eq_ind T u2 (\lambda (t: 
-T).(pr0 u1 t)) H2 u0 H9) in (let H11 \def (eq_ind C c3 (\lambda (c: C).(wcpr0 
-c0 c)) H0 e1 H7) in (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop 
-O O (CHead c0 k0 u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: 
-T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) c0 u1 
-(drop_refl (CHead c0 k0 u1)) H11 H10))) u2 (sym_eq T u2 u0 H9))) k (sym_eq K 
-k k0 H8))) c3 (sym_eq C c3 e1 H7))) H6)) H5)))]) in (H4 (refl_equal C (CHead 
-e1 k0 u0)))))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: 
-C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c3 k0 u2) (CHead e1 k1 
-u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c0 k0 
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c4 k u2) 
+(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C 
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow c4 | (CHead c _ _) 
+\Rightarrow c])) (CHead c4 k u2) (CHead e1 k0 u0) (drop_gen_refl (CHead c4 k 
+u2) (CHead e1 k0 u0) 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 _ k1 _) \Rightarrow k1])) (CHead c4 k u2) (CHead e1 k0 u0) 
+(drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in ((let H6 \def 
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
+[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c4 k u2) 
+(CHead e1 k0 u0) (drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in 
+(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c4 e1)).(eq_ind K k (\lambda 
+(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k 
+u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
+(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind T u2 (\lambda (t: 
+T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1) 
+(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda 
+(_: C).(\lambda (u3: T).(pr0 u3 t))))) (eq_ind C c4 (\lambda (c: C).(ex3_2 C 
+T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k 
+u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u3 u2))))) (ex3_2_intro C T (\lambda (e2: 
+C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k u3)))) (\lambda 
+(e2: C).(\lambda (_: T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0 
+u3 u2))) c3 u1 (drop_refl (CHead c3 k u1)) H0 H2) e1 H8) u0 H6) k0 H7)))) 
+H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: 
+C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c4 k0 u2) (CHead e1 k1 
+u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 k0 
 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
 (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) \to (\forall (e1: 
-C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c3 k0 u2) (CHead 
+C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c4 k0 u2) (CHead 
 e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O 
-(CHead c0 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
+(CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b: 
 B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall 
-(k0: K).((drop n O (CHead c3 (Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T 
-(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c0 (Bind b) u1) (CHead e2 
+(k0: K).((drop n O (CHead c4 (Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T 
+(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Bind b) u1) (CHead e2 
 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
 C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: 
-T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Bind b) u2) (CHead 
-e1 k0 u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n O c0 
-(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda 
-(_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda 
-(u3: T).(drop (S n) O (CHead c0 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda 
-(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 
-u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop n O c0 (CHead 
-x0 k0 x1))).(\lambda (H6: (wcpr0 x0 e1)).(\lambda (H7: (pr0 x1 
+T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Bind b) u2) (CHead 
+e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c4 (CHead e1 
+k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n 
+O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
+(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Bind b) u1) (CHead e2 k0 u3)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda 
+(u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop 
+n O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 
 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead 
-c0 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
+c3 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop 
-(Bind b) n c0 (CHead x0 k0 x1) H5 u1) H6 H7)))))) (H1 n e1 u0 k0 
-(drop_gen_drop (Bind b) c3 (CHead e1 k0 u0) u2 n H4)))))))))) (\lambda (f: 
+(Bind b) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) (\lambda (f: 
 F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall 
-(k0: K).((drop n O (CHead c3 (Flat f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T 
-(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c0 (Flat f) u1) (CHead e2 
+(k0: K).((drop n O (CHead c4 (Flat f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T 
+(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Flat f) u1) (CHead e2 
 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
 C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: 
-T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Flat f) u2) (CHead 
-e1 k0 u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O c0 
-(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda 
-(_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda 
-(u3: T).(drop (S n) O (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda 
-(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 
-u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop (S n) O c0 
-(CHead x0 k0 x1))).(\lambda (H6: (wcpr0 x0 e1)).(\lambda (H7: (pr0 x1 
-u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead 
-c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
-e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop 
-(Flat f) n c0 (CHead x0 k0 x1) H5 u1) H6 H7)))))) (H1 (S n) e1 u0 k0 
-(drop_gen_drop (Flat f) c3 (CHead e1 k0 u0) u2 n H4)))))))))) k) h)))))))))) 
-c2 c1 H))).
+T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Flat f) u2) (CHead 
+e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c4 (CHead 
+e1 k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop 
+(S n) O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 
+e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda 
+(u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop 
+(S n) O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 
+x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O 
+(CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: 
+T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 
+(drop_drop (Flat f) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k) 
+h)))))))))) c2 c1 H))).
 
 theorem wcpr0_getl:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: 
@@ -247,104 +233,104 @@ u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
 u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 
 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
 C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl 
-u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 
-c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: 
-T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u2: T).(getl h c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda 
+u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 
+c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: 
+T).(\forall (k: K).((getl h c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u2: T).(getl h c4 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda 
 (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 
 u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 
 u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall 
-(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c0 k u1) (CHead e1 
-k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k 
+(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 k u1) (CHead e1 
+k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c4 k 
 u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
 (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: 
-C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c0 k u1) 
-(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c0 k1 u1) (CHead e1 
-k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 k1 
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c3 k u1) 
+(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c3 k1 u1) (CHead e1 
+k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 k1 
 u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
 (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))))) (\lambda (b: B).(\lambda 
-(H4: (clear (CHead c0 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (f_equal C 
+(H4: (clear (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (f_equal C 
 C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c0 
-(Bind b) u1) (clear_gen_bind b c0 (CHead e1 k0 u0) u1 H4)) in ((let H6 \def 
+\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c3 
+(Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let H6 \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 e1 k0 u0) 
-(CHead c0 (Bind b) u1) (clear_gen_bind b c0 (CHead e1 k0 u0) u1 H4)) in ((let 
+(CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let 
 H7 \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 e1 k0 
-u0) (CHead c0 (Bind b) u1) (clear_gen_bind b c0 (CHead e1 k0 u0) u1 H4)) in 
-(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c0)).(eq_ind_r K 
+u0) (CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in 
+(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c3)).(eq_ind_r K 
 (Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl 
-O (CHead c3 (Bind b) u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: 
+O (CHead c4 (Bind b) u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: 
 T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind_r 
 T u1 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O 
-(CHead c3 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda 
+(CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda 
 (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 t u3))))) 
-(eq_ind_r C c0 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
-T).(getl O (CHead c3 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: 
+(eq_ind_r C c3 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
+T).(getl O (CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: 
 C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 
-u3))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 
+u3))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 
 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: 
-T).(wcpr0 c0 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c3 u2 
-(getl_refl b c3 u2) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f: 
-F).(\lambda (H4: (clear (CHead c0 (Flat f) u1) (CHead e1 k0 u0))).(let H5 
-\def (H1 O e1 u0 k0 (getl_intro O c0 (CHead e1 k0 u0) c0 (drop_refl c0) 
-(clear_gen_flat f c0 (CHead e1 k0 u0) u1 H4))) in (ex3_2_ind C T (\lambda 
-(e2: C).(\lambda (u3: T).(getl O c3 (CHead e2 k0 u3)))) (\lambda (e2: 
+T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c4 u2 
+(getl_refl b c4 u2) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f: 
+F).(\lambda (H4: (clear (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5 
+\def (H1 O e1 u0 k0 (getl_intro O c3 (CHead e1 k0 u0) c3 (drop_refl c3) 
+(clear_gen_flat f c3 (CHead e1 k0 u0) u1 H4))) in (ex3_2_ind C T (\lambda 
+(e2: C).(\lambda (u3: T).(getl O c4 (CHead e2 k0 u3)))) (\lambda (e2: 
 C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 
-u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f) 
+u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Flat f) 
 u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
 (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda 
-(x1: T).(\lambda (H6: (getl O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 
+(x1: T).(\lambda (H6: (getl O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 
 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda 
-(u3: T).(getl O (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: 
+(u3: T).(getl O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: 
 C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 
-u3))) x0 x1 (getl_flat c3 (CHead x0 k0 x1) O H6 f u2) H7 H8)))))) H5)))) k 
-(getl_gen_O (CHead c0 k u1) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0: 
+u3))) x0 x1 (getl_flat c4 (CHead x0 k0 x1) O H6 f u2) H7 H8)))))) H5)))) k 
+(getl_gen_O (CHead c3 k u1) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0: 
 K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1: 
-K).((getl n (CHead c0 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u4: T).(getl n (CHead c3 k0 u2) (CHead e2 k1 u4)))) (\lambda 
+K).((getl n (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u4: T).(getl n (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda 
 (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 
 u3 u4))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl 
-(S n) (CHead c0 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u4: T).(getl (S n) (CHead c3 k0 u2) (CHead e2 k1 u4)))) (\lambda 
+(S n) (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u4: T).(getl (S n) (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda 
 (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 
 u3 u4))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall 
-(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c0 (Bind b) u1) 
+(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Bind b) u1) 
 (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n 
-(CHead c3 (Bind b) u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: 
+(CHead c4 (Bind b) u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: 
 T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 
 u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: 
-(getl (S n) (CHead c0 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 
-u0 k0 (getl_gen_S (Bind b) c0 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T 
-(\lambda (e2: C).(\lambda (u3: T).(getl n c3 (CHead e2 k0 u3)))) (\lambda 
+(getl (S n) (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 
+u0 k0 (getl_gen_S (Bind b) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T 
+(\lambda (e2: C).(\lambda (u3: T).(getl n c4 (CHead e2 k0 u3)))) (\lambda 
 (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 
-u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 
+u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4 
 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 
 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (H6: (getl n c3 (CHead x0 k0 x1))).(\lambda (H7: 
+C).(\lambda (x1: T).(\lambda (H6: (getl n c4 (CHead x0 k0 x1))).(\lambda (H7: 
 (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: 
-C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u2) (CHead e2 k0 u3)))) 
+C).(\lambda (u3: T).(getl (S n) (CHead c4 (Bind b) u2) (CHead e2 k0 u3)))) 
 (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda 
-(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Bind b) n c3 (CHead x0 k0 x1) H6 u2) 
+(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Bind b) n c4 (CHead x0 k0 x1) H6 u2) 
 H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: 
-((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c0 (Flat 
+((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Flat 
 f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: 
-T).(getl n (CHead c3 (Flat f) u2) (CHead e2 k0 u4)))) (\lambda (e2: 
+T).(getl n (CHead c4 (Flat f) u2) (CHead e2 k0 u4)))) (\lambda (e2: 
 C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 
 u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: 
-(getl (S n) (CHead c0 (Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 (S n) 
-e1 u0 k0 (getl_gen_S (Flat f) c0 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T 
-(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c3 (CHead e2 k0 u3)))) (\lambda 
+(getl (S n) (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 (S n) 
+e1 u0 k0 (getl_gen_S (Flat f) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T 
+(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c4 (CHead e2 k0 u3)))) (\lambda 
 (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 
-u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 
+u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4 
 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 
 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead x0 k0 x1))).(\lambda 
+C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c4 (CHead x0 k0 x1))).(\lambda 
 (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: 
-C).(\lambda (u3: T).(getl (S n) (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) 
+C).(\lambda (u3: T).(getl (S n) (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) 
 (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda 
-(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Flat f) n c3 (CHead x0 k0 x1) H6 u2) 
+(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Flat f) n c4 (CHead x0 k0 x1) H6 u2) 
 H7 H8)))))) H5))))))))) k) h)))))))))) c1 c2 H))).
 
 theorem wcpr0_getl_back:
@@ -364,103 +350,103 @@ u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
 u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 
 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
 C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl 
-u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 
-c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: 
-T).(\forall (k: K).((getl h c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda 
+u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 
+c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: 
+T).(\forall (k: K).((getl h c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u2: T).(getl h c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda 
 (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 
 u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 
 u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall 
-(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 k u2) (CHead e1 
-k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c0 k 
+(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 k u2) (CHead e1 
+k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k 
 u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
 (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: 
-C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c3 k u2) 
-(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c3 k1 u2) (CHead e1 
-k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 k1 
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c4 k u2) 
+(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c4 k1 u2) (CHead e1 
+k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 k1 
 u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
 (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))))) (\lambda (b: B).(\lambda 
-(H4: (clear (CHead c3 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (f_equal C 
+(H4: (clear (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (f_equal C 
 C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c3 
-(Bind b) u2) (clear_gen_bind b c3 (CHead e1 k0 u0) u2 H4)) in ((let H6 \def 
+\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c4 
+(Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let H6 \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 e1 k0 u0) 
-(CHead c3 (Bind b) u2) (clear_gen_bind b c3 (CHead e1 k0 u0) u2 H4)) in ((let 
+(CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let 
 H7 \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 e1 k0 
-u0) (CHead c3 (Bind b) u2) (clear_gen_bind b c3 (CHead e1 k0 u0) u2 H4)) in 
-(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c3)).(eq_ind_r K 
+u0) (CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in 
+(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c4)).(eq_ind_r K 
 (Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl 
-O (CHead c0 (Bind b) u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: 
+O (CHead c3 (Bind b) u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: 
 T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind_r 
 T u2 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O 
-(CHead c0 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda 
+(CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda 
 (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 t))))) 
-(eq_ind_r C c3 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
-T).(getl O (CHead c0 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: 
+(eq_ind_r C c4 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
+T).(getl O (CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: 
 C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 
-u2))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 
+u2))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 
 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: 
-T).(wcpr0 e2 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))) c0 u1 
-(getl_refl b c0 u1) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f: 
-F).(\lambda (H4: (clear (CHead c3 (Flat f) u2) (CHead e1 k0 u0))).(let H5 
-\def (H1 O e1 u0 k0 (getl_intro O c3 (CHead e1 k0 u0) c3 (drop_refl c3) 
-(clear_gen_flat f c3 (CHead e1 k0 u0) u2 H4))) in (ex3_2_ind C T (\lambda 
-(e2: C).(\lambda (u3: T).(getl O c0 (CHead e2 k0 u3)))) (\lambda (e2: 
+T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))) c3 u1 
+(getl_refl b c3 u1) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f: 
+F).(\lambda (H4: (clear (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5 
+\def (H1 O e1 u0 k0 (getl_intro O c4 (CHead e1 k0 u0) c4 (drop_refl c4) 
+(clear_gen_flat f c4 (CHead e1 k0 u0) u2 H4))) in (ex3_2_ind C T (\lambda 
+(e2: C).(\lambda (u3: T).(getl O c3 (CHead e2 k0 u3)))) (\lambda (e2: 
 C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 
-u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (Flat f) 
+u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f) 
 u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
 (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda 
-(x1: T).(\lambda (H6: (getl O c0 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 
+(x1: T).(\lambda (H6: (getl O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 
 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda 
-(u3: T).(getl O (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: 
+(u3: T).(getl O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: 
 C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 
-u0))) x0 x1 (getl_flat c0 (CHead x0 k0 x1) O H6 f u1) H7 H8)))))) H5)))) k 
-(getl_gen_O (CHead c3 k u2) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0: 
+u0))) x0 x1 (getl_flat c3 (CHead x0 k0 x1) O H6 f u1) H7 H8)))))) H5)))) k 
+(getl_gen_O (CHead c4 k u2) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0: 
 K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1: 
-K).((getl n (CHead c3 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u4: T).(getl n (CHead c0 k0 u1) (CHead e2 k1 u4)))) (\lambda 
+K).((getl n (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u4: T).(getl n (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda 
 (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 
 u4 u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl 
-(S n) (CHead c3 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u4: T).(getl (S n) (CHead c0 k0 u1) (CHead e2 k1 u4)))) (\lambda 
+(S n) (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u4: T).(getl (S n) (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda 
 (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 
 u4 u3))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall 
-(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Bind b) u2) 
+(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Bind b) u2) 
 (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n 
-(CHead c0 (Bind b) u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: 
+(CHead c3 (Bind b) u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: 
 T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 
 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: 
-(getl (S n) (CHead c3 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 
-u0 k0 (getl_gen_S (Bind b) c3 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T 
-(\lambda (e2: C).(\lambda (u3: T).(getl n c0 (CHead e2 k0 u3)))) (\lambda 
+(getl (S n) (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 
+u0 k0 (getl_gen_S (Bind b) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T 
+(\lambda (e2: C).(\lambda (u3: T).(getl n c3 (CHead e2 k0 u3)))) (\lambda 
 (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 
-u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c0 
+u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 
 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 
 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (H6: (getl n c0 (CHead x0 k0 x1))).(\lambda (H7: 
+C).(\lambda (x1: T).(\lambda (H6: (getl n c3 (CHead x0 k0 x1))).(\lambda (H7: 
 (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: 
-C).(\lambda (u3: T).(getl (S n) (CHead c0 (Bind b) u1) (CHead e2 k0 u3)))) 
+C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u1) (CHead e2 k0 u3)))) 
 (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda 
-(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Bind b) n c0 (CHead x0 k0 x1) H6 u1) 
+(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Bind b) n c3 (CHead x0 k0 x1) H6 u1) 
 H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: 
-((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Flat 
+((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Flat 
 f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: 
-T).(getl n (CHead c0 (Flat f) u1) (CHead e2 k0 u4)))) (\lambda (e2: 
+T).(getl n (CHead c3 (Flat f) u1) (CHead e2 k0 u4)))) (\lambda (e2: 
 C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 
 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: 
-(getl (S n) (CHead c3 (Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 (S n) 
-e1 u0 k0 (getl_gen_S (Flat f) c3 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T 
-(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c0 (CHead e2 k0 u3)))) (\lambda 
+(getl (S n) (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 (S n) 
+e1 u0 k0 (getl_gen_S (Flat f) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T 
+(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c3 (CHead e2 k0 u3)))) (\lambda 
 (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 
-u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c0 
+u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 
 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 
 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c0 (CHead x0 k0 x1))).(\lambda 
+C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead x0 k0 x1))).(\lambda 
 (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: 
-C).(\lambda (u3: T).(getl (S n) (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) 
+C).(\lambda (u3: T).(getl (S n) (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) 
 (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda 
-(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Flat f) n c0 (CHead x0 k0 x1) H6 u1) 
+(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Flat f) n c3 (CHead x0 k0 x1) H6 u1) 
 H7 H8)))))) H5))))))))) k) h)))))))))) c2 c1 H))).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Makefile b/helm/software/matita/contribs/LAMBDA-TYPES/Makefile
index 17478c353..5f3bac93d 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Makefile
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Makefile
@@ -1,30 +1,34 @@
+H=@
+
+MATITAOPTIONS=-onepass
+
 DIR=$(shell basename $$PWD)
 
-MMAS = $(shell find -name "*.mma")
+MMAS = $(shell find Base-2 -name "*.mma")
 MAS = $(MMAS:%.mma=%.ma)
 
 %.ma: %.mma
-	../../matitac.opt $< 2> /dev/null
-	../../matitac.opt -dump $@ $< 2> /dev/null
-	$(MAKE) depend.opt
+	$(H)../../matitac.opt $(MATITAOPTIONS) $< 2> /dev/null
+	$(H)../../matitac.opt $(MATITAOPTIONS) -dump $@ $< 2> /dev/null
+	$(H)$(MAKE) depend.opt
 
 $(DIR) all: $(MAS)
-	../../matitac
+	$(H)../../matitac $(MATITAOPTIONS) 2> /dev/null
 $(DIR).opt opt all.opt: $(MAS)
-	../../matitac.opt
+	$(H)../../matitac.opt $(MATITAOPTIONS) 2> /dev/null
 clean:
-	../../matitaclean
-	rm -f $(MAS)
-	$(MAKE) depend
+	$(H)../../matitaclean
+	$(H)rm -f $(MAS)
+	$(H)$(MAKE) depend
 clean.opt:
-	../../matitaclean.opt
-	rm -f $(MAS)
-	$(MAKE) depend.opt
+	$(H)../../matitaclean.opt
+	$(H)rm -f $(MAS)
+	$(H)$(MAKE) depend.opt
 depend:
-	../../matitadep
-	cat Base-2/depends_mma >> depends
+	$(H)../../matitadep
+	$(H)cat Base-2/depends >> depends
 depend.opt:
-	../../matitadep.opt
-	cat Base-2/depends_mma >> depends
+	$(H)../../matitadep.opt
+	$(H)cat Base-2/depends >> depends
 
 include Base-2/.depend
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/defs.ma
index fb71d3976..32b8a5f67 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/defs.ma
@@ -12,13 +12,11 @@
 (*                                                                        *)
 (**************************************************************************)
 
-
-
 (* LIFT RELATION
    - Usage: invoke with positive polarity
 *)
 
-include "datatypes/Term.ma".
+include "Unified-Sub/datatypes/Term.ma".
 
 inductive Lift (l: Nat): Nat \to Term \to Term \to Prop \def
    | lift_sort   : \forall i,h. 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/fun.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/fun.ma
index d26bdb0e7..08d6704b2 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/fun.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/fun.ma
@@ -12,9 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-
-
-include "Lift/inv.ma".
+include "Unified-Sub/Lift/inv.ma".
 
 (* Functional properties ****************************************************)
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/inv.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/inv.ma
index bbd8d744e..152ec11d6 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/inv.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/inv.ma
@@ -12,9 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-
-
-include "Lift/defs.ma".
+include "Unified-Sub/Lift/defs.ma".
 
 (* Inversion properties *****************************************************)
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/props.ma
index 8e50c477b..3090e09e6 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/props.ma
@@ -12,9 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-
-
-include "Lift/fun.ma".
+include "Unified-Sub/Lift/fun.ma".
 
 (* NOTE: this holds because of nplus_comm_1 *)
 theorem lift_comp: \forall l1,i1,t1,t2. Lift l1 i1 t1 t2 \to
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/Makefile b/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/Makefile
deleted file mode 100644
index 7fd5e9ccb..000000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/Makefile
+++ /dev/null
@@ -1,14 +0,0 @@
-DIR=$(shell basename $$PWD)
-
-$(DIR) all:
-	../../../matitac
-$(DIR).opt opt all.opt:
-	../../../matitac.opt
-clean:
-	../../../matitaclean
-clean.opt:
-	../../../matitaclean.opt
-depend:
-	../matitadep
-depend.opt:
-	../matitadep.opt
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/datatypes/Context.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/datatypes/Context.ma
index d63ffcf2e..03d9d7610 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/datatypes/Context.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/datatypes/Context.ma
@@ -12,14 +12,12 @@
 (*                                                                        *)
 (**************************************************************************)
 
-
-
 (* FLAT CONTEXTS
    - Naming policy:
      - contexts: c d
 *)
 
-include "datatypes/Term.ma".
+include "Unified-Sub/datatypes/Term.ma".
 
 inductive Context: Type \def
    | leaf: Context
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/datatypes/Term.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/datatypes/Term.ma
index e1f090d1a..d1a98de5a 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/datatypes/Term.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/datatypes/Term.ma
@@ -12,8 +12,6 @@
 (*                                                                        *)
 (**************************************************************************)
 
-
-
 (* POLARIZED TERMS
    - Naming policy:
      - natural numbers      : sorts h k, local references i j, lengths l o
@@ -24,7 +22,7 @@
      - terms                : t u
 *)
 
-include "preamble4.ma".
+include "Unified-Sub/preamble.ma".
 
 inductive Bind: Type \def
    | abbr: Bind
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/depends b/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/depends
deleted file mode 100644
index 82206df5e..000000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/depends
+++ /dev/null
@@ -1,12 +0,0 @@
-preamble4.ma NLE/nplus.ma NLE/props.ma NPlus/monoid.ma datatypes/Bool.ma logic/equality.ma
-datatypes/Context.ma datatypes/Term.ma
-datatypes/Term.ma preamble4.ma
-Lift/props.ma Lift/fun.ma
-Lift/inv.ma Lift/defs.ma
-Lift/defs.ma datatypes/Term.ma
-Lift/fun.ma Lift/inv.ma
-NLE/nplus.ma 
-NLE/props.ma 
-NPlus/monoid.ma 
-datatypes/Bool.ma 
-logic/equality.ma 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/preamble.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/preamble.ma
new file mode 100644
index 000000000..f31ea2903
--- /dev/null
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/preamble.ma
@@ -0,0 +1,32 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* Project started Tue Aug 22, 2006 ***************************************)
+
+(* PREAMBLE
+*)
+
+include "logic/equality.ma".
+include "datatypes/Bool.ma".
+include "NPlus/monoid.ma".
+include "NLE/props.ma".
+include "NLE/nplus.ma".
+
+axiom f_equal_3: \forall (A,B,C,D:Set).
+                 \forall (f:A \to B \to C \to D). 
+                 \forall (x1,x2:A).
+                 \forall (y1,y2:B).
+                 \forall (z1,z2:C). 
+                 x1 = x2 \to y1 = y2 \to z1 = z2 \to 
+                 f x1 y1 z1 = f x2 y2 z2.  
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/preamble4.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/preamble4.ma
deleted file mode 100644
index 1449c154f..000000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/preamble4.ma
+++ /dev/null
@@ -1,34 +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                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* Project started Tue Aug 22, 2006 ***************************************)
-
-
-
-(* PREAMBLE
-*)
-
-include "logic/equality.ma".
-include "datatypes/Bool.ma".
-include "NPlus/monoid.ma".
-include "NLE/props.ma".
-include "NLE/nplus.ma".
-
-axiom f_equal_3: \forall (A,B,C,D:Set).
-                 \forall (f:A \to B \to C \to D). 
-                 \forall (x1,x2:A).
-                 \forall (y1,y2:B).
-                 \forall (z1,z2:C). 
-                 x1 = x2 \to y1 = y2 \to z1 = z2 \to 
-                 f x1 y1 z1 = f x2 y2 z2.  
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/root b/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/root
deleted file mode 100644
index e3402e0be..000000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Unified-Sub/root
+++ /dev/null
@@ -1,2 +0,0 @@
-baseuri=cic:/matita/LAMBDA-TYPES/Unified-Sub
-include_paths= ../../RELATIONAL/
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/depends b/helm/software/matita/contribs/LAMBDA-TYPES/depends
index 49056cf08..138d27e94 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/depends
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/depends
@@ -1,3 +1,211 @@
+Unified-Sub/preamble.ma NLE/nplus.ma NLE/props.ma NPlus/monoid.ma datatypes/Bool.ma logic/equality.ma
+Unified-Sub/datatypes/Term.ma Unified-Sub/preamble.ma
+Unified-Sub/datatypes/Context.ma Unified-Sub/datatypes/Term.ma
+Unified-Sub/Lift/inv.ma Unified-Sub/Lift/defs.ma
+Unified-Sub/Lift/defs.ma Unified-Sub/datatypes/Term.ma
+Unified-Sub/Lift/props.ma Unified-Sub/Lift/fun.ma
+Unified-Sub/Lift/fun.ma Unified-Sub/Lift/inv.ma
+LambdaDelta-2/preamble.ma Base-2/preamble.ma LambdaDelta-1/definitions.ma
+LambdaDelta-1/theory.ma LambdaDelta-1/ex0/props.ma LambdaDelta-1/ex1/props.ma LambdaDelta-1/ex2/props.ma LambdaDelta-1/pr3/wcpr0.ma LambdaDelta-1/subst0/tlt.ma LambdaDelta-1/tau1/cnt.ma LambdaDelta-1/ty3/dec.ma LambdaDelta-1/ty3/nf2.ma LambdaDelta-1/ty3/tau0.ma LambdaDelta-1/wcpr0/fwd.ma
+LambdaDelta-1/spare.ma LambdaDelta-1/theory.ma
+LambdaDelta-1/preamble.ma Base-1/theory.ma
+LambdaDelta-1/definitions.ma LambdaDelta-1/aprem/defs.ma LambdaDelta-1/cimp/defs.ma LambdaDelta-1/clen/defs.ma LambdaDelta-1/cnt/defs.ma LambdaDelta-1/csuba/defs.ma LambdaDelta-1/csubc/defs.ma LambdaDelta-1/csubst1/defs.ma LambdaDelta-1/csubt/defs.ma LambdaDelta-1/ex0/defs.ma LambdaDelta-1/ex1/defs.ma LambdaDelta-1/ex2/defs.ma LambdaDelta-1/flt/defs.ma LambdaDelta-1/fsubst0/defs.ma LambdaDelta-1/iso/defs.ma LambdaDelta-1/llt/defs.ma LambdaDelta-1/next_plus/defs.ma LambdaDelta-1/nf2/defs.ma LambdaDelta-1/pc1/defs.ma LambdaDelta-1/subst1/defs.ma LambdaDelta-1/tau1/defs.ma LambdaDelta-1/tlt/defs.ma LambdaDelta-1/wcpr0/defs.ma
+LambdaDelta-1/ex0/defs.ma LambdaDelta-1/A/defs.ma LambdaDelta-1/G/defs.ma
+LambdaDelta-1/ex0/props.ma LambdaDelta-1/aplus/props.ma LambdaDelta-1/ex0/defs.ma LambdaDelta-1/leq/defs.ma
+LambdaDelta-1/ex2/defs.ma LambdaDelta-1/C/defs.ma
+LambdaDelta-1/ex2/props.ma LambdaDelta-1/arity/fwd.ma LambdaDelta-1/ex2/defs.ma LambdaDelta-1/nf2/defs.ma LambdaDelta-1/pr2/fwd.ma
+LambdaDelta-1/tlist/defs.ma LambdaDelta-1/T/defs.ma
+LambdaDelta-1/tlist/props.ma LambdaDelta-1/tlist/defs.ma
+LambdaDelta-1/csubt/defs.ma LambdaDelta-1/ty3/defs.ma
+LambdaDelta-1/csubt/fwd.ma LambdaDelta-1/csubt/defs.ma
+LambdaDelta-1/csubt/props.ma LambdaDelta-1/csubt/defs.ma
+LambdaDelta-1/csubt/clear.ma LambdaDelta-1/clear/fwd.ma LambdaDelta-1/csubt/defs.ma
+LambdaDelta-1/csubt/drop.ma LambdaDelta-1/csubt/defs.ma LambdaDelta-1/drop/fwd.ma
+LambdaDelta-1/csubt/getl.ma LambdaDelta-1/csubt/clear.ma LambdaDelta-1/csubt/drop.ma LambdaDelta-1/csubt/fwd.ma LambdaDelta-1/getl/clear.ma
+LambdaDelta-1/csubt/pc3.ma LambdaDelta-1/csubt/getl.ma LambdaDelta-1/pc3/left.ma
+LambdaDelta-1/csubt/ty3.ma LambdaDelta-1/csubt/pc3.ma LambdaDelta-1/csubt/props.ma
+LambdaDelta-1/csubc/defs.ma LambdaDelta-1/sc3/defs.ma
+LambdaDelta-1/csubc/props.ma LambdaDelta-1/csubc/defs.ma LambdaDelta-1/sc3/props.ma
+LambdaDelta-1/csubc/csuba.ma LambdaDelta-1/csubc/defs.ma LambdaDelta-1/sc3/props.ma
+LambdaDelta-1/csubc/drop.ma LambdaDelta-1/csubc/fwd.ma LambdaDelta-1/sc3/props.ma
+LambdaDelta-1/csubc/drop1.ma LambdaDelta-1/csubc/drop.ma
+LambdaDelta-1/csubc/clear.ma LambdaDelta-1/csubc/fwd.ma
+LambdaDelta-1/csubc/getl.ma LambdaDelta-1/csubc/clear.ma LambdaDelta-1/csubc/drop.ma
+LambdaDelta-1/csubc/arity.ma LambdaDelta-1/csubc/csuba.ma
+LambdaDelta-1/csubc/fwd.ma LambdaDelta-1/csubc/defs.ma
+LambdaDelta-1/ty3/fsubst0.ma LambdaDelta-1/csubst0/props.ma LambdaDelta-1/getl/getl.ma LambdaDelta-1/pc3/fsubst0.ma LambdaDelta-1/ty3/props.ma
+LambdaDelta-1/ty3/defs.ma LambdaDelta-1/G/defs.ma LambdaDelta-1/pc3/defs.ma
+LambdaDelta-1/ty3/fwd.ma LambdaDelta-1/pc3/props.ma LambdaDelta-1/ty3/defs.ma
+LambdaDelta-1/ty3/props.ma LambdaDelta-1/pc3/fwd.ma LambdaDelta-1/ty3/fwd.ma
+LambdaDelta-1/ty3/subst1.ma LambdaDelta-1/getl/getl.ma LambdaDelta-1/pc3/subst1.ma LambdaDelta-1/ty3/props.ma
+LambdaDelta-1/ty3/pr3_props.ma LambdaDelta-1/ty3/pr3.ma
+LambdaDelta-1/ty3/pr3.ma LambdaDelta-1/csubt/ty3.ma LambdaDelta-1/pc1/props.ma LambdaDelta-1/pc3/pc1.ma LambdaDelta-1/pc3/wcpr0.ma LambdaDelta-1/ty3/fsubst0.ma LambdaDelta-1/ty3/subst1.ma
+LambdaDelta-1/ty3/tau0.ma LambdaDelta-1/tau0/defs.ma LambdaDelta-1/ty3/pr3_props.ma
+LambdaDelta-1/ty3/arity.ma LambdaDelta-1/arity/pr3.ma LambdaDelta-1/asucc/fwd.ma LambdaDelta-1/ty3/pr3_props.ma
+LambdaDelta-1/ty3/dec.ma LambdaDelta-1/getl/dec.ma LambdaDelta-1/getl/flt.ma LambdaDelta-1/pc3/dec.ma
+LambdaDelta-1/ty3/arity_props.ma LambdaDelta-1/sc3/arity.ma LambdaDelta-1/ty3/arity.ma
+LambdaDelta-1/ty3/nf2.ma LambdaDelta-1/nf2/arity.ma LambdaDelta-1/pc3/nf2.ma LambdaDelta-1/ty3/arity.ma
+LambdaDelta-1/pc3/defs.ma LambdaDelta-1/pr3/defs.ma
+LambdaDelta-1/pc3/props.ma LambdaDelta-1/pc3/defs.ma LambdaDelta-1/pr3/pr3.ma
+LambdaDelta-1/pc3/fsubst0.ma LambdaDelta-1/csubst0/getl.ma LambdaDelta-1/fsubst0/defs.ma LambdaDelta-1/pc3/left.ma
+LambdaDelta-1/pc3/pc1.ma LambdaDelta-1/pc1/defs.ma LambdaDelta-1/pc3/defs.ma LambdaDelta-1/pr3/pr1.ma
+LambdaDelta-1/pc3/wcpr0.ma LambdaDelta-1/pc3/props.ma LambdaDelta-1/wcpr0/getl.ma
+LambdaDelta-1/pc3/left.ma LambdaDelta-1/pc3/props.ma
+LambdaDelta-1/pc3/fwd.ma LambdaDelta-1/pc3/props.ma LambdaDelta-1/pr3/fwd.ma
+LambdaDelta-1/pc3/nf2.ma LambdaDelta-1/nf2/pr3.ma LambdaDelta-1/pc3/defs.ma
+LambdaDelta-1/pc3/subst1.ma LambdaDelta-1/pc3/props.ma LambdaDelta-1/pr3/subst1.ma
+LambdaDelta-1/pc3/dec.ma LambdaDelta-1/nf2/fwd.ma LambdaDelta-1/ty3/arity_props.ma
+LambdaDelta-1/pc1/defs.ma LambdaDelta-1/pr1/defs.ma
+LambdaDelta-1/pc1/props.ma LambdaDelta-1/pc1/defs.ma LambdaDelta-1/pr1/pr1.ma
+LambdaDelta-1/ex1/defs.ma LambdaDelta-1/C/defs.ma
+LambdaDelta-1/ex1/props.ma LambdaDelta-1/arity/defs.ma LambdaDelta-1/ex1/defs.ma LambdaDelta-1/leq/props.ma LambdaDelta-1/nf2/pr3.ma LambdaDelta-1/nf2/props.ma LambdaDelta-1/pc3/fwd.ma LambdaDelta-1/ty3/fwd.ma
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+LambdaDelta-1/cnt/defs.ma LambdaDelta-1/T/defs.ma
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+LambdaDelta-1/lift/fwd.ma LambdaDelta-1/lift/defs.ma
+LambdaDelta-1/lift/props.ma LambdaDelta-1/lift/fwd.ma LambdaDelta-1/s/props.ma
+LambdaDelta-1/lift/tlt.ma LambdaDelta-1/lift/fwd.ma LambdaDelta-1/tlt/props.ma
+LambdaDelta-1/iso/props.ma LambdaDelta-1/iso/fwd.ma
+LambdaDelta-1/iso/fwd.ma LambdaDelta-1/iso/defs.ma LambdaDelta-1/tlist/defs.ma
+LambdaDelta-1/iso/defs.ma LambdaDelta-1/T/defs.ma
+LambdaDelta-1/G/defs.ma LambdaDelta-1/preamble.ma
+LambdaDelta-1/flt/props.ma LambdaDelta-1/C/props.ma LambdaDelta-1/flt/defs.ma
+LambdaDelta-1/flt/defs.ma LambdaDelta-1/C/defs.ma
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+LambdaDelta-1/tlt/defs.ma LambdaDelta-1/T/defs.ma
+LambdaDelta-1/T/props.ma LambdaDelta-1/T/defs.ma
+LambdaDelta-1/T/dec.ma LambdaDelta-1/T/defs.ma
+LambdaDelta-1/T/defs.ma LambdaDelta-1/preamble.ma
+LambdaDelta-1/s/props.ma LambdaDelta-1/s/defs.ma
+LambdaDelta-1/s/defs.ma LambdaDelta-1/T/defs.ma
+LambdaDelta-1/C/props.ma LambdaDelta-1/C/defs.ma LambdaDelta-1/T/props.ma
+LambdaDelta-1/C/defs.ma LambdaDelta-1/T/defs.ma
+LambdaDelta-1/r/props.ma LambdaDelta-1/r/defs.ma LambdaDelta-1/s/defs.ma
+LambdaDelta-1/r/defs.ma LambdaDelta-1/T/defs.ma
+LambdaDelta-1/next_plus/props.ma LambdaDelta-1/next_plus/defs.ma
+LambdaDelta-1/next_plus/defs.ma LambdaDelta-1/G/defs.ma
+LambdaDelta-1/clen/getl.ma LambdaDelta-1/clen/defs.ma LambdaDelta-1/getl/props.ma
+LambdaDelta-1/clen/defs.ma LambdaDelta-1/C/defs.ma LambdaDelta-1/s/defs.ma
 Base-1/theory.ma Base-1/blt/props.ma Base-1/ext/arith.ma Base-1/ext/tactics.ma Base-1/plist/props.ma Base-1/types/props.ma
 Base-1/definitions.ma Base-1/blt/defs.ma Base-1/plist/defs.ma Base-1/types/defs.ma
 Base-1/preamble.ma coq.ma
@@ -20,7 +228,12 @@ Base-2/ext/arith.ma Base-2/preamble.ma
 Base-2/ext/tactics.ma Base-2/preamble.ma
 Base-2/blt/defs.ma Base-2/preamble.ma
 Base-2/blt/props.ma Base-2/blt/defs.ma
+NLE/nplus.ma 
+NLE/props.ma 
+NPlus/monoid.ma 
 coq.ma 
+datatypes/Bool.ma 
+logic/equality.ma 
 Base-2/theory.mma Base-2/preamble.ma Base-1/theory.ma
 Base-2/ext/tactics.mma Base-2/preamble.ma Base-1/ext/tactics.ma
 Base-2/ext/arith.mma Base-2/preamble.ma Base-1/ext/arith.ma
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/log.txt b/helm/software/matita/contribs/LAMBDA-TYPES/log.txt
deleted file mode 100644
index 2c20deb94..000000000
--- a/helm/software/matita/contribs/LAMBDA-TYPES/log.txt
+++ /dev/null
@@ -1,744 +0,0 @@
-Info:  execution of blt/defs.mma started:
-Debug: Executing: ``set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-2/blt ...''
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./preamble.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Debug: Executing: ``include "cic:/matita/LAMBDA-TYPES/Base-2/preamble" ...''
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./blt/defs.mma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Info:  execution of blt/defs.mma completed in 1''.
-Info:  execution of blt/props.mma started:
-Debug: Executing: ``set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-2/blt ...''
-Info:  baseuri cic:/matita/LAMBDA-TYPES/Base-2/blt/props is not empty
-Info:  cleaning baseuri cic:/matita/LAMBDA-TYPES/Base-2/blt/props
-Info:  Removing: cic:/matita/LAMBDA-TYPES/Base-2/blt/props/*
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./blt/defs.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Debug: Executing: ``include "cic:/matita/LAMBDA-TYPES/Base-2/blt/defs" ...''
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: lt_blt
-Pre Nodes : 257
-nat
-nat
-nat
-nat
-Warn:  Optimizer: remove 3
-Warn:  Optimizer: swap 2
-Warn:  Optimizer: nested application
-Warn:  Optimizer: anticipate 2
-Warn:  Optimizer: swap 2
-Optimized : (\lambda x:nat.(nat_ind (\lambda n:nat.(\forall y:nat.((lt y n)\to (eq bool (blt y n) true)))) (\lambda y:nat.(\lambda H:(lt y O). let H0 \def (le_ind (S y) (\lambda n:nat.((eq nat n O)\to (eq bool (blt y O) true))) (\lambda H0:(eq nat (S y) O). let H1 \def (eq_ind nat (S y) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with 
- [ O => False
- | (S (_:nat)) => True
-]) I O H0) in (False_ind (eq bool (blt y O) true) H1)) (\lambda m:nat.(\lambda H0:(le (S y) m).(\lambda _:((eq nat m O)\to (eq bool (blt y O) true)).(\lambda H1:(eq nat (S m) O). let H2 \def (eq_ind nat (S m) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with 
- [ O => False
- | (S (_:nat)) => True
-]) I O H1) in  let DEFINED \def (False_ind ((le (S y) m)\to (eq bool (blt y O) true)) H2) in (DEFINED H0))))) O H) in (H0 (refl_equal nat O)))) (\lambda n:nat.(\lambda H:(\forall y:nat.((lt y n)\to (eq bool (blt y n) true))).(\lambda y:nat.(nat_ind (\lambda n0:nat.((lt n0 (S n))\to (eq bool (blt n0 (S n)) true))) (\lambda _:(lt O (S n)).(refl_equal bool true)) (\lambda n0:nat.(\lambda _:((lt n0 (S n))\to (eq bool 
-match n0 return (\lambda n1:nat.bool) with 
- [ O => true
- | (S (m:nat)) => (blt m n)
-] true)).(\lambda H1:(lt (S n0) (S n)).(H n0 (le_S_n (S n0) n H1))))) y)))) x))
-Post Nodes: 272
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_bge
-Pre Nodes : 255
-nat
-nat
-nat
-nat
-Warn:  Optimizer: remove 3
-Warn:  Optimizer: swap 2
-Warn:  Optimizer: nested application
-Warn:  Optimizer: anticipate 2
-Warn:  Optimizer: swap 2
-Optimized : (\lambda x:nat.(nat_ind (\lambda n:nat.(\forall y:nat.((le n y)\to (eq bool (blt y n) false)))) (\lambda y:nat.(\lambda _:(le O y).(refl_equal bool false))) (\lambda n:nat.(\lambda H:(\forall y:nat.((le n y)\to (eq bool (blt y n) false))).(\lambda y:nat.(nat_ind (\lambda n0:nat.((le (S n) n0)\to (eq bool (blt n0 (S n)) false))) (\lambda H0:(le (S n) O). let H1 \def (le_ind (S n) (\lambda n0:nat.((eq nat n0 O)\to (eq bool (blt O (S n)) false))) (\lambda H1:(eq nat (S n) O). let H2 \def (eq_ind nat (S n) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with 
- [ O => False
- | (S (_:nat)) => True
-]) I O H1) in (False_ind (eq bool (blt O (S n)) false) H2)) (\lambda m:nat.(\lambda H1:(le (S n) m).(\lambda _:((eq nat m O)\to (eq bool (blt O (S n)) false)).(\lambda H2:(eq nat (S m) O). let H3 \def (eq_ind nat (S m) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with 
- [ O => False
- | (S (_:nat)) => True
-]) I O H2) in  let DEFINED \def (False_ind ((le (S n) m)\to (eq bool (blt O (S n)) false)) H3) in (DEFINED H1))))) O H0) in (H1 (refl_equal nat O))) (\lambda n0:nat.(\lambda _:((le (S n) n0)\to (eq bool (blt n0 (S n)) false)).(\lambda H1:(le (S n) (S n0)).(H n0 (le_S_n n n0 H1))))) y)))) x))
-Post Nodes: 272
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: blt_lt
-Pre Nodes : 221
-bool
-bool
-Warn:  Optimizer: remove 3
-Optimized : (\lambda x:nat.(nat_ind (\lambda n:nat.(\forall y:nat.((eq bool (blt y n) true)\to (lt y n)))) (\lambda y:nat.(\lambda H:(eq bool (blt y O) true). let H0 \def (eq_ind bool (blt y O) (\lambda b:bool.((eq bool b true)\to (lt y O))) (\lambda H0:(eq bool (blt y O) true). let H1 \def (eq_ind bool (blt y O) (\lambda e:bool.
-match e return (\lambda _:bool.Prop) with 
- [ true => False
- | false => True
-]) I true H0) in (False_ind (lt y O) H1)) true H) in (H0 (refl_equal bool true)))) (\lambda n:nat.(\lambda H:(\forall y:nat.((eq bool (blt y n) true)\to (lt y n))).(\lambda y:nat.(nat_ind (\lambda n0:nat.((eq bool (blt n0 (S n)) true)\to (lt n0 (S n)))) (\lambda _:(eq bool true true).(le_S_n (S O) (S n) (le_n_S (S O) (S n) (le_n_S O n (le_O_n n))))) (\lambda n0:nat.(\lambda _:((eq bool 
-match n0 return (\lambda n1:nat.bool) with 
- [ O => true
- | (S (m:nat)) => (blt m n)
-] true)\to (lt n0 (S n))).(\lambda H1:(eq bool (blt n0 n) true).(lt_le_S (S n0) (S n) (lt_n_S n0 n (H n0 H1)))))) y)))) x))
-Post Nodes: 219
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: bge_le
-Pre Nodes : 222
-bool
-bool
-Warn:  Optimizer: remove 3
-Optimized : (\lambda x:nat.(nat_ind (\lambda n:nat.(\forall y:nat.((eq bool (blt y n) false)\to (le n y)))) (\lambda y:nat.(\lambda _:(eq bool (blt y O) false).(le_O_n y))) (\lambda n:nat.(\lambda H:(\forall y:nat.((eq bool (blt y n) false)\to (le n y))).(\lambda y:nat.(nat_ind (\lambda n0:nat.((eq bool (blt n0 (S n)) false)\to (le (S n) n0))) (\lambda H0:(eq bool (blt O (S n)) false). let H1 \def (eq_ind bool (blt O (S n)) (\lambda b:bool.((eq bool b false)\to (le (S n) O))) (\lambda H1:(eq bool (blt O (S n)) false). let H2 \def (eq_ind bool (blt O (S n)) (\lambda e:bool.
-match e return (\lambda _:bool.Prop) with 
- [ true => True
- | false => False
-]) I false H1) in (False_ind (le (S n) O) H2)) false H0) in (H1 (refl_equal bool false))) (\lambda n0:nat.(\lambda _:((eq bool (blt n0 (S n)) false)\to (le (S n) n0)).(\lambda H1:(eq bool (blt (S n0) (S n)) false).(le_S_n (S n) (S n0) (le_n_S (S n) (S n0) (le_n_S n n0 (H n0 H1))))))) y)))) x))
-Post Nodes: 220
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./blt/props.mma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Info:  execution of blt/props.mma completed in 7''.
-Info:  execution of ext/arith.mma started:
-Debug: Executing: ``set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-2/ext ...''
-Info:  baseuri cic:/matita/LAMBDA-TYPES/Base-2/ext/arith is not empty
-Info:  cleaning baseuri cic:/matita/LAMBDA-TYPES/Base-2/ext/arith
-Info:  Removing: cic:/matita/LAMBDA-TYPES/Base-2/ext/arith/*
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./preamble.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Debug: Executing: ``include "cic:/matita/LAMBDA-TYPES/Base-2/preamble" ...''
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: nat_dec
-Pre Nodes : 474
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda n1:nat.(nat_ind (\lambda n:nat.(\forall n2:nat.(or (eq nat n n2) ((eq nat n n2)\to (\forall P:Prop.P))))) (\lambda n2:nat.(nat_ind (\lambda n:nat.(or (eq nat O n) ((eq nat O n)\to (\forall P:Prop.P)))) (or_introl (eq nat O O) ((eq nat O O)\to (\forall P:Prop.P)) (refl_equal nat O)) (\lambda n3:nat.(\lambda _:(or (eq nat O n3) ((eq nat O n3)\to (\forall P:Prop.P))).(or_intror (eq nat O (S n3)) ((eq nat O (S n3))\to (\forall P:Prop.P)) (\lambda H0:(eq nat O (S n3)).(\lambda P:Prop. let H1 \def (eq_ind nat O (\lambda ee:nat.
-match ee return (\lambda _:nat.Prop) with 
- [ O => True
- | (S (_:nat)) => False
-]) I (S n3) H0) in (False_ind P H1)))))) n2)) (\lambda n2:nat.(\lambda H:(\forall n2:nat.(or (eq nat n2 n2) ((eq nat n2 n2)\to (\forall P:Prop.P)))).(\lambda n3:nat.(nat_ind (\lambda n0:nat.(or (eq nat (S n2) n0) ((eq nat (S n2) n0)\to (\forall P:Prop.P)))) (or_intror (eq nat (S n2) O) ((eq nat (S n2) O)\to (\forall P:Prop.P)) (\lambda H0:(eq nat (S n2) O).(\lambda P:Prop. let H1 \def (eq_ind nat (S n2) (\lambda ee:nat.
-match ee return (\lambda _:nat.Prop) with 
- [ O => False
- | (S (_:nat)) => True
-]) I O H0) in (False_ind P H1)))) (\lambda n4:nat.(\lambda H0:(or (eq nat (S n2) n4) ((eq nat (S n2) n4)\to (\forall P:Prop.P))). let DEFINED \def (H n4) in (or_ind (eq nat n2 n4) ((eq nat n2 n4)\to (\forall P:Prop.P)) (or (eq nat (S n2) (S n4)) ((eq nat (S n2) (S n4))\to (\forall P:Prop.P))) (\lambda H1:(eq nat n2 n4).(eq_ind nat n2 (\lambda n3:nat.(or (eq nat (S n2) (S n3)) ((eq nat (S n2) (S n3))\to (\forall P:Prop.P)))) (or_introl (eq nat (S n2) (S n2)) ((eq nat (S n2) (S n2))\to (\forall P:Prop.P)) (refl_equal nat (S n2))) n4 H1)) (\lambda H1:((eq nat n2 n4)\to (\forall P:Prop.P)).(or_intror (eq nat (S n2) (S n4)) ((eq nat (S n2) (S n4))\to (\forall P:Prop.P)) (\lambda H2:(eq nat (S n2) (S n4)).(\lambda P:Prop. let H3 \def (f_equal nat nat (\lambda e:nat.
-match e return (\lambda _:nat.nat) with 
- [ O => n2
- | (S (n3:nat)) => n3
-]) (S n2) (S n4) H2) in  let H4 \def (eq_ind_r nat n4 (\lambda n3:nat.((eq nat n2 n3)\to (\forall P0:Prop.P0))) H1 n2 H3) in (H4 (refl_equal nat n2) P))))) DEFINED))) n3)))) n1))
-Post Nodes: 476
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: simpl_plus_r
-Pre Nodes : 85
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda n:nat.(\lambda m:nat.(\lambda p:nat.(\lambda H:(eq nat (plus m n) (plus p n)). let DEFINED \def (plus_comm n m) in (plus_reg_l n m p (eq_ind_r nat (plus m n) (\lambda n0:nat.(eq nat n0 (plus n p))) (eq_ind_r nat (plus p n) (\lambda n0:nat.(eq nat n0 (plus n p))) (sym_eq nat (plus n p) (plus p n) (plus_comm n p)) (plus m n) H) (plus n m) DEFINED))))))
-Post Nodes: 87
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: minus_plus_r
-Pre Nodes : 33
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda m:nat.(\lambda n:nat. let DEFINED \def (plus_comm m n) in (eq_ind_r nat (plus n m) (\lambda n0:nat.(eq nat (minus n0 n) m)) (minus_plus n m) (plus m n) DEFINED)))
-Post Nodes: 35
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: plus_permute_2_in_3
-Pre Nodes : 117
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda x:nat.(\lambda y:nat.(\lambda z:nat. let DEFINED \def (plus_assoc_reverse x z y) in  let DEFINED0 \def (plus_comm y z) in  let DEFINED1 \def (plus_assoc_reverse x y z) in (eq_ind_r nat (plus x (plus y z)) (\lambda n:nat.(eq nat n (plus (plus x z) y))) (eq_ind_r nat (plus z y) (\lambda n:nat.(eq nat (plus x n) (plus (plus x z) y))) (eq_ind nat (plus (plus x z) y) (\lambda n:nat.(eq nat n (plus (plus x z) y))) (refl_equal nat (plus (plus x z) y)) (plus x (plus z y)) DEFINED) (plus y z) DEFINED0) (plus (plus x y) z) DEFINED1))))
-Post Nodes: 123
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: plus_permute_2_in_3_assoc
-Pre Nodes : 86
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda n:nat.(\lambda h:nat.(\lambda k:nat. let DEFINED \def (plus_assoc n k h) in  let DEFINED0 \def (plus_permute_2_in_3 n h k) in (eq_ind_r nat (plus (plus n k) h) (\lambda n0:nat.(eq nat n0 (plus n (plus k h)))) (eq_ind_r nat (plus (plus n k) h) (\lambda n0:nat.(eq nat (plus (plus n k) h) n0)) (refl_equal nat (plus (plus n k) h)) (plus n (plus k h)) DEFINED) (plus (plus n h) k) DEFINED0))))
-Post Nodes: 90
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: plus_O
-Pre Nodes : 191
-nat
-nat
-Warn:  Optimizer: remove 3
-Optimized : (\lambda x:nat.(nat_ind (\lambda n:nat.(\forall y:nat.((eq nat (plus n y) O)\to (and (eq nat n O) (eq nat y O))))) (\lambda y:nat.(\lambda H:(eq nat (plus O y) O).(conj (eq nat O O) (eq nat y O) (refl_equal nat O) H))) (\lambda n:nat.(\lambda _:(\forall y:nat.((eq nat (plus n y) O)\to (and (eq nat n O) (eq nat y O)))).(\lambda y:nat.(\lambda H0:(eq nat (plus (S n) y) O). let H1 \def (eq_ind nat (plus (S n) y) (\lambda n0:nat.((eq nat n0 O)\to (and (eq nat (S n) O) (eq nat y O)))) (\lambda H1:(eq nat (plus (S n) y) O). let H2 \def (eq_ind nat (plus (S n) y) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with 
- [ O => False
- | (S (_:nat)) => True
-]) I O H1) in (False_ind (and (eq nat (S n) O) (eq nat y O)) H2)) O H0) in (H1 (refl_equal nat O)))))) x))
-Post Nodes: 189
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: minus_Sx_SO
-Pre Nodes : 24
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda x:nat. let DEFINED \def (minus_n_O x) in (eq_ind nat x (\lambda n:nat.(eq nat n x)) (refl_equal nat x) (minus x O) DEFINED))
-Post Nodes: 26
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: eq_nat_dec
-Pre Nodes : 303
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda i:nat.(nat_ind (\lambda n:nat.(\forall j:nat.(or (not (eq nat n j)) (eq nat n j)))) (\lambda j:nat.(nat_ind (\lambda n:nat.(or (not (eq nat O n)) (eq nat O n))) (or_intror (not (eq nat O O)) (eq nat O O) (refl_equal nat O)) (\lambda n:nat.(\lambda _:(or (not (eq nat O n)) (eq nat O n)).(or_introl (not (eq nat O (S n))) (eq nat O (S n)) (O_S n)))) j)) (\lambda n:nat.(\lambda H:(\forall j:nat.(or (not (eq nat n j)) (eq nat n j))).(\lambda j:nat.(nat_ind (\lambda n0:nat.(or (not (eq nat (S n) n0)) (eq nat (S n) n0))) (or_introl (not (eq nat (S n) O)) (eq nat (S n) O) (sym_not_eq nat O (S n) (O_S n))) (\lambda n0:nat.(\lambda _:(or (not (eq nat (S n) n0)) (eq nat (S n) n0)). let DEFINED \def (H n0) in (or_ind (not (eq nat n n0)) (eq nat n n0) (or (not (eq nat (S n) (S n0))) (eq nat (S n) (S n0))) (\lambda H1:(not (eq nat n n0)).(or_introl (not (eq nat (S n) (S n0))) (eq nat (S n) (S n0)) (not_eq_S n n0 H1))) (\lambda H1:(eq nat n n0).(or_intror (not (eq nat (S n) (S n0))) (eq nat (S n) (S n0)) (f_equal nat nat S n n0 H1))) DEFINED))) j)))) i))
-Post Nodes: 305
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: neq_eq_e
-Pre Nodes : 47
-Optimized : (\lambda i:nat.(\lambda j:nat.(\lambda P:Prop.(\lambda H:((not (eq nat i j))\to P).(\lambda H0:((eq nat i j)\to P). let o \def (eq_nat_dec i j) in (or_ind (not (eq nat i j)) (eq nat i j) P H H0 o))))))
-Post Nodes: 47
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_false
-Pre Nodes : 382
-nat
-nat
-nat
-nat
-Warn:  Optimizer: remove 3
-Warn:  Optimizer: swap 2
-Warn:  Optimizer: nested application
-Warn:  Optimizer: anticipate 2
-Warn:  Optimizer: swap 2
-nat
-nat
-nat
-nat
-Warn:  Optimizer: remove 3
-Warn:  Optimizer: swap 2
-Warn:  Optimizer: nested application
-Warn:  Optimizer: anticipate 2
-Warn:  Optimizer: swap 2
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda m:nat.(nat_ind (\lambda n:nat.(\forall n0:nat.(\forall P:Prop.((le n n0)\to ((le (S n0) n)\to P))))) (\lambda n:nat.(\lambda P:Prop.(\lambda _:(le O n).(\lambda H0:(le (S n) O). let H1 \def (le_ind (S n) (\lambda n0:nat.((eq nat n0 O)\to P)) (\lambda H1:(eq nat (S n) O). let H2 \def (eq_ind nat (S n) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with 
- [ O => False
- | (S (_:nat)) => True
-]) I O H1) in (False_ind P H2)) (\lambda m0:nat.(\lambda H1:(le (S n) m0).(\lambda _:((eq nat m0 O)\to P).(\lambda H2:(eq nat (S m0) O). let H3 \def (eq_ind nat (S m0) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with 
- [ O => False
- | (S (_:nat)) => True
-]) I O H2) in  let DEFINED \def (False_ind ((le (S n) m0)\to P) H3) in (DEFINED H1))))) O H0) in (H1 (refl_equal nat O)))))) (\lambda n:nat.(\lambda H:(\forall n0:nat.(\forall P:Prop.((le n n0)\to ((le (S n0) n)\to P)))).(\lambda n0:nat.(nat_ind (\lambda n1:nat.(\forall P:Prop.((le (S n) n1)\to ((le (S n1) (S n))\to P)))) (\lambda P:Prop.(\lambda H0:(le (S n) O).(\lambda _:(le (S O) (S n)). let H2 \def (le_ind (S n) (\lambda n1:nat.((eq nat n1 O)\to P)) (\lambda H2:(eq nat (S n) O). let H3 \def (eq_ind nat (S n) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with 
- [ O => False
- | (S (_:nat)) => True
-]) I O H2) in (False_ind P H3)) (\lambda m0:nat.(\lambda H2:(le (S n) m0).(\lambda _:((eq nat m0 O)\to P).(\lambda H3:(eq nat (S m0) O). let H4 \def (eq_ind nat (S m0) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with 
- [ O => False
- | (S (_:nat)) => True
-]) I O H3) in  let DEFINED \def (False_ind ((le (S n) m0)\to P) H4) in (DEFINED H2))))) O H0) in (H2 (refl_equal nat O))))) (\lambda n1:nat.(\lambda _:(\forall P:Prop.((le (S n) n1)\to ((le (S n1) (S n))\to P))).(\lambda P:Prop.(\lambda H1:(le (S n) (S n1)).(\lambda H2:(le (S (S n1)) (S n)). let DEFINED \def (le_S_n (S n1) n H2) in (H n1 P (le_S_n n n1 H1) DEFINED)))))) n0)))) m))
-Post Nodes: 400
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_Sx_x
-Pre Nodes : 20
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda x:nat.(\lambda H:(le (S x) x).(\lambda P:Prop. let H0 \def le_Sn_n in  let DEFINED \def (H0 x H) in (False_ind P DEFINED))))
-Post Nodes: 22
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: minus_le
-Pre Nodes : 88
-Optimized : (\lambda x:nat.(nat_ind (\lambda n:nat.(\forall y:nat.(le (minus n y) n))) (\lambda _:nat.(le_n O)) (\lambda n:nat.(\lambda H:(\forall y:nat.(le (minus n y) n)).(\lambda y:nat.(nat_ind (\lambda n0:nat.(le (minus (S n) n0) (S n))) (le_n (S n)) (\lambda n0:nat.(\lambda _:(le 
-match n0 return (\lambda n1:nat.nat) with 
- [ O => (S n)
- | (S (l:nat)) => (minus n l)
-] (S n)).(le_S (minus n n0) n (H n0)))) y)))) x))
-Post Nodes: 88
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_plus_minus_sym
-Pre Nodes : 45
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda n:nat.(\lambda m:nat.(\lambda H:(le n m). let DEFINED \def (plus_comm (minus m n) n) in (eq_ind_r nat (plus n (minus m n)) (\lambda n0:nat.(eq nat m n0)) (le_plus_minus n m H) (plus (minus m n) n) DEFINED))))
-Post Nodes: 47
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_minus_minus
-Pre Nodes : 84
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda x:nat.(\lambda y:nat.(\lambda H:(le x y).(\lambda z:nat.(\lambda H0:(le y z). let DEFINED \def (le_plus_minus_r x z (le_trans x y z H H0)) in  let DEFINED0 \def (le_plus_minus_r x y H) in (plus_le_reg_l x (minus y x) (minus z x) (eq_ind_r nat y (\lambda n:nat.(le n (plus x (minus z x)))) (eq_ind_r nat z (\lambda n:nat.(le y n)) H0 (plus x (minus z x)) DEFINED) (plus x (minus y x)) DEFINED0)))))))
-Post Nodes: 88
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_minus_plus
-Pre Nodes : 505
-Warn:  Optimizer: remove 3
-Warn:  Optimizer: anticipate 2
-Warn:  Optimizer: swap 2
-nat
-nat
-nat
-nat
-Warn:  Optimizer: remove 3
-Warn:  Optimizer: swap 2
-Warn:  Optimizer: nested application
-Warn:  Optimizer: anticipate 2
-Warn:  Optimizer: swap 2
-Optimized : (\lambda z:nat.(nat_ind (\lambda n:nat.(\forall x:nat.((le n x)\to (\forall y:nat.(eq nat (minus (plus x y) n) (plus (minus x n) y)))))) (\lambda x:nat.(\lambda H:(le O x). let H0 \def (le_ind O (\lambda n:nat.((eq nat n x)\to (\forall y:nat.(eq nat (minus (plus x y) O) (plus (minus x O) y))))) (\lambda H0:(eq nat O x).(eq_ind nat O (\lambda n:nat.(\forall y:nat.(eq nat (minus (plus n y) O) (plus (minus n O) y)))) (\lambda y:nat.(sym_eq nat (plus (minus O O) y) (minus (plus O y) O) (minus_n_O (plus O y)))) x H0)) (\lambda m:nat.(\lambda H0:(le O m).(\lambda _:((eq nat m x)\to (\forall y:nat.(eq nat (minus (plus x y) O) (plus (minus x O) y)))).(\lambda H1:(eq nat (S m) x). let DEFINED \def (eq_ind nat (S m) (\lambda n:nat.((le O m)\to (\forall y:nat.(eq nat (minus (plus n y) O) (plus (minus n O) y))))) (\lambda _:(le O m).(\lambda y:nat.(refl_equal nat (plus (minus (S m) O) y)))) x H1) in (DEFINED H0))))) x H) in (H0 (refl_equal nat x)))) (\lambda z0:nat.(\lambda H:(\forall x:nat.((le z0 x)\to (\forall y:nat.(eq nat (minus (plus x y) z0) (plus (minus x z0) y))))).(\lambda x:nat.(nat_ind (\lambda n:nat.((le (S z0) n)\to (\forall y:nat.(eq nat (minus (plus n y) (S z0)) (plus (minus n (S z0)) y))))) (\lambda H0:(le (S z0) O).(\lambda y:nat. let H1 \def (le_ind (S z0) (\lambda n:nat.((eq nat n O)\to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y)))) (\lambda H1:(eq nat (S z0) O). let H2 \def (eq_ind nat (S z0) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with 
- [ O => False
- | (S (_:nat)) => True
-]) I O H1) in (False_ind (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y)) H2)) (\lambda m:nat.(\lambda H1:(le (S z0) m).(\lambda _:((eq nat m O)\to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))).(\lambda H2:(eq nat (S m) O). let H3 \def (eq_ind nat (S m) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with 
- [ O => False
- | (S (_:nat)) => True
-]) I O H2) in  let DEFINED \def (False_ind ((le (S z0) m)\to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))) H3) in (DEFINED H1))))) O H0) in (H1 (refl_equal nat O)))) (\lambda n:nat.(\lambda _:((le (S z0) n)\to (\forall y:nat.(eq nat (minus (plus n y) (S z0)) (plus (minus n (S z0)) y)))).(\lambda H1:(le (S z0) (S n)).(\lambda y:nat.(H n (le_S_n z0 n H1) y))))) x)))) z))
-Post Nodes: 560
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_minus
-Pre Nodes : 51
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda x:nat.(\lambda z:nat.(\lambda y:nat.(\lambda H:(le (plus x y) z). let DEFINED \def (minus_plus_r x y) in (eq_ind nat (minus (plus x y) y) (\lambda n:nat.(le n (minus z y))) (le_minus_minus y (plus x y) (le_plus_r x y) z H) x DEFINED)))))
-Post Nodes: 53
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_trans_plus_r
-Pre Nodes : 27
-Optimized : (\lambda x:nat.(\lambda y:nat.(\lambda z:nat.(\lambda H:(le (plus x y) z).(le_trans y (plus x y) z (le_plus_r x y) H)))))
-Post Nodes: 27
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_gen_S
-Pre Nodes : 217
-Warn:  Optimizer: remove 3
-Warn:  Optimizer: anticipate 2
-Warn:  Optimizer: swap 2
-Optimized : (\lambda m:nat.(\lambda x:nat.(\lambda H:(le (S m) x). let H0 \def (le_ind (S m) (\lambda n:nat.((eq nat n x)\to (ex2 nat (\lambda n0:nat.(eq nat x (S n0))) (\lambda n0:nat.(le m n0))))) (\lambda H0:(eq nat (S m) x).(eq_ind nat (S m) (\lambda n:nat.(ex2 nat (\lambda n0:nat.(eq nat n (S n0))) (\lambda n0:nat.(le m n0)))) (ex_intro2 nat (\lambda n:nat.(eq nat (S m) (S n))) (\lambda n:nat.(le m n)) m (refl_equal nat (S m)) (le_n m)) x H0)) (\lambda m0:nat.(\lambda H0:(le (S m) m0).(\lambda _:((eq nat m0 x)\to (ex2 nat (\lambda n0:nat.(eq nat x (S n0))) (\lambda n0:nat.(le m n0)))).(\lambda H1:(eq nat (S m0) x). let DEFINED \def (eq_ind nat (S m0) (\lambda n:nat.((le (S m) m0)\to (ex2 nat (\lambda n0:nat.(eq nat n (S n0))) (\lambda n0:nat.(le m n0))))) (\lambda H2:(le (S m) m0).(ex_intro2 nat (\lambda n:nat.(eq nat (S m0) (S n))) (\lambda n:nat.(le m n)) m0 (refl_equal nat (S m0)) (le_S_n m m0 (le_S (S m) m0 H2)))) x H1) in (DEFINED H0))))) x H) in (H0 (refl_equal nat x)))))
-Post Nodes: 243
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: lt_x_plus_x_Sy
-Pre Nodes : 64
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda x:nat.(\lambda y:nat. let DEFINED \def (plus_comm x (S y)) in (eq_ind_r nat (plus (S y) x) (\lambda n:nat.(lt x n)) (le_S_n (S x) (S (plus y x)) (le_n_S (S x) (S (plus y x)) (le_n_S x (plus y x) (le_plus_r y x)))) (plus x (S y)) DEFINED)))
-Post Nodes: 66
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: simpl_lt_plus_r
-Pre Nodes : 75
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 1
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 1
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda p:nat.(\lambda n:nat.(\lambda m:nat.(\lambda H:(lt (plus n p) (plus m p)). let DEFINED \def (plus_comm n p) in  let H0 \def (eq_ind nat (plus n p) (\lambda n0:nat.(lt n0 (plus m p))) H (plus p n) DEFINED) in  let DEFINED0 \def (plus_comm m p) in  let H1 \def (eq_ind nat (plus m p) (\lambda n0:nat.(lt (plus p n) n0)) H0 (plus p m) DEFINED0) in (plus_lt_reg_l n m p H1)))))
-Post Nodes: 79
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: minus_x_Sy
-Pre Nodes : 330
-nat
-nat
-nat
-nat
-Warn:  Optimizer: remove 3
-Warn:  Optimizer: swap 2
-Warn:  Optimizer: nested application
-Warn:  Optimizer: anticipate 2
-Warn:  Optimizer: swap 2
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda x:nat.(nat_ind (\lambda n:nat.(\forall y:nat.((lt y n)\to (eq nat (minus n y) (S (minus n (S y))))))) (\lambda y:nat.(\lambda H:(lt y O). let H0 \def (le_ind (S y) (\lambda n:nat.((eq nat n O)\to (eq nat (minus O y) (S (minus O (S y)))))) (\lambda H0:(eq nat (S y) O). let H1 \def (eq_ind nat (S y) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with 
- [ O => False
- | (S (_:nat)) => True
-]) I O H0) in (False_ind (eq nat (minus O y) (S (minus O (S y)))) H1)) (\lambda m:nat.(\lambda H0:(le (S y) m).(\lambda _:((eq nat m O)\to (eq nat (minus O y) (S (minus O (S y))))).(\lambda H1:(eq nat (S m) O). let H2 \def (eq_ind nat (S m) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with 
- [ O => False
- | (S (_:nat)) => True
-]) I O H1) in  let DEFINED \def (False_ind ((le (S y) m)\to (eq nat (minus O y) (S (minus O (S y))))) H2) in (DEFINED H0))))) O H) in (H0 (refl_equal nat O)))) (\lambda n:nat.(\lambda H:(\forall y:nat.((lt y n)\to (eq nat (minus n y) (S (minus n (S y)))))).(\lambda y:nat.(nat_ind (\lambda n0:nat.((lt n0 (S n))\to (eq nat (minus (S n) n0) (S (minus (S n) (S n0)))))) (\lambda _:(lt O (S n)). let DEFINED \def (minus_n_O n) in (eq_ind nat n (\lambda n0:nat.(eq nat (S n) (S n0))) (refl_equal nat (S n)) (minus n O) DEFINED)) (\lambda n0:nat.(\lambda _:((lt n0 (S n))\to (eq nat (minus (S n) n0) (S (minus (S n) (S n0))))).(\lambda H1:(lt (S n0) (S n)). let H2 \def (le_S_n (S n0) n H1) in (H n0 H2)))) y)))) x))
-Post Nodes: 354
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: lt_plus_minus
-Pre Nodes : 16
-Optimized : (\lambda x:nat.(\lambda y:nat.(\lambda H:(lt x y).(le_plus_minus (S x) y H))))
-Post Nodes: 16
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: lt_plus_minus_r
-Pre Nodes : 53
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda x:nat.(\lambda y:nat.(\lambda H:(lt x y). let DEFINED \def (plus_comm (minus y (S x)) x) in (eq_ind_r nat (plus x (minus y (S x))) (\lambda n:nat.(eq nat y (S n))) (lt_plus_minus x y H) (plus (minus y (S x)) x) DEFINED))))
-Post Nodes: 55
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: minus_x_SO
-Pre Nodes : 56
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda x:nat.(\lambda H:(lt O x). let DEFINED \def (minus_n_O x) in  let DEFINED0 \def (minus_x_Sy x O H) in (eq_ind nat (minus x O) (\lambda n:nat.(eq nat x n)) (eq_ind nat x (\lambda n:nat.(eq nat x n)) (refl_equal nat x) (minus x O) DEFINED) (S (minus x (S O))) DEFINED0)))
-Post Nodes: 60
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_x_pred_y
-Pre Nodes : 175
-nat
-nat
-nat
-nat
-Warn:  Optimizer: remove 3
-Warn:  Optimizer: swap 2
-Warn:  Optimizer: nested application
-Warn:  Optimizer: anticipate 2
-Warn:  Optimizer: swap 2
-Optimized : (\lambda y:nat.(nat_ind (\lambda n:nat.(\forall x:nat.((lt x n)\to (le x (pred n))))) (\lambda x:nat.(\lambda H:(lt x O). let H0 \def (le_ind (S x) (\lambda n:nat.((eq nat n O)\to (le x O))) (\lambda H0:(eq nat (S x) O). let H1 \def (eq_ind nat (S x) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with 
- [ O => False
- | (S (_:nat)) => True
-]) I O H0) in (False_ind (le x O) H1)) (\lambda m:nat.(\lambda H0:(le (S x) m).(\lambda _:((eq nat m O)\to (le x O)).(\lambda H1:(eq nat (S m) O). let H2 \def (eq_ind nat (S m) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with 
- [ O => False
- | (S (_:nat)) => True
-]) I O H1) in  let DEFINED \def (False_ind ((le (S x) m)\to (le x O)) H2) in (DEFINED H0))))) O H) in (H0 (refl_equal nat O)))) (\lambda n:nat.(\lambda _:(\forall x:nat.((lt x n)\to (le x (pred n)))).(\lambda x:nat.(\lambda H0:(lt x (S n)).(le_S_n x n H0))))) y))
-Post Nodes: 186
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: lt_le_minus
-Pre Nodes : 44
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda x:nat.(\lambda y:nat.(\lambda H:(lt x y). let DEFINED \def (plus_comm x (S O)) in (le_minus x y (S O) (eq_ind_r nat (plus (S O) x) (\lambda n:nat.(le n y)) H (plus x (S O)) DEFINED)))))
-Post Nodes: 46
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: lt_le_e
-Pre Nodes : 39
-Optimized : (\lambda n:nat.(\lambda d:nat.(\lambda P:Prop.(\lambda H:((lt n d)\to P).(\lambda H0:((le d n)\to P). let H1 \def (le_or_lt d n) in (or_ind (le d n) (lt n d) P H0 H H1))))))
-Post Nodes: 39
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: lt_eq_e
-Pre Nodes : 45
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda x:nat.(\lambda y:nat.(\lambda P:Prop.(\lambda H:((lt x y)\to P).(\lambda H0:((eq nat x y)\to P).(\lambda H1:(le x y). let DEFINED \def (le_lt_or_eq x y H1) in (or_ind (lt x y) (eq nat x y) P H H0 DEFINED)))))))
-Post Nodes: 47
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: lt_eq_gt_e
-Pre Nodes : 60
-Optimized : (\lambda x:nat.(\lambda y:nat.(\lambda P:Prop.(\lambda H:((lt x y)\to P).(\lambda H0:((eq nat x y)\to P).(\lambda H1:((lt y x)\to P).(lt_le_e x y P H (\lambda H2:(le y x).(lt_eq_e y x P H1 (\lambda H3:(eq nat y x).(H0 (sym_eq nat y x H3))) H2)))))))))
-Post Nodes: 60
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: lt_gen_xS
-Pre Nodes : 190
-Optimized : (\lambda x:nat.(nat_ind (\lambda n:nat.(\forall n0:nat.((lt n (S n0))\to (or (eq nat n O) (ex2 nat (\lambda m:nat.(eq nat n (S m))) (\lambda m:nat.(lt m n0))))))) (\lambda n:nat.(\lambda _:(lt O (S n)).(or_introl (eq nat O O) (ex2 nat (\lambda m:nat.(eq nat O (S m))) (\lambda m:nat.(lt m n))) (refl_equal nat O)))) (\lambda n:nat.(\lambda _:(\forall n0:nat.((lt n (S n0))\to (or (eq nat n O) (ex2 nat (\lambda m:nat.(eq nat n (S m))) (\lambda m:nat.(lt m n0)))))).(\lambda n0:nat.(\lambda H0:(lt (S n) (S n0)).(or_intror (eq nat (S n) O) (ex2 nat (\lambda m:nat.(eq nat (S n) (S m))) (\lambda m:nat.(lt m n0))) (ex_intro2 nat (\lambda m:nat.(eq nat (S n) (S m))) (\lambda m:nat.(lt m n0)) n (refl_equal nat (S n)) (le_S_n (S n) n0 H0))))))) x))
-Post Nodes: 190
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_lt_false
-Pre Nodes : 25
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda x:nat.(\lambda y:nat.(\lambda H:(le x y).(\lambda H0:(lt y x).(\lambda P:Prop. let DEFINED \def (le_not_lt x y H H0) in (False_ind P DEFINED))))))
-Post Nodes: 27
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: lt_neq
-Pre Nodes : 33
-Optimized : (\lambda x:nat.(\lambda y:nat.(\lambda H:(lt x y).(\lambda H0:(eq nat x y). let H1 \def (eq_ind nat x (\lambda n:nat.(lt n y)) H y H0) in (lt_irrefl y H1)))))
-Post Nodes: 33
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: arith0
-Pre Nodes : 185
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda h2:nat.(\lambda d2:nat.(\lambda n:nat.(\lambda H:(le (plus d2 h2) n).(\lambda h3:nat. let DEFINED \def (plus_comm h2 d2) in  let DEFINED0 \def (plus_assoc h2 d2 h3) in  let DEFINED1 \def (minus_plus h2 (plus d2 h3)) in (eq_ind nat (minus (plus h2 (plus d2 h3)) h2) (\lambda n0:nat.(le n0 (minus (plus n h3) h2))) (le_minus_minus h2 (plus h2 (plus d2 h3)) (le_plus_l h2 (plus d2 h3)) (plus n h3) (eq_ind_r nat (plus (plus h2 d2) h3) (\lambda n0:nat.(le n0 (plus n h3))) (eq_ind_r nat (plus d2 h2) (\lambda n0:nat.(le (plus n0 h3) (plus n h3))) (le_S_n (plus (plus d2 h2) h3) (plus n h3) (lt_le_S (plus (plus d2 h2) h3) (S (plus n h3)) (le_lt_n_Sm (plus (plus d2 h2) h3) (plus n h3) (plus_le_compat (plus d2 h2) n h3 h3 H (le_n h3))))) (plus h2 d2) DEFINED) (plus h2 (plus d2 h3)) DEFINED0)) (plus d2 h3) DEFINED1))))))
-Post Nodes: 191
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: O_minus
-Pre Nodes : 211
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda x:nat.(nat_ind (\lambda n:nat.(\forall y:nat.((le n y)\to (eq nat (minus n y) O)))) (\lambda y:nat.(\lambda _:(le O y).(refl_equal nat O))) (\lambda x0:nat.(\lambda H:(\forall y:nat.((le x0 y)\to (eq nat (minus x0 y) O))).(\lambda y:nat.(nat_ind (\lambda n:nat.((le (S x0) n)\to (eq nat 
-match n return (\lambda n1:nat.nat) with 
- [ O => (S x0)
- | (S (l:nat)) => (minus x0 l)
-] O))) (\lambda H0:(le (S x0) O). let DEFINED \def (le_gen_S x0 O H0) in (ex2_ind nat (\lambda n:nat.(eq nat O (S n))) (\lambda n:nat.(le x0 n)) (eq nat (S x0) O) (\lambda x1:nat.(\lambda H1:(eq nat O (S x1)).(\lambda _:(le x0 x1). let H3 \def (eq_ind nat O (\lambda ee:nat.
-match ee return (\lambda _:nat.Prop) with 
- [ O => True
- | (S (_:nat)) => False
-]) I (S x1) H1) in (False_ind (eq nat (S x0) O) H3)))) DEFINED)) (\lambda n:nat.(\lambda _:((le (S x0) n)\to (eq nat 
-match n return (\lambda n1:nat.nat) with 
- [ O => (S x0)
- | (S (l:nat)) => (minus x0 l)
-] O)).(\lambda H1:(le (S x0) (S n)).(H n (le_S_n x0 n H1))))) y)))) x))
-Post Nodes: 213
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: minus_minus
-Pre Nodes : 592
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 1
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 1
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda z:nat.(nat_ind (\lambda n:nat.(\forall x:nat.(\forall y:nat.((le n x)\to ((le n y)\to ((eq nat (minus x n) (minus y n))\to (eq nat x y))))))) (\lambda x:nat.(\lambda y:nat.(\lambda _:(le O x).(\lambda _:(le O y).(\lambda H1:(eq nat (minus x O) (minus y O)). let DEFINED \def (minus_n_O x) in  let H2 \def (eq_ind_r nat (minus x O) (\lambda n:nat.(eq nat n (minus y O))) H1 x DEFINED) in  let DEFINED0 \def (minus_n_O y) in  let H3 \def (eq_ind_r nat (minus y O) (\lambda n:nat.(eq nat x n)) H2 y DEFINED0) in H3))))) (\lambda z0:nat.(\lambda IH:(\forall x:nat.(\forall y:nat.((le z0 x)\to ((le z0 y)\to ((eq nat (minus x z0) (minus y z0))\to (eq nat x y)))))).(\lambda x:nat.(nat_ind (\lambda n:nat.(\forall y:nat.((le (S z0) n)\to ((le (S z0) y)\to ((eq nat (minus n (S z0)) (minus y (S z0)))\to (eq nat n y)))))) (\lambda y:nat.(\lambda H:(le (S z0) O).(\lambda _:(le (S z0) y).(\lambda _:(eq nat (minus O (S z0)) (minus y (S z0))). let DEFINED \def (le_gen_S z0 O H) in (ex2_ind nat (\lambda n:nat.(eq nat O (S n))) (\lambda n:nat.(le z0 n)) (eq nat O y) (\lambda x0:nat.(\lambda H2:(eq nat O (S x0)).(\lambda _:(le z0 x0). let H4 \def (eq_ind nat O (\lambda ee:nat.
-match ee return (\lambda _:nat.Prop) with 
- [ O => True
- | (S (_:nat)) => False
-]) I (S x0) H2) in (False_ind (eq nat O y) H4)))) DEFINED))))) (\lambda x0:nat.(\lambda _:(\forall y:nat.((le (S z0) x0)\to ((le (S z0) y)\to ((eq nat (minus x0 (S z0)) (minus y (S z0)))\to (eq nat x0 y))))).(\lambda y:nat.(nat_ind (\lambda n:nat.((le (S z0) (S x0))\to ((le (S z0) n)\to ((eq nat (minus (S x0) (S z0)) (minus n (S z0)))\to (eq nat (S x0) n))))) (\lambda _:(le (S z0) (S x0)).(\lambda H0:(le (S z0) O).(\lambda _:(eq nat (minus (S x0) (S z0)) (minus O (S z0))). let DEFINED \def (le_gen_S z0 O H0) in (ex2_ind nat (\lambda n:nat.(eq nat O (S n))) (\lambda n:nat.(le z0 n)) (eq nat (S x0) O) (\lambda x1:nat.(\lambda H2:(eq nat O (S x1)).(\lambda _:(le z0 x1). let H4 \def (eq_ind nat O (\lambda ee:nat.
-match ee return (\lambda _:nat.Prop) with 
- [ O => True
- | (S (_:nat)) => False
-]) I (S x1) H2) in (False_ind (eq nat (S x0) O) H4)))) DEFINED)))) (\lambda y0:nat.(\lambda _:((le (S z0) (S x0))\to ((le (S z0) y0)\to ((eq nat (minus (S x0) (S z0)) (minus y0 (S z0)))\to (eq nat (S x0) y0)))).(\lambda H:(le (S z0) (S x0)).(\lambda H0:(le (S z0) (S y0)).(\lambda H1:(eq nat (minus (S x0) (S z0)) (minus (S y0) (S z0))).(f_equal nat nat S x0 y0 (IH x0 y0 (le_S_n z0 x0 H) (le_S_n z0 y0 H0) H1))))))) y)))) x)))) z))
-Post Nodes: 600
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: plus_plus
-Pre Nodes : 1222
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 1
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 1
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 1
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 1
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 1
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 1
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Warn:  Optimizer: swap 1
-Optimized : (\lambda z:nat.(nat_ind (\lambda n:nat.(\forall x1:nat.(\forall x2:nat.(\forall y1:nat.(\forall y2:nat.((le x1 n)\to ((le x2 n)\to ((eq nat (plus (minus n x1) y1) (plus (minus n x2) y2))\to (eq nat (plus x1 y2) (plus x2 y1)))))))))) (\lambda x1:nat.(\lambda x2:nat.(\lambda y1:nat.(\lambda y2:nat.(\lambda H:(le x1 O).(\lambda H0:(le x2 O).(\lambda H1:(eq nat y1 y2). let H_y \def (le_n_O_eq x2 H0) in  let H_y0 \def (le_n_O_eq x1 H) in (eq_ind nat y1 (\lambda n:nat.(eq nat (plus x1 n) (plus x2 y1))) (eq_ind nat O (\lambda n:nat.(eq nat (plus x1 y1) (plus n y1))) (eq_ind nat O (\lambda n:nat.(eq nat (plus n y1) (plus O y1))) (refl_equal nat (plus O y1)) x1 H_y0) x2 H_y) y2 H1)))))))) (\lambda z0:nat.(\lambda IH:(\forall x1:nat.(\forall x2:nat.(\forall y1:nat.(\forall y2:nat.((le x1 z0)\to ((le x2 z0)\to ((eq nat (plus (minus z0 x1) y1) (plus (minus z0 x2) y2))\to (eq nat (plus x1 y2) (plus x2 y1))))))))).(\lambda x1:nat.(nat_ind (\lambda n:nat.(\forall x2:nat.(\forall y1:nat.(\forall y2:nat.((le n (S z0))\to ((le x2 (S z0))\to ((eq nat (plus (minus (S z0) n) y1) (plus (minus (S z0) x2) y2))\to (eq nat (plus n y2) (plus x2 y1))))))))) (\lambda x2:nat.(nat_ind (\lambda n:nat.(\forall y1:nat.(\forall y2:nat.((le O (S z0))\to ((le n (S z0))\to ((eq nat (plus (minus (S z0) O) y1) (plus (minus (S z0) n) y2))\to (eq nat (plus O y2) (plus n y1)))))))) (\lambda y1:nat.(\lambda y2:nat.(\lambda _:(le O (S z0)).(\lambda _:(le O (S z0)).(\lambda H1:(eq nat (S (plus z0 y1)) (S (plus z0 y2))). let H_y \def (IH O O) in  let DEFINED \def (minus_n_O z0) in  let H2 \def (eq_ind_r nat (minus z0 O) (\lambda n:nat.(\forall y3:nat.(\forall y4:nat.((le O z0)\to ((le O z0)\to ((eq nat (plus n y3) (plus n y4))\to (eq nat y4 y3))))))) H_y z0 DEFINED) in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (H2 (plus z0 y2) (plus z0 y1) (le_O_n z0) (le_O_n z0) (f_equal nat nat (plus z0) (plus z0 y2) (plus z0 y1) (sym_eq nat (plus z0 y1) (plus z0 y2) (eq_add_S (plus z0 y1) (plus z0 y2) H1)))))))))) (\lambda x3:nat.(\lambda _:(\forall y1:nat.(\forall y2:nat.((le O (S z0))\to ((le x3 (S z0))\to ((eq nat (S (plus z0 y1)) (plus 
-match x3 return (\lambda n:nat.nat) with 
- [ O => (S z0)
- | (S (l:nat)) => (minus z0 l)
-] y2))\to (eq nat y2 (plus x3 y1))))))).(\lambda y1:nat.(\lambda y2:nat.(\lambda _:(le O (S z0)).(\lambda H0:(le (S x3) (S z0)).(\lambda H1:(eq nat (S (plus z0 y1)) (plus (minus z0 x3) y2)). let H_y \def (IH O x3 (S y1)) in  let DEFINED \def (minus_n_O z0) in  let H2 \def (eq_ind_r nat (minus z0 O) (\lambda n:nat.(\forall y3:nat.((le O z0)\to ((le x3 z0)\to ((eq nat (plus n (S y1)) (plus (minus z0 x3) y3))\to (eq nat y3 (plus x3 (S y1)))))))) H_y z0 DEFINED) in  let DEFINED0 \def (plus_n_Sm z0 y1) in  let H3 \def (eq_ind_r nat (plus z0 (S y1)) (\lambda n:nat.(\forall y3:nat.((le O z0)\to ((le x3 z0)\to ((eq nat n (plus (minus z0 x3) y3))\to (eq nat y3 (plus x3 (S y1)))))))) H2 (S (plus z0 y1)) DEFINED0) in  let DEFINED1 \def (plus_n_Sm x3 y1) in  let H4 \def (eq_ind_r nat (plus x3 (S y1)) (\lambda n:nat.(\forall y3:nat.((le O z0)\to ((le x3 z0)\to ((eq nat (S (plus z0 y1)) (plus (minus z0 x3) y3))\to (eq nat y3 n)))))) H3 (S (plus x3 y1)) DEFINED1) in (H4 y2 (le_O_n z0) (le_S_n x3 z0 H0) H1)))))))) x2)) (\lambda x2:nat.(\lambda _:(\forall x3:nat.(\forall y1:nat.(\forall y2:nat.((le x2 (S z0))\to ((le x3 (S z0))\to ((eq nat (plus (minus (S z0) x2) y1) (plus (minus (S z0) x3) y2))\to (eq nat (plus x2 y2) (plus x3 y1)))))))).(\lambda x3:nat.(nat_ind (\lambda n:nat.(\forall y1:nat.(\forall y2:nat.((le (S x2) (S z0))\to ((le n (S z0))\to ((eq nat (plus (minus (S z0) (S x2)) y1) (plus (minus (S z0) n) y2))\to (eq nat (plus (S x2) y2) (plus n y1)))))))) (\lambda y1:nat.(\lambda y2:nat.(\lambda H:(le (S x2) (S z0)).(\lambda _:(le O (S z0)).(\lambda H1:(eq nat (plus (minus z0 x2) y1) (S (plus z0 y2))). let H_y \def (IH x2 O y1 (S y2)) in  let DEFINED \def (minus_n_O z0) in  let H2 \def (eq_ind_r nat (minus z0 O) (\lambda n:nat.((le x2 z0)\to ((le O z0)\to ((eq nat (plus (minus z0 x2) y1) (plus n (S y2)))\to (eq nat (plus x2 (S y2)) y1))))) H_y z0 DEFINED) in  let DEFINED0 \def (plus_n_Sm z0 y2) in  let H3 \def (eq_ind_r nat (plus z0 (S y2)) (\lambda n:nat.((le x2 z0)\to ((le O z0)\to ((eq nat (plus (minus z0 x2) y1) n)\to (eq nat (plus x2 (S y2)) y1))))) H2 (S (plus z0 y2)) DEFINED0) in  let DEFINED1 \def (plus_n_Sm x2 y2) in  let H4 \def (eq_ind_r nat (plus x2 (S y2)) (\lambda n:nat.((le x2 z0)\to ((le O z0)\to ((eq nat (plus (minus z0 x2) y1) (S (plus z0 y2)))\to (eq nat n y1))))) H3 (S (plus x2 y2)) DEFINED1) in (H4 (le_S_n x2 z0 H) (le_O_n z0) H1)))))) (\lambda x4:nat.(\lambda _:(\forall y1:nat.(\forall y2:nat.((le (S x2) (S z0))\to ((le x4 (S z0))\to ((eq nat (plus (minus z0 x2) y1) (plus 
-match x4 return (\lambda n:nat.nat) with 
- [ O => (S z0)
- | (S (l:nat)) => (minus z0 l)
-] y2))\to (eq nat (S (plus x2 y2)) (plus x4 y1))))))).(\lambda y1:nat.(\lambda y2:nat.(\lambda H:(le (S x2) (S z0)).(\lambda H0:(le (S x4) (S z0)).(\lambda H1:(eq nat (plus (minus z0 x2) y1) (plus (minus z0 x4) y2)).(f_equal nat nat S (plus x2 y2) (plus x4 y1) (IH x2 x4 y1 y2 (le_S_n x2 z0 H) (le_S_n x4 z0 H0) H1))))))))) x3)))) x1)))) z))
-Post Nodes: 1236
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_S_minus
-Pre Nodes : 41
-Warn:  Optimizer: remove 1
-Optimized : (\lambda d:nat.(\lambda h:nat.(\lambda n:nat.(\lambda H:(le (plus d h) n).(le_S d (minus n h) (le_minus d n h H))))))
-Post Nodes: 27
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./ext/arith.mma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Info:  execution of ext/arith.mma completed in 44''.
-Info:  execution of ext/tactics.mma started:
-Debug: Executing: ``set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-2/ext ...''
-Info:  baseuri cic:/matita/LAMBDA-TYPES/Base-2/ext/tactics is not empty
-Info:  cleaning baseuri cic:/matita/LAMBDA-TYPES/Base-2/ext/tactics
-Info:  Removing: cic:/matita/LAMBDA-TYPES/Base-2/ext/tactics/*
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./preamble.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Debug: Executing: ``include "cic:/matita/LAMBDA-TYPES/Base-2/preamble" ...''
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: insert_eq
-Pre Nodes : 36
-Optimized : (\lambda S:Set.(\lambda x:S.(\lambda P:(S\to Prop).(\lambda G:Prop.(\lambda H:(\forall y:S.((P y)\to ((eq S y x)\to G))).(\lambda H0:(P x).(H x H0 (refl_equal S x))))))))
-Post Nodes: 36
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: unintro
-Pre Nodes : 17
-Optimized : (\lambda A:Set.(\lambda a:A.(\lambda P:(A\to Prop).(\lambda H:(\forall x:A.(P x)).(H a)))))
-Post Nodes: 17
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: xinduction
-Pre Nodes : 27
-Optimized : (\lambda A:Set.(\lambda t:A.(\lambda P:(A\to Prop).(\lambda H:(\forall x:A.((eq A t x)\to (P x))).(H t (refl_equal A t))))))
-Post Nodes: 27
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./ext/tactics.mma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Info:  execution of ext/tactics.mma completed in 4''.
-Info:  execution of pippo.mma started:
-Debug: Executing: ``set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-2/pip ...''
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./preamble.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Debug: Executing: ``include "cic:/matita/LAMBDA-TYPES/Base-2/preamble" ...''
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: minus_x_Sy
-Pre Nodes : 330
-nat
-nat
-nat
-nat
-Warn:  Optimizer: remove 3
-Warn:  Optimizer: swap 2
-Warn:  Optimizer: nested application
-Warn:  Optimizer: anticipate 2
-Warn:  Optimizer: swap 2
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\lambda x:nat.(nat_ind (\lambda n:nat.(\forall y:nat.((lt y n)\to (eq nat (minus n y) (S (minus n (S y))))))) (\lambda y:nat.(\lambda H:(lt y O). let H0 \def (le_ind (S y) (\lambda n:nat.((eq nat n O)\to (eq nat (minus O y) (S (minus O (S y)))))) (\lambda H0:(eq nat (S y) O). let H1 \def (eq_ind nat (S y) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with 
- [ O => False
- | (S (_:nat)) => True
-]) I O H0) in (False_ind (eq nat (minus O y) (S (minus O (S y)))) H1)) (\lambda m:nat.(\lambda H0:(le (S y) m).(\lambda _:((eq nat m O)\to (eq nat (minus O y) (S (minus O (S y))))).(\lambda H1:(eq nat (S m) O). let H2 \def (eq_ind nat (S m) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with 
- [ O => False
- | (S (_:nat)) => True
-]) I O H1) in  let DEFINED \def (False_ind ((le (S y) m)\to (eq nat (minus O y) (S (minus O (S y))))) H2) in (DEFINED H0))))) O H) in (H0 (refl_equal nat O)))) (\lambda n:nat.(\lambda H:(\forall y:nat.((lt y n)\to (eq nat (minus n y) (S (minus n (S y)))))).(\lambda y:nat.(nat_ind (\lambda n0:nat.((lt n0 (S n))\to (eq nat (minus (S n) n0) (S (minus (S n) (S n0)))))) (\lambda _:(lt O (S n)). let DEFINED \def (minus_n_O n) in (eq_ind nat n (\lambda n0:nat.(eq nat (S n) (S n0))) (refl_equal nat (S n)) (minus n O) DEFINED)) (\lambda n0:nat.(\lambda _:((lt n0 (S n))\to (eq nat (minus (S n) n0) (S (minus (S n) (S n0))))).(\lambda H1:(lt (S n0) (S n)). let H2 \def (le_S_n (S n0) n H1) in (H n0 H2)))) y)))) x))
-Post Nodes: 354
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./pippo.mma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Info:  execution of pippo.mma completed in 2''.
-Info:  execution of plist/defs.mma started:
-Debug: Executing: ``set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-2/pli ...''
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./preamble.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Debug: Executing: ``include "cic:/matita/LAMBDA-TYPES/Base-2/preamble" ...''
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./plist/defs.mma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Info:  execution of plist/defs.mma completed in 1''.
-Info:  execution of plist/props.mma started:
-Debug: Executing: ``set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-2/pli ...''
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./plist/defs.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Debug: Executing: ``include "cic:/matita/LAMBDA-TYPES/Base-2/plist/def ...''
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: papp_ss
-Pre Nodes : 121
-Warn:  Optimizer: anticipate 3
-Warn:  Optimizer: swap 3
-Optimized : (\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. let DEFINED \def (H is2) in (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)) DEFINED)))))) is1))
-Post Nodes: 123
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./plist/props.mma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Info:  execution of plist/props.mma completed in 2''.
-Info:  execution of types/defs.mma started:
-Debug: Executing: ``set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-2/typ ...''
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./preamble.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Debug: Executing: ``include "cic:/matita/LAMBDA-TYPES/Base-2/preamble" ...''
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./types/defs.mma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Info:  execution of types/defs.mma completed in 0''.
-Info:  execution of types/props.mma started:
-Debug: Executing: ``set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-2/typ ...''
-Info:  baseuri cic:/matita/LAMBDA-TYPES/Base-2/types/props is not empty
-Info:  cleaning baseuri cic:/matita/LAMBDA-TYPES/Base-2/types/props
-Info:  Removing: cic:/matita/LAMBDA-TYPES/Base-2/types/props/*
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./types/defs.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Debug: Executing: ``include "cic:/matita/LAMBDA-TYPES/Base-2/types/def ...''
-Debug: Executing: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: ex2_sym
-Pre Nodes : 77
-Optimized : (\lambda A:Set.(\lambda P:(A\to Prop).(\lambda Q:(A\to Prop).(\lambda H:(ex2 A (\lambda x:A.(P x)) (\lambda x:A.(Q x))).(ex2_ind A (\lambda x:A.(P x)) (\lambda x:A.(Q x)) (ex2 A (\lambda x:A.(Q x)) (\lambda x:A.(P x))) (\lambda x:A.(\lambda H0:(P x).(\lambda H1:(Q x).(ex_intro2 A (\lambda x0:A.(Q x0)) (\lambda x0:A.(P x0)) x H1 H0)))) H)))))
-Post Nodes: 77
-Debug: Procedural: level 2 transformation
-Debug: Procedural: grafite rendering
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./types/props.mma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Info:  execution of types/props.mma completed in 2''.
-Info:  execution of theory.mma started:
-Debug: Executing: ``set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-2/the ...''
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./ext/tactics.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Debug: Executing: ``include "cic:/matita/LAMBDA-TYPES/Base-2/ext/tacti ...''
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./ext/arith.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Debug: Executing: ``include "cic:/matita/LAMBDA-TYPES/Base-2/ext/arith ...''
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./types/props.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Debug: Executing: ``include "cic:/matita/LAMBDA-TYPES/Base-2/types/pro ...''
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./blt/props.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Debug: Executing: ``include "cic:/matita/LAMBDA-TYPES/Base-2/blt/props ...''
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./plist/props.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Debug: Executing: ``include "cic:/matita/LAMBDA-TYPES/Base-2/plist/pro ...''
-Debug: Including /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./theory.mma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-Info:  execution of theory.mma completed in 1''.
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/root b/helm/software/matita/contribs/LAMBDA-TYPES/root
index d3e398271..092311539 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/root
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/root
@@ -1,2 +1,2 @@
 baseuri=cic:/matita/LAMBDA-TYPES
-include_paths= ../../legacy 
+include_paths= ../../legacy ../RELATIONAL