From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Thu, 14 Sep 2006 17:53:17 +0000 (+0000)
Subject: ok up to pc1
X-Git-Tag: 0.4.95@7852~1031
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=4ce09491ef9001883737418ca8b572267ac0a219;p=helm.git

ok up to pc1
---

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/arity/pr3.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/arity/pr3.ma
new file mode 100644
index 000000000..26a96fc7b
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/arity/pr3.ma
@@ -0,0 +1,625 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/arity/pr3".
+
+include "csuba/arity.ma".
+
+include "pr3/defs.ma".
+
+include "pr1/defs.ma".
+
+include "wcpr0/getl.ma".
+
+include "pr0/fwd.ma".
+
+include "arity/subst0.ma".
+
+theorem arity_sred_wcpr0_pr0:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g 
+c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 
+t2) \to (arity g c2 t2 a)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda 
+(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda 
+(a0: A).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to 
+(arity g c2 t2 a0)))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2: 
+C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort n) 
+t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n))) 
+(arity_sort g c2 n) t2 (pr0_gen_sort t2 n H1)))))))) (\lambda (c: C).(\lambda 
+(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d 
+(Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda 
+(H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to 
+(arity g c2 t2 a0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c 
+c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i) t2)).(eq_ind_r T (TLRef i) 
+(\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T (\lambda (e2: C).(\lambda 
+(u2: T).(getl i c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: 
+T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (arity g c2 
+(TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl i c2 
+(CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u 
+x1)).(arity_abbr g c2 x0 x1 i H5 a0 (H2 x0 H6 x1 H7))))))) (wcpr0_getl c c2 
+H3 i d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 i H4)))))))))))))) (\lambda (c: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c 
+(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g 
+a0))).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: 
+T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (c2: 
+C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i) 
+t2)).(eq_ind_r T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T 
+(\lambda (e2: C).(\lambda (u2: T).(getl i c2 (CHead e2 (Bind Abst) u2)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: 
+T).(pr0 u u2))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (H5: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (H6: (wcpr0 
+d x0)).(\lambda (H7: (pr0 u x1)).(arity_abst g c2 x0 x1 i H5 a0 (H2 x0 H6 x1 
+H7))))))) (wcpr0_getl c c2 H3 i d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 i 
+H4)))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda 
+(c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u 
+a1)).(\lambda (H2: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 
+u t2) \to (arity g c2 t2 a1))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda 
+(H3: (arity g (CHead c (Bind b) u) t a2)).(\lambda (H4: ((\forall (c2: 
+C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t t2) \to 
+(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H5: (wcpr0 c 
+c2)).(\lambda (t2: T).(\lambda (H6: (pr0 (THead (Bind b) u t) t2)).(insert_eq 
+T (THead (Bind b) u t) (\lambda (t0: T).(pr0 t0 t2)) (arity g c2 t2 a2) 
+(\lambda (y: T).(\lambda (H7: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda 
+(t3: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t3 a2)))) (\lambda 
+(t0: T).(\lambda (H8: (eq T t0 (THead (Bind b) u t))).(let H9 \def (f_equal T 
+T (\lambda (e: T).e) t0 (THead (Bind b) u t) H8) in (eq_ind_r T (THead (Bind 
+b) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_bind g b H0 c2 u a1 (H2 
+c2 H5 u (pr0_refl u)) t a2 (H4 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H5 u u 
+(pr0_refl u) (Bind b)) t (pr0_refl t))) t0 H9)))) (\lambda (u1: T).(\lambda 
+(u2: T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1 (THead (Bind b) u 
+t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda 
+(H10: (pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b) u t)) \to (arity 
+g c2 t4 a2)))).(\lambda (k: K).(\lambda (H12: (eq T (THead k u1 t3) (THead 
+(Bind b) u t))).(let H13 \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 t) H12) in ((let H14 \def (f_equal T T (\lambda (e: T).(match e in T 
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
+\Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) (THead 
+(Bind b) u t) H12) in ((let H15 \def (f_equal T T (\lambda (e: T).(match e in 
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) 
+\Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3) (THead 
+(Bind b) u t) H12) in (\lambda (H16: (eq T u1 u)).(\lambda (H17: (eq K k 
+(Bind b))).(eq_ind_r K (Bind b) (\lambda (k0: K).(arity g c2 (THead k0 u2 t4) 
+a2)) (let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind b) u 
+t)) \to (arity g c2 t4 a2))) H11 t H15) in (let H19 \def (eq_ind T t3 
+(\lambda (t0: T).(pr0 t0 t4)) H10 t H15) in (let H20 \def (eq_ind T u1 
+(\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 u2 a2))) H9 
+u H16) in (let H21 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) H8 u H16) 
+in (arity_bind g b H0 c2 u2 a1 (H2 c2 H5 u2 H21) t4 a2 (H4 (CHead c2 (Bind b) 
+u2) (wcpr0_comp c c2 H5 u u2 H21 (Bind b)) t4 H19)))))) k H17)))) H14)) 
+H13)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda 
+(_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u t)) \to (arity g 
+c2 v2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 
+t4)).(\lambda (_: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 
+a2)))).(\lambda (H12: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) 
+(THead (Bind b) u t))).(let H13 \def (eq_ind T (THead (Flat Appl) v1 (THead 
+(Bind Abst) u0 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) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) v2 t4) a2) 
+H13)))))))))))) (\lambda (b0: B).(\lambda (_: (not (eq B b0 Abst))).(\lambda 
+(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 
+(THead (Bind b) u t)) \to (arity g c2 v2 a2)))).(\lambda (u1: T).(\lambda 
+(u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Bind b) u 
+t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: 
+(pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 
+a2)))).(\lambda (H15: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1 t3)) 
+(THead (Bind b) u t))).(let H16 \def (eq_ind T (THead (Flat Appl) v1 (THead 
+(Bind b0) 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 
+b) u t) H15) in (False_ind (arity g c2 (THead (Bind b0) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) a2) H16))))))))))))))))) (\lambda (u1: T).(\lambda 
+(u2: T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1 (THead (Bind b) u 
+t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda 
+(H10: (pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b) u t)) \to (arity 
+g c2 t4 a2)))).(\lambda (w: T).(\lambda (H12: (subst0 O u2 t4 w)).(\lambda 
+(H13: (eq T (THead (Bind Abbr) u1 t3) (THead (Bind b) u t))).(let H14 \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 t) H13) in ((let H15 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | 
+(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind 
+Abbr) u1 t3) (THead (Bind b) u t) H13) in ((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 _ _ t0) \Rightarrow t0])) 
+(THead (Bind Abbr) u1 t3) (THead (Bind b) u t) H13) in (\lambda (H17: (eq T 
+u1 u)).(\lambda (H18: (eq B Abbr b)).(let H19 \def (eq_ind T t3 (\lambda (t0: 
+T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t4 a2))) H11 t H16) in 
+(let H20 \def (eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H10 t H16) in (let 
+H21 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to 
+(arity g c2 u2 a2))) H9 u H17) in (let H22 \def (eq_ind T u1 (\lambda (t0: 
+T).(pr0 t0 u2)) H8 u H17) in (let H23 \def (eq_ind_r B b (\lambda (b0: 
+B).((eq T t (THead (Bind b0) u t)) \to (arity g c2 t4 a2))) H19 Abbr H18) in 
+(let H24 \def (eq_ind_r B b (\lambda (b0: B).((eq T u (THead (Bind b0) u t)) 
+\to (arity g c2 u2 a2))) H21 Abbr H18) in (let H25 \def (eq_ind_r B b 
+(\lambda (b0: B).(\forall (c3: C).((wcpr0 (CHead c (Bind b0) u) c3) \to 
+(\forall (t5: T).((pr0 t t5) \to (arity g c3 t5 a2)))))) H4 Abbr H18) in (let 
+H26 \def (eq_ind_r B b (\lambda (b0: B).(arity g (CHead c (Bind b0) u) t a2)) 
+H3 Abbr H18) in (let H27 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 
+Abst))) H0 Abbr H18) in (arity_bind g Abbr H27 c2 u2 a1 (H2 c2 H5 u2 H22) w 
+a2 (arity_subst0 g (CHead c2 (Bind Abbr) u2) t4 a2 (H25 (CHead c2 (Bind Abbr) 
+u2) (wcpr0_comp c c2 H5 u u2 H22 (Bind Abbr)) t4 H20) c2 u2 O (getl_refl Abbr 
+c2 u2) w H12)))))))))))))) H15)) H14))))))))))))) (\lambda (b0: B).(\lambda 
+(H8: (not (eq B b0 Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H9: 
+(pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 
+t4 a2)))).(\lambda (u0: T).(\lambda (H11: (eq T (THead (Bind b0) u0 (lift (S 
+O) O t3)) (THead (Bind b) u t))).(let H12 \def (f_equal T B (\lambda (e: 
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | 
+(TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k in K return 
+(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow 
+b0])])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u t) H11) in 
+((let H13 \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 (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u 
+t) H11) in ((let H14 \def (f_equal T T (\lambda (e: T).(match e in T return 
+(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat 
+\to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u1 t5) 
+\Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t5))]) in 
+lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow 
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
+t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
+(THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) 
+t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t0) 
+\Rightarrow t0])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u t) 
+H11) in (\lambda (_: (eq T u0 u)).(\lambda (H16: (eq B b0 b)).(let H17 \def 
+(eq_ind B b0 (\lambda (b1: B).(not (eq B b1 Abst))) H8 b H16) in (let H18 
+\def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead (Bind b) u t0)) \to 
+(arity g c2 t4 a2))) H10 (lift (S O) O t3) H14) in (let H19 \def (eq_ind_r T 
+t (\lambda (t0: T).(\forall (c3: C).((wcpr0 (CHead c (Bind b) u) c3) \to 
+(\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 a2)))))) H4 (lift (S O) O 
+t3) H14) in (let H20 \def (eq_ind_r T t (\lambda (t0: T).(arity g (CHead c 
+(Bind b) u) t0 a2)) H3 (lift (S O) O t3) H14) in (arity_gen_lift g (CHead c2 
+(Bind b) u) t4 a2 (S O) O (H19 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H5 u u 
+(pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t3 t4 H9 (S O) O)) c2 
+(drop_drop (Bind b) O c2 c2 (drop_refl c2) u))))))))) H13)) H12)))))))))) 
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: 
+(((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0: 
+T).(\lambda (H10: (eq T (THead (Flat Cast) u0 t3) (THead (Bind b) u t))).(let 
+H11 \def (eq_ind T (THead (Flat Cast) u0 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) u t) H10) in (False_ind (arity g c2 t4 a2) 
+H11)))))))) y t2 H7))) H6)))))))))))))))) (\lambda (c: C).(\lambda (u: 
+T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: 
+((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to (arity g 
+c2 t2 (asucc g a1)))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (H2: 
+(arity g (CHead c (Bind Abst) u) t a2)).(\lambda (H3: ((\forall (c2: 
+C).((wcpr0 (CHead c (Bind Abst) u) c2) \to (\forall (t2: T).((pr0 t t2) \to 
+(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c 
+c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) u t) 
+t2)).(insert_eq T (THead (Bind Abst) u t) (\lambda (t0: T).(pr0 t0 t2)) 
+(arity g c2 t2 (AHead a1 a2)) (\lambda (y: T).(\lambda (H6: (pr0 y 
+t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq T t0 (THead (Bind Abst) 
+u t)) \to (arity g c2 t3 (AHead a1 a2))))) (\lambda (t0: T).(\lambda (H7: (eq 
+T t0 (THead (Bind Abst) u t))).(let H8 \def (f_equal T T (\lambda (e: T).e) 
+t0 (THead (Bind Abst) u t) H7) in (eq_ind_r T (THead (Bind Abst) u t) 
+(\lambda (t3: T).(arity g c2 t3 (AHead a1 a2))) (arity_head g c2 u a1 (H1 c2 
+H4 u (pr0_refl u)) t a2 (H3 (CHead c2 (Bind Abst) u) (wcpr0_comp c c2 H4 u u 
+(pr0_refl u) (Bind Abst)) t (pr0_refl t))) t0 H8)))) (\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1 
+(THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2))))).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 
+(THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (k: 
+K).(\lambda (H11: (eq T (THead k u1 t3) (THead (Bind Abst) u t))).(let H12 
+\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) 
+with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
+\Rightarrow k0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in ((let H13 
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
+with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) 
+\Rightarrow t0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) 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 _ _ t0) 
+\Rightarrow t0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in (\lambda 
+(H15: (eq T u1 u)).(\lambda (H16: (eq K k (Bind Abst))).(eq_ind_r K (Bind 
+Abst) (\lambda (k0: K).(arity g c2 (THead k0 u2 t4) (AHead a1 a2))) (let H17 
+\def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to 
+(arity g c2 t4 (AHead a1 a2)))) H10 t H14) in (let H18 \def (eq_ind T t3 
+(\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind T u1 
+(\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead 
+a1 a2)))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 
+u2)) H7 u H15) in (arity_head g c2 u2 a1 (H1 c2 H4 u2 H20) t4 a2 (H3 (CHead 
+c2 (Bind Abst) u2) (wcpr0_comp c c2 H4 u u2 H20 (Bind Abst)) t4 H18)))))) k 
+H16)))) H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2: 
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u t)) 
+\to (arity g c2 v2 (AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) 
+\to (arity g c2 t4 (AHead a1 a2))))).(\lambda (H11: (eq T (THead (Flat Appl) 
+v1 (THead (Bind Abst) u0 t3)) (THead (Bind Abst) u t))).(let H12 \def (eq_ind 
+T (THead (Flat Appl) v1 (THead (Bind Abst) u0 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 Abst) u t) H11) in (False_ind (arity g c2 (THead 
+(Bind Abbr) v2 t4) (AHead a1 a2)) H12)))))))))))) (\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) u t)) \to (arity g c2 v2 
+(AHead a1 a2))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 
+u2)).(\lambda (_: (((eq T u1 (THead (Bind Abst) u t)) \to (arity g c2 u2 
+(AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 
+t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) \to (arity g c2 t4 
+(AHead a1 a2))))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) 
+u1 t3)) (THead (Bind Abst) u t))).(let H15 \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 
+Abst) u t) H14) in (False_ind (arity g c2 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) (AHead a1 a2)) H15))))))))))))))))) (\lambda 
+(u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 
+(THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2))))).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead 
+(Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (w: 
+T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T (THead (Bind Abbr) 
+u1 t3) (THead (Bind Abst) u t))).(let H13 \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 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 t) H12) in (False_ind (arity g 
+c2 (THead (Bind Abbr) u2 w) (AHead a1 a2)) H13))))))))))))) (\lambda (b: 
+B).(\lambda (H7: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (_: (pr0 t3 t4)).(\lambda (H9: (((eq T t3 (THead (Bind Abst) u 
+t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (u0: T).(\lambda (H10: (eq 
+T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Bind Abst) u t))).(let H11 
+\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) u0 (lift (S O) O 
+t3)) (THead (Bind Abst) u t) H10) in ((let H12 \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 (Bind b) 
+u0 (lift (S O) O t3)) (THead (Bind Abst) u t) H10) in ((let H13 \def (f_equal 
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T 
+\def (match t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow 
+(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) 
+| (THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) 
+t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) 
+\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T 
+\def (match t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow 
+(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) 
+| (THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) 
+t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t0) 
+\Rightarrow t0])) (THead (Bind b) u0 (lift (S O) O t3)) (THead (Bind Abst) u 
+t) H10) in (\lambda (_: (eq T u0 u)).(\lambda (H15: (eq B b Abst)).(let H16 
+\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 Abst H15) in (let 
+H17 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead (Bind Abst) u t0)) 
+\to (arity g c2 t4 (AHead a1 a2)))) H9 (lift (S O) O t3) H13) in (let H18 
+\def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 (CHead c (Bind 
+Abst) u) c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 a2)))))) H3 
+(lift (S O) O t3) H13) in (let H19 \def (eq_ind_r T t (\lambda (t0: T).(arity 
+g (CHead c (Bind Abst) u) t0 a2)) H2 (lift (S O) O t3) H13) in (let H20 \def 
+(match (H16 (refl_equal B Abst)) in False return (\lambda (_: False).(arity g 
+c2 t4 (AHead a1 a2))) with []) in H20)))))))) H12)) H11)))))))))) (\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 
+(THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (u0: 
+T).(\lambda (H9: (eq T (THead (Flat Cast) u0 t3) (THead (Bind Abst) u 
+t))).(let H10 \def (eq_ind T (THead (Flat Cast) u0 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 Abst) u t) H9) in (False_ind (arity g c2 
+t4 (AHead a1 a2)) H10)))))))) y t2 H6))) H5)))))))))))))) (\lambda (c: 
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda 
+(H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to 
+(arity g c2 t2 a1))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (H2: 
+(arity g c t (AHead a1 a2))).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) 
+\to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 (AHead a1 
+a2)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: 
+T).(\lambda (H5: (pr0 (THead (Flat Appl) u t) t2)).(insert_eq T (THead (Flat 
+Appl) u t) (\lambda (t0: T).(pr0 t0 t2)) (arity g c2 t2 a2) (\lambda (y: 
+T).(\lambda (H6: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq 
+T t0 (THead (Flat Appl) u t)) \to (arity g c2 t3 a2)))) (\lambda (t0: 
+T).(\lambda (H7: (eq T t0 (THead (Flat Appl) u t))).(let H8 \def (f_equal T T 
+(\lambda (e: T).e) t0 (THead (Flat Appl) u t) H7) in (eq_ind_r T (THead (Flat 
+Appl) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_appl g c2 u a1 (H1 c2 
+H4 u (pr0_refl u)) t a2 (H3 c2 H4 t (pr0_refl t))) t0 H8)))) (\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1 
+(THead (Flat Appl) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda 
+(t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Flat 
+Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (k: K).(\lambda (H11: (eq T 
+(THead k u1 t3) (THead (Flat Appl) u t))).(let H12 \def (f_equal T K (\lambda 
+(e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k 
+| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) 
+(THead (Flat Appl) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | 
+(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) 
+(THead (Flat Appl) u t) H11) 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 _ _ t0) \Rightarrow t0])) (THead k u1 t3) 
+(THead (Flat Appl) u t) H11) in (\lambda (H15: (eq T u1 u)).(\lambda (H16: 
+(eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda (k0: K).(arity g c2 
+(THead k0 u2 t4) a2)) (let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 
+(THead (Flat Appl) u t)) \to (arity g c2 t4 a2))) H10 t H14) in (let H18 \def 
+(eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind 
+T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 u2 
+a2))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) 
+H7 u H15) in (arity_appl g c2 u2 a1 (H1 c2 H4 u2 H20) t4 a2 (H3 c2 H4 t4 
+H18)))))) k H16)))) H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1: 
+T).(\lambda (v2: T).(\lambda (H7: (pr0 v1 v2)).(\lambda (H8: (((eq T v1 
+(THead (Flat Appl) u t)) \to (arity g c2 v2 a2)))).(\lambda (t3: T).(\lambda 
+(t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Flat 
+Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (H11: (eq T (THead (Flat Appl) 
+v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) u t))).(let H12 \def 
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
+[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t0 _) 
+\Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead 
+(Flat Appl) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e 
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind Abst) 
+u0 t3) | (TLRef _) \Rightarrow (THead (Bind Abst) u0 t3) | (THead _ _ t0) 
+\Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead 
+(Flat Appl) u t) H11) in (\lambda (H14: (eq T v1 u)).(let H15 \def (eq_ind T 
+v1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 v2 
+a2))) H8 u H14) in (let H16 \def (eq_ind T v1 (\lambda (t0: T).(pr0 t0 v2)) 
+H7 u H14) in (let H17 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead 
+(Flat Appl) u t0)) \to (arity g c2 t4 a2))) H10 (THead (Bind Abst) u0 t3) 
+H13) in (let H18 \def (eq_ind_r T t (\lambda (t0: T).((eq T u (THead (Flat 
+Appl) u t0)) \to (arity g c2 v2 a2))) H15 (THead (Bind Abst) u0 t3) H13) in 
+(let H19 \def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 c c3) 
+\to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 (AHead a1 a2))))))) H3 
+(THead (Bind Abst) u0 t3) H13) in (let H20 \def (eq_ind_r T t (\lambda (t0: 
+T).(arity g c t0 (AHead a1 a2))) H2 (THead (Bind Abst) u0 t3) H13) in (let 
+H21 \def (H1 c2 H4 v2 H16) in (let H22 \def (H19 c2 H4 (THead (Bind Abst) u0 
+t4) (pr0_comp u0 u0 (pr0_refl u0) t3 t4 H9 (Bind Abst))) in (let H23 \def 
+(arity_gen_abst g c2 u0 t4 (AHead a1 a2) H22) in (ex3_2_ind A A (\lambda (a3: 
+A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4)))) (\lambda (a3: 
+A).(\lambda (_: A).(arity g c2 u0 (asucc g a3)))) (\lambda (_: A).(\lambda 
+(a4: A).(arity g (CHead c2 (Bind Abst) u0) t4 a4))) (arity g c2 (THead (Bind 
+Abbr) v2 t4) a2) (\lambda (x0: A).(\lambda (x1: A).(\lambda (H24: (eq A 
+(AHead a1 a2) (AHead x0 x1))).(\lambda (H25: (arity g c2 u0 (asucc g 
+x0))).(\lambda (H26: (arity g (CHead c2 (Bind Abst) u0) t4 x1)).(let H27 \def 
+(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with 
+[(ASort _ _) \Rightarrow a1 | (AHead a0 _) \Rightarrow a0])) (AHead a1 a2) 
+(AHead x0 x1) H24) in ((let H28 \def (f_equal A A (\lambda (e: A).(match e in 
+A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a2 | (AHead _ a0) 
+\Rightarrow a0])) (AHead a1 a2) (AHead x0 x1) H24) in (\lambda (H29: (eq A a1 
+x0)).(let H30 \def (eq_ind_r A x1 (\lambda (a0: A).(arity g (CHead c2 (Bind 
+Abst) u0) t4 a0)) H26 a2 H28) in (let H31 \def (eq_ind_r A x0 (\lambda (a0: 
+A).(arity g c2 u0 (asucc g a0))) H25 a1 H29) in (arity_bind g Abbr 
+not_abbr_abst c2 v2 a1 H21 t4 a2 (csuba_arity g (CHead c2 (Bind Abst) u0) t4 
+a2 H30 (CHead c2 (Bind Abbr) v2) (csuba_abst g c2 c2 (csuba_refl g c2) u0 a1 
+H31 v2 H21))))))) H27))))))) H23)))))))))))) H12)))))))))))) (\lambda (b: 
+B).(\lambda (H7: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
+T).(\lambda (H8: (pr0 v1 v2)).(\lambda (H9: (((eq T v1 (THead (Flat Appl) u 
+t)) \to (arity g c2 v2 a2)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda 
+(H10: (pr0 u1 u2)).(\lambda (H11: (((eq T u1 (THead (Flat Appl) u t)) \to 
+(arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H12: (pr0 
+t3 t4)).(\lambda (H13: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 
+a2)))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) 
+(THead (Flat Appl) u t))).(let H15 \def (f_equal T T (\lambda (e: T).(match e 
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) 
+\Rightarrow v1 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (THead (Flat Appl) u t) H14) in ((let H16 \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 _ _ t0) \Rightarrow t0])) (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (THead (Flat Appl) u t) H14) in (\lambda (H17: (eq T 
+v1 u)).(let H18 \def (eq_ind T v1 (\lambda (t0: T).((eq T t0 (THead (Flat 
+Appl) u t)) \to (arity g c2 v2 a2))) H9 u H17) in (let H19 \def (eq_ind T v1 
+(\lambda (t0: T).(pr0 t0 v2)) H8 u H17) in (let H20 \def (eq_ind_r T t 
+(\lambda (t0: T).((eq T t3 (THead (Flat Appl) u t0)) \to (arity g c2 t4 a2))) 
+H13 (THead (Bind b) u1 t3) H16) in (let H21 \def (eq_ind_r T t (\lambda (t0: 
+T).((eq T u1 (THead (Flat Appl) u t0)) \to (arity g c2 u2 a2))) H11 (THead 
+(Bind b) u1 t3) H16) in (let H22 \def (eq_ind_r T t (\lambda (t0: T).((eq T u 
+(THead (Flat Appl) u t0)) \to (arity g c2 v2 a2))) H18 (THead (Bind b) u1 t3) 
+H16) in (let H23 \def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 
+c c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 (AHead a1 
+a2))))))) H3 (THead (Bind b) u1 t3) H16) in (let H24 \def (eq_ind_r T t 
+(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) H2 (THead (Bind b) u1 t3) H16) 
+in (let H25 \def (H1 c2 H4 v2 H19) in (let H26 \def (H23 c2 H4 (THead (Bind 
+b) u2 t4) (pr0_comp u1 u2 H10 t3 t4 H12 (Bind b))) in (let H27 \def 
+(arity_gen_bind b H7 g c2 u2 t4 (AHead a1 a2) H26) in (ex2_ind A (\lambda 
+(a3: A).(arity g c2 u2 a3)) (\lambda (_: A).(arity g (CHead c2 (Bind b) u2) 
+t4 (AHead a1 a2))) (arity g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) a2) (\lambda (x: A).(\lambda (H28: (arity g c2 u2 x)).(\lambda 
+(H29: (arity g (CHead c2 (Bind b) u2) t4 (AHead a1 a2))).(arity_bind g b H7 
+c2 u2 x H28 (THead (Flat Appl) (lift (S O) O v2) t4) a2 (arity_appl g (CHead 
+c2 (Bind b) u2) (lift (S O) O v2) a1 (arity_lift g c2 v2 a1 H25 (CHead c2 
+(Bind b) u2) (S O) O (drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) t4 a2 
+H29))))) H27))))))))))))) H15))))))))))))))))) (\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Flat Appl) u t)) 
+\to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 
+t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 
+a2)))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T 
+(THead (Bind Abbr) u1 t3) (THead (Flat Appl) u t))).(let H13 \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) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) u2 w) a2) 
+H13))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 
+(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0: T).(\lambda 
+(H10: (eq T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Flat Appl) u 
+t))).(let H11 \def (eq_ind T (THead (Bind b) u0 (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) u t) H10) in (False_ind 
+(arity g c2 t4 a2) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda 
+(_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity 
+g c2 t4 a2)))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat Cast) u0 t3) 
+(THead (Flat Appl) u t))).(let H10 \def (eq_ind T (THead (Flat Cast) u0 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) u t) 
+H9) in (False_ind (arity g c2 t4 a2) H10)))))))) y t2 H6))) H5)))))))))))))) 
+(\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c u 
+(asucc g a0))).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall 
+(t2: T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (t: 
+T).(\lambda (_: (arity g c t a0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c 
+c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a0))))))).(\lambda 
+(c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H5: (pr0 
+(THead (Flat Cast) u t) t2)).(insert_eq T (THead (Flat Cast) u t) (\lambda 
+(t0: T).(pr0 t0 t2)) (arity g c2 t2 a0) (\lambda (y: T).(\lambda (H6: (pr0 y 
+t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq T t0 (THead (Flat Cast) 
+u t)) \to (arity g c2 t3 a0)))) (\lambda (t0: T).(\lambda (H7: (eq T t0 
+(THead (Flat Cast) u t))).(let H8 \def (f_equal T T (\lambda (e: T).e) t0 
+(THead (Flat Cast) u t) H7) in (eq_ind_r T (THead (Flat Cast) u t) (\lambda 
+(t3: T).(arity g c2 t3 a0)) (arity_cast g c2 u a0 (H1 c2 H4 u (pr0_refl u)) t 
+(H3 c2 H4 t (pr0_refl t))) t0 H8)))) (\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1 (THead (Flat Cast) u 
+t)) \to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H9: 
+(pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Flat Cast) u t)) \to (arity g 
+c2 t4 a0)))).(\lambda (k: K).(\lambda (H11: (eq T (THead k u1 t3) (THead 
+(Flat Cast) u t))).(let H12 \def (f_equal T K (\lambda (e: T).(match e in T 
+return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat 
+Cast) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e in T 
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
+\Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) (THead 
+(Flat Cast) u t) H11) 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 _ _ t0) \Rightarrow t0])) (THead k u1 t3) (THead 
+(Flat Cast) u t) H11) in (\lambda (H15: (eq T u1 u)).(\lambda (H16: (eq K k 
+(Flat Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(arity g c2 (THead k0 
+u2 t4) a0)) (let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead 
+(Flat Cast) u t)) \to (arity g c2 t4 a0))) H10 t H14) in (let H18 \def 
+(eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind 
+T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Cast) u t)) \to (arity g c2 u2 
+a0))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) 
+H7 u H15) in (arity_cast g c2 u2 a0 (H1 c2 H4 u2 H20) t4 (H3 c2 H4 t4 
+H18)))))) k H16)))) H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1: 
+T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead 
+(Flat Cast) u t)) \to (arity g c2 v2 a0)))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) 
+\to (arity g c2 t4 a0)))).(\lambda (H11: (eq T (THead (Flat Appl) v1 (THead 
+(Bind Abst) u0 t3)) (THead (Flat Cast) u t))).(let H12 \def (eq_ind T (THead 
+(Flat Appl) v1 (THead (Bind Abst) u0 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 t) H11) in (False_ind (arity 
+g c2 (THead (Bind Abbr) v2 t4) a0) H12)))))))))))) (\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) u t)) \to (arity g c2 v2 
+a0)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(_: (((eq T u1 (THead (Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 
+(THead (Flat Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (H14: (eq T 
+(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Cast) u t))).(let 
+H15 \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 t) 
+H14) in (False_ind (arity g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) a0) H15))))))))))))))))) (\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Flat Cast) u t)) 
+\to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 
+t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) \to (arity g c2 t4 
+a0)))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T 
+(THead (Bind Abbr) u1 t3) (THead (Flat Cast) u t))).(let H13 \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 t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) u2 w) a0) 
+H13))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 
+(THead (Flat Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (u0: T).(\lambda 
+(H10: (eq T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Flat Cast) u 
+t))).(let H11 \def (eq_ind T (THead (Bind b) u0 (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) u t) H10) in (False_ind 
+(arity g c2 t4 a0) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda 
+(H7: (pr0 t3 t4)).(\lambda (H8: (((eq T t3 (THead (Flat Cast) u t)) \to 
+(arity g c2 t4 a0)))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat Cast) 
+u0 t3) (THead (Flat Cast) u t))).(let H10 \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 (Flat 
+Cast) u0 t3) (THead (Flat Cast) u t) H9) in ((let H11 \def (f_equal T T 
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) 
+(THead (Flat Cast) u0 t3) (THead (Flat Cast) u t) H9) in (\lambda (_: (eq T 
+u0 u)).(let H13 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Flat 
+Cast) u t)) \to (arity g c2 t4 a0))) H8 t H11) in (let H14 \def (eq_ind T t3 
+(\lambda (t0: T).(pr0 t0 t4)) H7 t H11) in (H3 c2 H4 t4 H14))))) H10)))))))) 
+y t2 H6))) H5))))))))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (a1: 
+A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c 
+c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a1))))))).(\lambda 
+(a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (wcpr0 c 
+c2)).(\lambda (t2: T).(\lambda (H4: (pr0 t t2)).(arity_repl g c2 t2 a1 (H1 c2 
+H3 t2 H4) a2 H2)))))))))))) c1 t1 a H))))).
+
+theorem arity_sred_wcpr0_pr1:
+ \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall 
+(c1: C).(\forall (a: A).((arity g c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 
+c2) \to (arity g c2 t2 a)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda 
+(t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c1: C).(\forall (a: 
+A).((arity g c1 t a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t0 
+a))))))))) (\lambda (t: T).(\lambda (g: G).(\lambda (c1: C).(\lambda (a: 
+A).(\lambda (H0: (arity g c1 t a)).(\lambda (c2: C).(\lambda (H1: (wcpr0 c1 
+c2)).(arity_sred_wcpr0_pr0 g c1 t a H0 c2 H1 t (pr0_refl t))))))))) (\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5: T).(\lambda 
+(_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c1: C).(\forall (a: 
+A).((arity g c1 t3 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t5 
+a))))))))).(\lambda (g: G).(\lambda (c1: C).(\lambda (a: A).(\lambda (H3: 
+(arity g c1 t4 a)).(\lambda (c2: C).(\lambda (H4: (wcpr0 c1 c2)).(H2 g c2 a 
+(arity_sred_wcpr0_pr0 g c1 t4 a H3 c2 H4 t3 H0) c2 (wcpr0_refl 
+c2)))))))))))))) t1 t2 H))).
+
+theorem arity_sred_pr2:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
+(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a)))))))
+\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 (g: 
+G).(\forall (a: A).((arity g c0 t a) \to (arity g c0 t0 a))))))) (\lambda 
+(c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda 
+(g: G).(\lambda (a: A).(\lambda (H1: (arity g c0 t3 a)).(arity_sred_wcpr0_pr0 
+g c0 t3 a H1 c0 (wcpr0_refl c0) 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 (g: 
+G).(\lambda (a: A).(\lambda (H3: (arity g c0 t3 a)).(arity_subst0 g c0 t4 a 
+(arity_sred_wcpr0_pr0 g c0 t3 a H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t 
+H2)))))))))))))) c t1 t2 H)))).
+
+theorem arity_sred_pr3:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
+(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall (a: 
+A).((arity g c t a) \to (arity g c t0 a)))))) (\lambda (t: T).(\lambda (g: 
+G).(\lambda (a: A).(\lambda (H0: (arity g c t a)).H0)))) (\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda 
+(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (a: A).((arity g c 
+t3 a) \to (arity g c t5 a)))))).(\lambda (g: G).(\lambda (a: A).(\lambda (H3: 
+(arity g c t4 a)).(H2 g a (arity_sred_pr2 c t4 t3 H0 g a H3))))))))))) t1 t2 
+H)))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ceqc/defs.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ceqc/defs.ma
new file mode 100644
index 000000000..91ae5de91
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ceqc/defs.ma
@@ -0,0 +1,35 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ceqc/defs".
+
+include "sc3/defs.ma".
+
+inductive csubc (g:G): C \to (C \to Prop) \def
+| csubc_sort: \forall (n: nat).(csubc g (CSort n) (CSort n))
+| csubc_head: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall 
+(k: K).(\forall (v: T).(csubc g (CHead c1 k v) (CHead c2 k v))))))
+| csubc_abst: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall 
+(v: T).(\forall (a: A).((sc3 g (asucc g a) c1 v) \to (\forall (w: T).((sc3 g 
+a c2 w) \to (csubc g (CHead c1 (Bind Abst) v) (CHead c2 (Bind Abbr) 
+w))))))))).
+
+definition ceqc:
+ G \to (C \to (C \to Prop))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(or (csubc g c1 c2) (csubc 
+g c2 c1)))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ceqc/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ceqc/props.ma
new file mode 100644
index 000000000..7ed7c50d2
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ceqc/props.ma
@@ -0,0 +1,563 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ceqc/props".
+
+include "ceqc/defs.ma".
+
+include "sc3/props.ma".
+
+theorem csubc_refl:
+ \forall (g: G).(\forall (c: C).(csubc g c c))
+\def
+ \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubc g c0 c0)) 
+(\lambda (n: nat).(csubc_sort g n)) (\lambda (c0: C).(\lambda (H: (csubc g c0 
+c0)).(\lambda (k: K).(\lambda (t: T).(csubc_head g c0 c0 H k t))))) c)).
+
+theorem ceqc_sym:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((ceqc g c1 c2) \to (ceqc g 
+c2 c1))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (ceqc g c1 
+c2)).(let H0 \def H in (or_ind (csubc g c1 c2) (csubc g c2 c1) (ceqc g c2 c1) 
+(\lambda (H1: (csubc g c1 c2)).(or_intror (csubc g c2 c1) (csubc g c1 c2) 
+H1)) (\lambda (H1: (csubc g c2 c1)).(or_introl (csubc g c2 c1) (csubc g c1 
+c2) H1)) H0))))).
+
+theorem drop_csubc_trans:
+ \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall 
+(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C 
+(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))))))))
+\def
+ \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: 
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: 
+C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda 
+(c1: C).(csubc g c c1)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda 
+(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda 
+(e1: C).(\lambda (H0: (csubc g e2 e1)).(and3_ind (eq C e2 (CSort n)) (eq nat 
+h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: 
+C).(csubc g (CSort n) c1))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: 
+(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: 
+nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g 
+(CSort n) c1)))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: 
+C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)))) (let H4 \def 
+(eq_ind C e2 (\lambda (c: C).(csubc g c e1)) H0 (CSort n) H1) in (ex_intro2 C 
+(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)) 
+e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) 
+(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: 
+nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C 
+(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c 
+c1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: 
+nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) 
+e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h 
+n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) (\lambda (h: 
+nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall 
+(e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) 
+(\lambda (c1: C).(csubc g (CHead c k t) c1))))))) (\lambda (H0: (drop O O 
+(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2 
+\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g c0 e1)) H1 (CHead c k t) 
+(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O 
+O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)) e1 (drop_refl e1) 
+H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to 
+(\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 
+e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))).(\lambda (H1: (drop 
+(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 
+e1)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in 
+(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) 
+(\lambda (c1: C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 
+e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))) (\lambda (x: C).(\lambda 
+(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g c x)).(ex_intro2 C 
+(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k 
+t) c1)) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g c x H5 k t))))) 
+H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n 
+(CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda 
+(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) 
+c1))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) 
+e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 e1)).(ex3_2_ind C T (\lambda 
+(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: 
+T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k 
+n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: 
+C).(csubc g (CHead c k t) c1))) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
+(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) 
+x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda 
+(c0: C).(csubc g c0 e1)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 
+(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to 
+(\forall (e3: C).((csubc g c0 e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 
+e3)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) H0 (CHead x0 k x1) 
+H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 
+n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g (CHead x0 k 
+x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: 
+C).(csubc g (CHead c k t0) c1)))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r 
+T (lift h (r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) 
+c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t0) c1)))) (let H9 \def (match 
+H6 in csubc return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csubc ? c0 
+c1)).((eq C c0 (CHead x0 k x1)) \to ((eq C c1 e1) \to (ex2 C (\lambda (c3: 
+C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) 
+x1)) c3)))))))) with [(csubc_sort n0) \Rightarrow (\lambda (H9: (eq C (CSort 
+n0) (CHead x0 k x1))).(\lambda (H10: (eq C (CSort n0) e1)).((let H11 \def 
+(eq_ind C (CSort n0) (\lambda (e: C).(match e in C return (\lambda (_: 
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
+False])) I (CHead x0 k x1) H9) in (False_ind ((eq C (CSort n0) e1) \to (ex2 C 
+(\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g (CHead c k 
+(lift h (r k n) x1)) c1)))) H11)) H10))) | (csubc_head c1 c0 H9 k0 v) 
+\Rightarrow (\lambda (H10: (eq C (CHead c1 k0 v) (CHead x0 k x1))).(\lambda 
+(H11: (eq C (CHead c0 k0 v) e1)).((let H12 \def (f_equal C T (\lambda (e: 
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | 
+(CHead _ _ t0) \Rightarrow t0])) (CHead c1 k0 v) (CHead x0 k x1) H10) in 
+((let H13 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: 
+C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) 
+(CHead c1 k0 v) (CHead x0 k x1) H10) in ((let H14 \def (f_equal C C (\lambda 
+(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 
+| (CHead c3 _ _) \Rightarrow c3])) (CHead c1 k0 v) (CHead x0 k x1) H10) in 
+(eq_ind C x0 (\lambda (c3: C).((eq K k0 k) \to ((eq T v x1) \to ((eq C (CHead 
+c0 k0 v) e1) \to ((csubc g c3 c0) \to (ex2 C (\lambda (c4: C).(drop h (S n) 
+c4 e1)) (\lambda (c4: C).(csubc g (CHead c k (lift h (r k n) x1)) c4)))))))) 
+(\lambda (H15: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T v x1) \to 
+((eq C (CHead c0 k1 v) e1) \to ((csubc g x0 c0) \to (ex2 C (\lambda (c3: 
+C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) 
+x1)) c3))))))) (\lambda (H16: (eq T v x1)).(eq_ind T x1 (\lambda (t0: T).((eq 
+C (CHead c0 k t0) e1) \to ((csubc g x0 c0) \to (ex2 C (\lambda (c3: C).(drop 
+h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) 
+c3)))))) (\lambda (H17: (eq C (CHead c0 k x1) e1)).(eq_ind C (CHead c0 k x1) 
+(\lambda (c3: C).((csubc g x0 c0) \to (ex2 C (\lambda (c4: C).(drop h (S n) 
+c4 c3)) (\lambda (c4: C).(csubc g (CHead c k (lift h (r k n) x1)) c4))))) 
+(\lambda (H18: (csubc g x0 c0)).(let H_x \def (H x0 (r k n) h H5 c0 H18) in 
+(let H19 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r k n) c3 c0)) 
+(\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 
+(CHead c0 k x1))) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) 
+c3))) (\lambda (x: C).(\lambda (H20: (drop h (r k n) x c0)).(\lambda (H21: 
+(csubc g c x)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c0 k 
+x1))) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3)) (CHead x 
+k (lift h (r k n) x1)) (drop_skip k h n x c0 H20 x1) (csubc_head g c x H21 k 
+(lift h (r k n) x1)))))) H19)))) e1 H17)) v (sym_eq T v x1 H16))) k0 (sym_eq 
+K k0 k H15))) c1 (sym_eq C c1 x0 H14))) H13)) H12)) H11 H9))) | (csubc_abst 
+c1 c0 H9 v a H10 w H11) \Rightarrow (\lambda (H12: (eq C (CHead c1 (Bind 
+Abst) v) (CHead x0 k x1))).(\lambda (H13: (eq C (CHead c0 (Bind Abbr) w) 
+e1)).((let H14 \def (f_equal C T (\lambda (e: C).(match e in C return 
+(\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t0) \Rightarrow 
+t0])) (CHead c1 (Bind Abst) v) (CHead x0 k x1) H12) in ((let H15 \def 
+(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with 
+[(CSort _) \Rightarrow (Bind Abst) | (CHead _ k0 _) \Rightarrow k0])) (CHead 
+c1 (Bind Abst) v) (CHead x0 k x1) H12) in ((let H16 \def (f_equal C C 
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
+\Rightarrow c1 | (CHead c3 _ _) \Rightarrow c3])) (CHead c1 (Bind Abst) v) 
+(CHead x0 k x1) H12) in (eq_ind C x0 (\lambda (c3: C).((eq K (Bind Abst) k) 
+\to ((eq T v x1) \to ((eq C (CHead c0 (Bind Abbr) w) e1) \to ((csubc g c3 c0) 
+\to ((sc3 g (asucc g a) c3 v) \to ((sc3 g a c0 w) \to (ex2 C (\lambda (c4: 
+C).(drop h (S n) c4 e1)) (\lambda (c4: C).(csubc g (CHead c k (lift h (r k n) 
+x1)) c4)))))))))) (\lambda (H17: (eq K (Bind Abst) k)).(eq_ind K (Bind Abst) 
+(\lambda (k0: K).((eq T v x1) \to ((eq C (CHead c0 (Bind Abbr) w) e1) \to 
+((csubc g x0 c0) \to ((sc3 g (asucc g a) x0 v) \to ((sc3 g a c0 w) \to (ex2 C 
+(\lambda (c3: C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k0 
+(lift h (r k0 n) x1)) c3))))))))) (\lambda (H18: (eq T v x1)).(eq_ind T x1 
+(\lambda (t0: T).((eq C (CHead c0 (Bind Abbr) w) e1) \to ((csubc g x0 c0) \to 
+((sc3 g (asucc g a) x0 t0) \to ((sc3 g a c0 w) \to (ex2 C (\lambda (c3: 
+C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c (Bind Abst) (lift 
+h (r (Bind Abst) n) x1)) c3)))))))) (\lambda (H19: (eq C (CHead c0 (Bind 
+Abbr) w) e1)).(eq_ind C (CHead c0 (Bind Abbr) w) (\lambda (c3: C).((csubc g 
+x0 c0) \to ((sc3 g (asucc g a) x0 x1) \to ((sc3 g a c0 w) \to (ex2 C (\lambda 
+(c4: C).(drop h (S n) c4 c3)) (\lambda (c4: C).(csubc g (CHead c (Bind Abst) 
+(lift h (r (Bind Abst) n) x1)) c4))))))) (\lambda (H20: (csubc g x0 
+c0)).(\lambda (H21: (sc3 g (asucc g a) x0 x1)).(\lambda (H22: (sc3 g a c0 
+w)).(let H23 \def (eq_ind_r K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 
+n (CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: 
+C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c3: C).(drop h0 n c3 
+e3)) (\lambda (c3: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c3)))))))) 
+H8 (Bind Abst) H17) in (let H24 \def (eq_ind_r K k (\lambda (k0: K).(drop h 
+(r k0 n) c x0)) H5 (Bind Abst) H17) in (let H_x \def (H x0 (r (Bind Abst) n) 
+h H24 c0 H20) in (let H25 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r 
+(Bind Abst) n) c3 c0)) (\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (c3: 
+C).(drop h (S n) c3 (CHead c0 (Bind Abbr) w))) (\lambda (c3: C).(csubc g 
+(CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c3))) (\lambda (x: 
+C).(\lambda (H26: (drop h (r (Bind Abst) n) x c0)).(\lambda (H27: (csubc g c 
+x)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c0 (Bind Abbr) w))) 
+(\lambda (c3: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) 
+c3)) (CHead x (Bind Abbr) (lift h n w)) (drop_skip_bind h n x c0 H26 Abbr w) 
+(csubc_abst g c x H27 (lift h (r (Bind Abst) n) x1) a (sc3_lift g (asucc g a) 
+x0 x1 H21 c h (r (Bind Abst) n) H24) (lift h n w) (sc3_lift g a c0 w H22 x h 
+n H26)))))) H25)))))))) e1 H19)) v (sym_eq T v x1 H18))) k H17)) c1 (sym_eq C 
+c1 x0 H16))) H15)) H14)) H13 H9 H10 H11)))]) in (H9 (refl_equal C (CHead x0 k 
+x1)) (refl_equal C e1))) t H4))))))))) (drop_gen_skip_l c e2 t h n k 
+H1)))))))) d))))))) c2)).
+
+theorem csubc_drop_conf_rev:
+ \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall 
+(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C 
+(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))))))))
+\def
+ \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: 
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: 
+C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda 
+(c1: C).(csubc g c1 c)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda 
+(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda 
+(e1: C).(\lambda (H0: (csubc g e1 e2)).(and3_ind (eq C e2 (CSort n)) (eq nat 
+h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: 
+C).(csubc g c1 (CSort n)))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: 
+(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: 
+nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g c1 
+(CSort n))))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: 
+C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))))) (let H4 \def 
+(eq_ind C e2 (\lambda (c: C).(csubc g e1 c)) H0 (CSort n) H1) in (ex_intro2 C 
+(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))) 
+e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) 
+(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: 
+nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C 
+(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 
+c))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: 
+nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) 
+e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h 
+n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) (\lambda (h: 
+nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall 
+(e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) 
+(\lambda (c1: C).(csubc g c1 (CHead c k t)))))))) (\lambda (H0: (drop O O 
+(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2 
+\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g e1 c0)) H1 (CHead c k t) 
+(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O 
+O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))) e1 (drop_refl e1) 
+H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to 
+(\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 
+e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))).(\lambda (H1: (drop 
+(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 
+e2)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in 
+(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) 
+(\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 
+e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))) (\lambda (x: C).(\lambda 
+(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g x c)).(ex_intro2 C 
+(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c 
+k t))) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g x c H5 k t))))) 
+H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n 
+(CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda 
+(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k 
+t)))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) 
+e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 e2)).(ex3_2_ind C T (\lambda 
+(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: 
+T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k 
+n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: 
+C).(csubc g c1 (CHead c k t)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
+(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) 
+x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda 
+(c0: C).(csubc g e1 c0)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 
+(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to 
+(\forall (e3: C).((csubc g e3 c0) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 
+e3)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) H0 (CHead x0 k x1) 
+H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 
+n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g e3 (CHead x0 
+k x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc 
+g c1 (CHead c k t0))))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h 
+(r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) 
+(\lambda (c1: C).(csubc g c1 (CHead c k t0))))) (let H9 \def (match H6 in 
+csubc return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csubc ? c0 
+c1)).((eq C c0 e1) \to ((eq C c1 (CHead x0 k x1)) \to (ex2 C (\lambda (c3: 
+C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k 
+n) x1)))))))))) with [(csubc_sort n0) \Rightarrow (\lambda (H9: (eq C (CSort 
+n0) e1)).(\lambda (H10: (eq C (CSort n0) (CHead x0 k x1))).(eq_ind C (CSort 
+n0) (\lambda (c0: C).((eq C (CSort n0) (CHead x0 k x1)) \to (ex2 C (\lambda 
+(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h 
+(r k n) x1))))))) (\lambda (H11: (eq C (CSort n0) (CHead x0 k x1))).(let H12 
+\def (eq_ind C (CSort n0) (\lambda (e: C).(match e in C return (\lambda (_: 
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
+False])) I (CHead x0 k x1) H11) in (False_ind (ex2 C (\lambda (c1: C).(drop h 
+(S n) c1 (CSort n0))) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) 
+x1))))) H12))) e1 H9 H10))) | (csubc_head c1 c0 H9 k0 v) \Rightarrow (\lambda 
+(H10: (eq C (CHead c1 k0 v) e1)).(\lambda (H11: (eq C (CHead c0 k0 v) (CHead 
+x0 k x1))).(eq_ind C (CHead c1 k0 v) (\lambda (c3: C).((eq C (CHead c0 k0 v) 
+(CHead x0 k x1)) \to ((csubc g c1 c0) \to (ex2 C (\lambda (c4: C).(drop h (S 
+n) c4 c3)) (\lambda (c4: C).(csubc g c4 (CHead c k (lift h (r k n) x1)))))))) 
+(\lambda (H12: (eq C (CHead c0 k0 v) (CHead x0 k x1))).(let H13 \def (f_equal 
+C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
+\Rightarrow v | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k0 v) (CHead x0 k 
+x1) H12) in ((let H14 \def (f_equal C K (\lambda (e: C).(match e in C return 
+(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) 
+\Rightarrow k1])) (CHead c0 k0 v) (CHead x0 k x1) H12) in ((let H15 \def 
+(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with 
+[(CSort _) \Rightarrow c0 | (CHead c3 _ _) \Rightarrow c3])) (CHead c0 k0 v) 
+(CHead x0 k x1) H12) in (eq_ind C x0 (\lambda (c3: C).((eq K k0 k) \to ((eq T 
+v x1) \to ((csubc g c1 c3) \to (ex2 C (\lambda (c4: C).(drop h (S n) c4 
+(CHead c1 k0 v))) (\lambda (c4: C).(csubc g c4 (CHead c k (lift h (r k n) 
+x1))))))))) (\lambda (H16: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T v 
+x1) \to ((csubc g c1 x0) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead 
+c1 k1 v))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))))))) 
+(\lambda (H17: (eq T v x1)).(eq_ind T x1 (\lambda (t0: T).((csubc g c1 x0) 
+\to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k t0))) (\lambda (c3: 
+C).(csubc g c3 (CHead c k (lift h (r k n) x1))))))) (\lambda (H18: (csubc g 
+c1 x0)).(let H19 \def (eq_ind T v (\lambda (t0: T).(eq C (CHead c1 k0 t0) 
+e1)) H10 x1 H17) in (let H20 \def (eq_ind K k0 (\lambda (k1: K).(eq C (CHead 
+c1 k1 x1) e1)) H19 k H16) in (let H_x \def (H x0 (r k n) h H5 c1 H18) in (let 
+H21 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r k n) c3 c1)) (\lambda 
+(c3: C).(csubc g c3 c)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k 
+x1))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1))))) 
+(\lambda (x: C).(\lambda (H22: (drop h (r k n) x c1)).(\lambda (H23: (csubc g 
+x c)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k x1))) 
+(\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))) (CHead x k 
+(lift h (r k n) x1)) (drop_skip k h n x c1 H22 x1) (csubc_head g x c H23 k 
+(lift h (r k n) x1)))))) H21)))))) v (sym_eq T v x1 H17))) k0 (sym_eq K k0 k 
+H16))) c0 (sym_eq C c0 x0 H15))) H14)) H13))) e1 H10 H11 H9))) | (csubc_abst 
+c1 c0 H9 v a H10 w H11) \Rightarrow (\lambda (H12: (eq C (CHead c1 (Bind 
+Abst) v) e1)).(\lambda (H13: (eq C (CHead c0 (Bind Abbr) w) (CHead x0 k 
+x1))).(eq_ind C (CHead c1 (Bind Abst) v) (\lambda (c3: C).((eq C (CHead c0 
+(Bind Abbr) w) (CHead x0 k x1)) \to ((csubc g c1 c0) \to ((sc3 g (asucc g a) 
+c1 v) \to ((sc3 g a c0 w) \to (ex2 C (\lambda (c4: C).(drop h (S n) c4 c3)) 
+(\lambda (c4: C).(csubc g c4 (CHead c k (lift h (r k n) x1)))))))))) (\lambda 
+(H14: (eq C (CHead c0 (Bind Abbr) w) (CHead x0 k x1))).(let H15 \def (f_equal 
+C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
+\Rightarrow w | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 (Bind Abbr) w) 
+(CHead x0 k x1) H14) in ((let H16 \def (f_equal C K (\lambda (e: C).(match e 
+in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow (Bind Abbr) | 
+(CHead _ k0 _) \Rightarrow k0])) (CHead c0 (Bind Abbr) w) (CHead x0 k x1) 
+H14) in ((let H17 \def (f_equal C C (\lambda (e: C).(match e in C return 
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c3 _ _) 
+\Rightarrow c3])) (CHead c0 (Bind Abbr) w) (CHead x0 k x1) H14) in (eq_ind C 
+x0 (\lambda (c3: C).((eq K (Bind Abbr) k) \to ((eq T w x1) \to ((csubc g c1 
+c3) \to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a c3 w) \to (ex2 C (\lambda 
+(c4: C).(drop h (S n) c4 (CHead c1 (Bind Abst) v))) (\lambda (c4: C).(csubc g 
+c4 (CHead c k (lift h (r k n) x1))))))))))) (\lambda (H18: (eq K (Bind Abbr) 
+k)).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T w x1) \to ((csubc g c1 x0) 
+\to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a x0 w) \to (ex2 C (\lambda (c3: 
+C).(drop h (S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: C).(csubc g c3 
+(CHead c k0 (lift h (r k0 n) x1)))))))))) (\lambda (H19: (eq T w x1)).(eq_ind 
+T x1 (\lambda (t0: T).((csubc g c1 x0) \to ((sc3 g (asucc g a) c1 v) \to 
+((sc3 g a x0 t0) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 (Bind 
+Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c (Bind Abbr) (lift h (r (Bind 
+Abbr) n) x1))))))))) (\lambda (H20: (csubc g c1 x0)).(\lambda (H21: (sc3 g 
+(asucc g a) c1 v)).(\lambda (H22: (sc3 g a x0 x1)).(let H23 \def (eq_ind_r K 
+k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c k0 (lift h (r k0 
+n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3 (CHead x0 k0 x1)) 
+\to (ex2 C (\lambda (c3: C).(drop h0 n c3 e3)) (\lambda (c3: C).(csubc g c3 
+(CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind Abbr) H18) in (let H24 
+\def (eq_ind_r K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5 (Bind Abbr) 
+H18) in (let H_x \def (H x0 (r (Bind Abbr) n) h H24 c1 H20) in (let H25 \def 
+H_x in (ex2_ind C (\lambda (c3: C).(drop h (r (Bind Abbr) n) c3 c1)) (\lambda 
+(c3: C).(csubc g c3 c)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 
+(Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c (Bind Abbr) (lift h (r 
+(Bind Abbr) n) x1))))) (\lambda (x: C).(\lambda (H26: (drop h (r (Bind Abbr) 
+n) x c1)).(\lambda (H27: (csubc g x c)).(ex_intro2 C (\lambda (c3: C).(drop h 
+(S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c 
+(Bind Abbr) (lift h (r (Bind Abbr) n) x1)))) (CHead x (Bind Abst) (lift h n 
+v)) (drop_skip_bind h n x c1 H26 Abst v) (csubc_abst g x c H27 (lift h n v) a 
+(sc3_lift g (asucc g a) c1 v H21 x h n H26) (lift h (r (Bind Abbr) n) x1) 
+(sc3_lift g a x0 x1 H22 c h (r (Bind Abbr) n) H24)))))) H25)))))))) w (sym_eq 
+T w x1 H19))) k H18)) c0 (sym_eq C c0 x0 H17))) H16)) H15))) e1 H12 H13 H9 
+H10 H11)))]) in (H9 (refl_equal C e1) (refl_equal C (CHead x0 k x1)))) t 
+H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
+
+theorem drop1_csubc_trans:
+ \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: 
+C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C 
+(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))
+\def
+ \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
+(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 
+e1) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 
+c1))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 
+e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e2 e1)).(let H1 \def (match H in 
+drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda 
+(_: (drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 e2) \to 
+(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c2 
+c1)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil 
+PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c e2)).(eq_ind C c2 
+(\lambda (c0: C).((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1 
+e1)) (\lambda (c1: C).(csubc g c2 c1))))) (\lambda (H4: (eq C c2 e2)).(eq_ind 
+C e2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda 
+(c1: C).(csubc g c0 c1)))) (let H5 \def (eq_ind_r C e2 (\lambda (c0: 
+C).(csubc g c0 e1)) H0 c2 H4) in (eq_ind C c2 (\lambda (c0: C).(ex2 C 
+(\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c0 c1)))) 
+(ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g 
+c2 c1)) e1 (drop1_nil e1) H5) e2 H4)) c2 (sym_eq C c2 e2 H4))) c (sym_eq C c 
+c2 H2) H3)))) | (drop1_cons c1 c0 h d H1 c3 hds0 H2) \Rightarrow (\lambda 
+(H3: (eq PList (PCons h d hds0) PNil)).(\lambda (H4: (eq C c1 c2)).(\lambda 
+(H5: (eq C c3 e2)).((let H6 \def (eq_ind PList (PCons h d hds0) (\lambda (e: 
+PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil 
+\Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in 
+(False_ind ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop h d c1 c0) \to ((drop1 
+hds0 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 PNil c4 e1)) (\lambda (c4: 
+C).(csubc g c2 c4))))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList 
+PNil) (refl_equal C c2) (refl_equal C e2)))))))) (\lambda (n: nat).(\lambda 
+(n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: C).(\forall (e2: 
+C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda 
+(c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))).(\lambda 
+(c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n n0 p) c2 e2)).(\lambda 
+(e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2 \def (match H0 in drop1 return 
+(\lambda (p0: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p0 
+c c0)).((eq PList p0 (PCons n n0 p)) \to ((eq C c c2) \to ((eq C c0 e2) \to 
+(ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc 
+g c2 c1)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList 
+PNil (PCons n n0 p))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c 
+e2)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e in PList 
+return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) 
+\Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c2) \to ((eq 
+C c e2) \to (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda 
+(c1: C).(csubc g c2 c1))))) H5)) H3 H4)))) | (drop1_cons c1 c0 h d H2 c3 hds0 
+H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds0) (PCons n n0 
+p))).(\lambda (H5: (eq C c1 c2)).(\lambda (H6: (eq C c3 e2)).((let H7 \def 
+(f_equal PList PList (\lambda (e: PList).(match e in PList return (\lambda 
+(_: PList).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p0) \Rightarrow 
+p0])) (PCons h d hds0) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat 
+(\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with 
+[PNil \Rightarrow d | (PCons _ n1 _) \Rightarrow n1])) (PCons h d hds0) 
+(PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e: 
+PList).(match e in PList return (\lambda (_: PList).nat) with [PNil 
+\Rightarrow h | (PCons n1 _ _) \Rightarrow n1])) (PCons h d hds0) (PCons n n0 
+p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds0 
+p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n1 d c1 c0) \to ((drop1 
+hds0 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) 
+(\lambda (c4: C).(csubc g c2 c4)))))))))) (\lambda (H10: (eq nat d 
+n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds0 p) \to ((eq C c1 c2) 
+\to ((eq C c3 e2) \to ((drop n n1 c1 c0) \to ((drop1 hds0 c0 c3) \to (ex2 C 
+(\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c2 
+c4))))))))) (\lambda (H11: (eq PList hds0 p)).(eq_ind PList p (\lambda (p0: 
+PList).((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n n0 c1 c0) \to ((drop1 p0 
+c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda 
+(c4: C).(csubc g c2 c4)))))))) (\lambda (H12: (eq C c1 c2)).(eq_ind C c2 
+(\lambda (c: C).((eq C c3 e2) \to ((drop n n0 c c0) \to ((drop1 p c0 c3) \to 
+(ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc 
+g c2 c4))))))) (\lambda (H13: (eq C c3 e2)).(eq_ind C e2 (\lambda (c: 
+C).((drop n n0 c2 c0) \to ((drop1 p c0 c) \to (ex2 C (\lambda (c4: C).(drop1 
+(PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c2 c4)))))) (\lambda (H14: 
+(drop n n0 c2 c0)).(\lambda (H15: (drop1 p c0 e2)).(let H_x \def (H c0 e2 H15 
+e1 H1) in (let H16 \def H_x in (ex2_ind C (\lambda (c4: C).(drop1 p c4 e1)) 
+(\lambda (c4: C).(csubc g c0 c4)) (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 
+p) c4 e1)) (\lambda (c4: C).(csubc g c2 c4))) (\lambda (x: C).(\lambda (H17: 
+(drop1 p x e1)).(\lambda (H18: (csubc g c0 x)).(let H_x0 \def 
+(drop_csubc_trans g c2 c0 n0 n H14 x H18) in (let H19 \def H_x0 in (ex2_ind C 
+(\lambda (c4: C).(drop n n0 c4 x)) (\lambda (c4: C).(csubc g c2 c4)) (ex2 C 
+(\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c2 
+c4))) (\lambda (x0: C).(\lambda (H20: (drop n n0 x0 x)).(\lambda (H21: (csubc 
+g c2 x0)).(ex_intro2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) 
+(\lambda (c4: C).(csubc g c2 c4)) x0 (drop1_cons x0 x n n0 H20 e1 p H17) 
+H21)))) H19)))))) H16))))) c3 (sym_eq C c3 e2 H13))) c1 (sym_eq C c1 c2 
+H12))) hds0 (sym_eq PList hds0 p H11))) d (sym_eq nat d n0 H10))) h (sym_eq 
+nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n 
+n0 p)) (refl_equal C c2) (refl_equal C e2)))))))))))) hds)).
+
+theorem csubc_drop1_conf_rev:
+ \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: 
+C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C 
+(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))
+\def
+ \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
+(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 
+e2) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 
+c2))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 
+e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e1 e2)).(let H1 \def (match H in 
+drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda 
+(_: (drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 e2) \to 
+(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 
+c2)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil 
+PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c e2)).(eq_ind C c2 
+(\lambda (c0: C).((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1 
+e1)) (\lambda (c1: C).(csubc g c1 c2))))) (\lambda (H4: (eq C c2 e2)).(eq_ind 
+C e2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda 
+(c1: C).(csubc g c1 c0)))) (let H5 \def (eq_ind_r C e2 (\lambda (c0: 
+C).(csubc g e1 c0)) H0 c2 H4) in (eq_ind C c2 (\lambda (c0: C).(ex2 C 
+(\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 c0)))) 
+(ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g 
+c1 c2)) e1 (drop1_nil e1) H5) e2 H4)) c2 (sym_eq C c2 e2 H4))) c (sym_eq C c 
+c2 H2) H3)))) | (drop1_cons c1 c0 h d H1 c3 hds0 H2) \Rightarrow (\lambda 
+(H3: (eq PList (PCons h d hds0) PNil)).(\lambda (H4: (eq C c1 c2)).(\lambda 
+(H5: (eq C c3 e2)).((let H6 \def (eq_ind PList (PCons h d hds0) (\lambda (e: 
+PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil 
+\Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in 
+(False_ind ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop h d c1 c0) \to ((drop1 
+hds0 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 PNil c4 e1)) (\lambda (c4: 
+C).(csubc g c4 c2))))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList 
+PNil) (refl_equal C c2) (refl_equal C e2)))))))) (\lambda (n: nat).(\lambda 
+(n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: C).(\forall (e2: 
+C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda 
+(c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))).(\lambda 
+(c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n n0 p) c2 e2)).(\lambda 
+(e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2 \def (match H0 in drop1 return 
+(\lambda (p0: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p0 
+c c0)).((eq PList p0 (PCons n n0 p)) \to ((eq C c c2) \to ((eq C c0 e2) \to 
+(ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc 
+g c1 c2)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList 
+PNil (PCons n n0 p))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c 
+e2)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e in PList 
+return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) 
+\Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c2) \to ((eq 
+C c e2) \to (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda 
+(c1: C).(csubc g c1 c2))))) H5)) H3 H4)))) | (drop1_cons c1 c0 h d H2 c3 hds0 
+H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds0) (PCons n n0 
+p))).(\lambda (H5: (eq C c1 c2)).(\lambda (H6: (eq C c3 e2)).((let H7 \def 
+(f_equal PList PList (\lambda (e: PList).(match e in PList return (\lambda 
+(_: PList).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p0) \Rightarrow 
+p0])) (PCons h d hds0) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat 
+(\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with 
+[PNil \Rightarrow d | (PCons _ n1 _) \Rightarrow n1])) (PCons h d hds0) 
+(PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e: 
+PList).(match e in PList return (\lambda (_: PList).nat) with [PNil 
+\Rightarrow h | (PCons n1 _ _) \Rightarrow n1])) (PCons h d hds0) (PCons n n0 
+p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds0 
+p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n1 d c1 c0) \to ((drop1 
+hds0 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) 
+(\lambda (c4: C).(csubc g c4 c2)))))))))) (\lambda (H10: (eq nat d 
+n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds0 p) \to ((eq C c1 c2) 
+\to ((eq C c3 e2) \to ((drop n n1 c1 c0) \to ((drop1 hds0 c0 c3) \to (ex2 C 
+(\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c4 
+c2))))))))) (\lambda (H11: (eq PList hds0 p)).(eq_ind PList p (\lambda (p0: 
+PList).((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n n0 c1 c0) \to ((drop1 p0 
+c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda 
+(c4: C).(csubc g c4 c2)))))))) (\lambda (H12: (eq C c1 c2)).(eq_ind C c2 
+(\lambda (c: C).((eq C c3 e2) \to ((drop n n0 c c0) \to ((drop1 p c0 c3) \to 
+(ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc 
+g c4 c2))))))) (\lambda (H13: (eq C c3 e2)).(eq_ind C e2 (\lambda (c: 
+C).((drop n n0 c2 c0) \to ((drop1 p c0 c) \to (ex2 C (\lambda (c4: C).(drop1 
+(PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c4 c2)))))) (\lambda (H14: 
+(drop n n0 c2 c0)).(\lambda (H15: (drop1 p c0 e2)).(let H_x \def (H c0 e2 H15 
+e1 H1) in (let H16 \def H_x in (ex2_ind C (\lambda (c4: C).(drop1 p c4 e1)) 
+(\lambda (c4: C).(csubc g c4 c0)) (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 
+p) c4 e1)) (\lambda (c4: C).(csubc g c4 c2))) (\lambda (x: C).(\lambda (H17: 
+(drop1 p x e1)).(\lambda (H18: (csubc g x c0)).(let H_x0 \def 
+(csubc_drop_conf_rev g c2 c0 n0 n H14 x H18) in (let H19 \def H_x0 in 
+(ex2_ind C (\lambda (c4: C).(drop n n0 c4 x)) (\lambda (c4: C).(csubc g c4 
+c2)) (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: 
+C).(csubc g c4 c2))) (\lambda (x0: C).(\lambda (H20: (drop n n0 x0 
+x)).(\lambda (H21: (csubc g x0 c2)).(ex_intro2 C (\lambda (c4: C).(drop1 
+(PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c4 c2)) x0 (drop1_cons x0 x 
+n n0 H20 e1 p H17) H21)))) H19)))))) H16))))) c3 (sym_eq C c3 e2 H13))) c1 
+(sym_eq C c1 c2 H12))) hds0 (sym_eq PList hds0 p H11))) d (sym_eq nat d n0 
+H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal 
+PList (PCons n n0 p)) (refl_equal C c2) (refl_equal C e2)))))))))))) hds)).
+
+theorem drop1_ceqc_trans:
+ \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: 
+C).((drop1 hds c2 e2) \to (\forall (e1: C).((ceqc g e2 e1) \to (ex2 C 
+(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(ceqc g c2 c1)))))))))
+\def
+ \lambda (g: G).(\lambda (hds: PList).(\lambda (c2: C).(\lambda (e2: 
+C).(\lambda (H: (drop1 hds c2 e2)).(\lambda (e1: C).(\lambda (H0: (ceqc g e2 
+e1)).(let H1 \def H0 in (or_ind (csubc g e2 e1) (csubc g e1 e2) (ex2 C 
+(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(ceqc g c2 c1))) 
+(\lambda (H2: (csubc g e2 e1)).(let H_x \def (drop1_csubc_trans g hds c2 e2 H 
+e1 H2) in (let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop1 hds c1 e1)) 
+(\lambda (c1: C).(csubc g c2 c1)) (ex2 C (\lambda (c1: C).(drop1 hds c1 e1)) 
+(\lambda (c1: C).(ceqc g c2 c1))) (\lambda (x: C).(\lambda (H4: (drop1 hds x 
+e1)).(\lambda (H5: (csubc g c2 x)).(ex_intro2 C (\lambda (c1: C).(drop1 hds 
+c1 e1)) (\lambda (c1: C).(ceqc g c2 c1)) x H4 (or_introl (csubc g c2 x) 
+(csubc g x c2) H5))))) H3)))) (\lambda (H2: (csubc g e1 e2)).(let H_x \def 
+(csubc_drop1_conf_rev g hds c2 e2 H e1 H2) in (let H3 \def H_x in (ex2_ind C 
+(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c1 c2)) (ex2 C 
+(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(ceqc g c2 c1))) 
+(\lambda (x: C).(\lambda (H4: (drop1 hds x e1)).(\lambda (H5: (csubc g x 
+c2)).(ex_intro2 C (\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(ceqc 
+g c2 c1)) x H4 (or_intror (csubc g c2 x) (csubc g x c2) H5))))) H3)))) 
+H1)))))))).
+
+axiom sc3_ceqc_trans:
+ \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c1: 
+C).(\forall (t: T).((sc3 g a c1 (THeads (Flat Appl) vs t)) \to (\forall (c2: 
+C).((ceqc g c2 c1) \to (sc3 g a c2 (THeads (Flat Appl) vs t)))))))))
+.
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/arity.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/arity.ma
new file mode 100644
index 000000000..acd564afb
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/arity.ma
@@ -0,0 +1,146 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/arity".
+
+include "csuba/getl.ma".
+
+include "csuba/props.ma".
+
+include "arity/props.ma".
+
+include "T/props.ma".
+
+theorem csuba_arity:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 
+t a) \to (\forall (c2: C).((csuba g c1 c2) \to (arity g c2 t a)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
+(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: 
+A).(\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0)))))) (\lambda (c: 
+C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (csuba g c 
+c2)).(arity_sort g c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) 
+u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall 
+(c2: C).((csuba g d c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda 
+(H3: (csuba g c c2)).(let H4 \def (csuba_getl_abbr g c d u i H0 c2 H3) in 
+(ex2_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda 
+(d2: C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda 
+(H5: (getl i c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (csuba g d 
+x)).(arity_abbr g c2 x u i H5 a0 (H2 x H6))))) H4)))))))))))) (\lambda (c: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c 
+(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc 
+g a0))).(\lambda (H2: ((\forall (c2: C).((csuba g d c2) \to (arity g c2 u 
+(asucc g a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c c2)).(let H4 \def 
+(csuba_getl_abst g c d u i H0 c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(getl 
+i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2))) (ex4_3 C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc 
+g a1))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 
+a1))))) (arity g c2 (TLRef i) a0) (\lambda (H5: (ex2 C (\lambda (d2: C).(getl 
+i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2)))).(ex2_ind C 
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H6: 
+(getl i c2 (CHead x (Bind Abst) u))).(\lambda (H7: (csuba g d x)).(arity_abst 
+g c2 x u i H6 a0 (H2 x H7))))) H5)) (\lambda (H5: (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d d2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc g a1))))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 
+a1)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a1: A).(arity g d u (asucc g a1))))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a1: A).(arity g d2 u2 a1)))) (arity g c2 (TLRef i) a0) (\lambda 
+(x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H6: (getl i c2 (CHead x0 
+(Bind Abbr) x1))).(\lambda (_: (csuba g d x0)).(\lambda (H8: (arity g d u 
+(asucc g x2))).(\lambda (H9: (arity g x0 x1 x2)).(arity_repl g c2 (TLRef i) 
+x2 (arity_abbr g c2 x0 x1 i H6 x2 H9) a0 (asucc_inj g x2 a0 (arity_mono g d u 
+(asucc g x2) H8 (asucc g a0) H1)))))))))) H5)) H4)))))))))))) (\lambda (b: 
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: 
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall 
+(c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda 
+(a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: 
+((\forall (c2: C).((csuba g (CHead c (Bind b) u) c2) \to (arity g c2 t0 
+a2))))).(\lambda (c2: C).(\lambda (H5: (csuba g c c2)).(arity_bind g b H0 c2 
+u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u) (csuba_head g c c2 H5 (Bind 
+b) u)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: 
+A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c2: 
+C).((csuba g c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda (t0: 
+T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0 
+a2)).(\lambda (H3: ((\forall (c2: C).((csuba g (CHead c (Bind Abst) u) c2) 
+\to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (csuba g c 
+c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u) 
+(csuba_head g c c2 H4 (Bind Abst) u)))))))))))))) (\lambda (c: C).(\lambda 
+(u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: 
+((\forall (c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0: 
+T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3: 
+((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 (AHead a1 
+a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c c2)).(arity_appl g c2 u a1 
+(H1 c2 H4) t0 a2 (H3 c2 H4))))))))))))) (\lambda (c: C).(\lambda (u: 
+T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: 
+((\forall (c2: C).((csuba g c c2) \to (arity g c2 u (asucc g 
+a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3: 
+((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0))))).(\lambda (c2: 
+C).(\lambda (H4: (csuba g c c2)).(arity_cast g c2 u a0 (H1 c2 H4) t0 (H3 c2 
+H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: 
+(arity g c t0 a1)).(\lambda (H1: ((\forall (c2: C).((csuba g c c2) \to (arity 
+g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: 
+C).(\lambda (H3: (csuba g c c2)).(arity_repl g c2 t0 a1 (H1 c2 H3) a2 
+H2)))))))))) c1 t a H))))).
+
+axiom csuba_arity_rev:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 
+t a) \to (\forall (c2: C).((csuba g c2 c1) \to (arity g c2 t a)))))))
+.
+
+theorem arity_appls_appl:
+ \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (a1: A).((arity g c 
+v a1) \to (\forall (u: T).((arity g c u (asucc g a1)) \to (\forall (t: 
+T).(\forall (vs: TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) vs 
+(THead (Bind Abbr) v t)) a2) \to (arity g c (THeads (Flat Appl) vs (THead 
+(Flat Appl) v (THead (Bind Abst) u t))) a2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H: 
+(arity g c v a1)).(\lambda (u: T).(\lambda (H0: (arity g c u (asucc g 
+a1))).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: 
+TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) t0 (THead (Bind Abbr) 
+v t)) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead 
+(Bind Abst) u t))) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g c (THead 
+(Bind Abbr) v t) a2)).(let H_x \def (arity_gen_bind Abbr (\lambda (H2: (eq B 
+Abbr Abst)).(not_abbr_abst H2)) g c v t a2 H1) in (let H2 \def H_x in 
+(ex2_ind A (\lambda (a3: A).(arity g c v a3)) (\lambda (_: A).(arity g (CHead 
+c (Bind Abbr) v) t a2)) (arity g c (THead (Flat Appl) v (THead (Bind Abst) u 
+t)) a2) (\lambda (x: A).(\lambda (_: (arity g c v x)).(\lambda (H4: (arity g 
+(CHead c (Bind Abbr) v) t a2)).(arity_appl g c v a1 H (THead (Bind Abst) u t) 
+a2 (arity_head g c u a1 H0 t a2 (csuba_arity_rev g (CHead c (Bind Abbr) v) t 
+a2 H4 (CHead c (Bind Abst) u) (csuba_abst g c c (csuba_refl g c) u a1 H0 v 
+H))))))) H2))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H1: 
+((\forall (a2: A).((arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) 
+a2) \to (arity g c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind 
+Abst) u t))) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g c (THead (Flat 
+Appl) t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t))) a2)).(let H3 \def 
+(arity_gen_appl g c t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) a2 H2) 
+in (ex2_ind A (\lambda (a3: A).(arity g c t0 a3)) (\lambda (a3: A).(arity g c 
+(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) (AHead a3 a2))) (arity g c 
+(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead 
+(Bind Abst) u t)))) a2) (\lambda (x: A).(\lambda (H4: (arity g c t0 
+x)).(\lambda (H5: (arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) 
+(AHead x a2))).(arity_appl g c t0 x H4 (THeads (Flat Appl) t1 (THead (Flat 
+Appl) v (THead (Bind Abst) u t))) a2 (H1 (AHead x a2) H5))))) H3))))))) 
+vs))))))))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/clear.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/clear.ma
new file mode 100644
index 000000000..8e22a5484
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/clear.ma
@@ -0,0 +1,64 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/clear".
+
+include "csuba/defs.ma".
+
+include "clear/fwd.ma".
+
+theorem csuba_clear_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to 
+(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) 
+(\lambda (e2: C).(clear c2 e2))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c1 
+c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c 
+e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c0 
+e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) 
+e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e1 e2)) 
+(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: 
+C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 
+e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c4 
+e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: 
+(clear (CHead c3 k u) e1)).((match k in K return (\lambda (k0: K).((clear 
+(CHead c3 k0 u) e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda 
+(e2: C).(clear (CHead c4 k0 u) e2))))) with [(Bind b) \Rightarrow (\lambda 
+(H3: (clear (CHead c3 (Bind b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) 
+(\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g c e2)) (\lambda (e2: 
+C).(clear (CHead c4 (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g 
+(CHead c3 (Bind b) u) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)) 
+(CHead c4 (Bind b) u) (csuba_head g c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) 
+e1 (clear_gen_bind b c3 e1 u H3))) | (Flat f) \Rightarrow (\lambda (H3: 
+(clear (CHead c3 (Flat f) u) e1)).(let H4 \def (H1 e1 (clear_gen_flat f c3 e1 
+u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: 
+C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: 
+C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: C).(\lambda (H5: (csuba g 
+e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C (\lambda (e2: C).(csuba g e1 
+e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2)) x H5 (clear_flat c4 x 
+H6 f u))))) H4)))]) H2))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda 
+(H0: (csuba g c3 c4)).(\lambda (_: ((\forall (e1: C).((clear c3 e1) \to (ex2 
+C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c4 
+e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H2: (arity g c3 t (asucc 
+g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u a)).(\lambda (e1: 
+C).(\lambda (H4: (clear (CHead c3 (Bind Abst) t) e1)).(eq_ind_r C (CHead c3 
+(Bind Abst) t) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g c e2)) 
+(\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)))) (ex_intro2 C (\lambda 
+(e2: C).(csuba g (CHead c3 (Bind Abst) t) e2)) (\lambda (e2: C).(clear (CHead 
+c4 (Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) u) (csuba_abst g c3 c4 H0 t a H2 
+u H3) (clear_bind Abbr c4 u)) e1 (clear_gen_bind Abst c3 e1 t H4))))))))))))) 
+c1 c2 H)))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/defs.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/defs.ma
new file mode 100644
index 000000000..d108b9bba
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/defs.ma
@@ -0,0 +1,29 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/defs".
+
+include "arity/defs.ma".
+
+inductive csuba (g:G): C \to (C \to Prop) \def
+| csuba_sort: \forall (n: nat).(csuba g (CSort n) (CSort n))
+| csuba_head: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall 
+(k: K).(\forall (u: T).(csuba g (CHead c1 k u) (CHead c2 k u))))))
+| csuba_abst: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall 
+(t: T).(\forall (a: A).((arity g c1 t (asucc g a)) \to (\forall (u: 
+T).((arity g c2 u a) \to (csuba g (CHead c1 (Bind Abst) t) (CHead c2 (Bind 
+Abbr) u))))))))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/drop.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/drop.ma
new file mode 100644
index 000000000..c3a51fe8f
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/drop.ma
@@ -0,0 +1,813 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/drop".
+
+include "csuba/fwd.ma".
+
+include "drop/fwd.ma".
+
+theorem csuba_drop_abbr:
+ \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i 
+O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g 
+c1 c2) \to (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u))) 
+(\lambda (d2: C).(csuba g d1 d2))))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: 
+C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: 
+G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2 
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))) 
+(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 
+(CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: 
+(csuba g c1 c2)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c2)) H0 
+(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H)) in 
+(let H2 \def (csuba_gen_abbr g d1 c2 u H1) in (ex2_ind C (\lambda (d2: C).(eq 
+C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C 
+(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
+C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x (Bind 
+Abbr) u))).(\lambda (H4: (csuba g d1 x)).(eq_ind_r C (CHead x (Bind Abbr) u) 
+(\lambda (c: C).(ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) 
+u))) (\lambda (d2: C).(csuba g d1 d2)))) (ex_intro2 C (\lambda (d2: C).(drop 
+O O (CHead x (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
+C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abbr) u)) H4) c2 H3)))) 
+H2)))))))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: 
+C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: 
+G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2 
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
+d2)))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: 
+C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall 
+(g: G).(\forall (c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S 
+n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))))) 
+(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n) 
+O (CSort n0) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: 
+C).(\lambda (_: (csuba g (CSort n0) c2)).(and3_ind (eq C (CHead d1 (Bind 
+Abbr) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (ex2 C (\lambda (d2: 
+C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
+d2))) (\lambda (H2: (eq C (CHead d1 (Bind Abbr) u) (CSort n0))).(\lambda (_: 
+(eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (match H2 in eq 
+return (\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c (CSort n0)) \to (ex2 
+C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
+C).(csuba g d1 d2)))))) with [refl_equal \Rightarrow (\lambda (H5: (eq C 
+(CHead d1 (Bind Abbr) u) (CSort n0))).(let H6 \def (eq_ind C (CHead d1 (Bind 
+Abbr) u) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with 
+[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n0) 
+H5) in (False_ind (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind 
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H6)))]) in (H5 (refl_equal C 
+(CSort n0))))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abbr) u) 
+H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u: 
+T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall 
+(c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead 
+d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))).(\lambda (k: 
+K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n) 
+O (CHead c k t) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: 
+C).(\lambda (H2: (csuba g (CHead c k t) c2)).(K_ind (\lambda (k0: K).((csuba 
+g (CHead c k0 t) c2) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) u)) \to 
+(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda 
+(d2: C).(csuba g d1 d2)))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c 
+(Bind b) t) c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abbr) 
+u))).(B_ind (\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop 
+(r (Bind b0) n) O c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: 
+C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
+d2)))))) (\lambda (H5: (csuba g (CHead c (Bind Abbr) t) c2)).(\lambda (H6: 
+(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abbr) u))).(let H7 \def 
+(csuba_gen_abbr g c c2 t H5) in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead 
+d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g c d2)) (ex2 C (\lambda (d2: 
+C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
+d2))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Abbr) 
+t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abbr) t) 
+(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind 
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def (H c d1 u H6 g x 
+H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u))) 
+(\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O 
+(CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g 
+d1 d2))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead x0 (Bind Abbr) 
+u))).(\lambda (H12: (csuba g d1 x0)).(let H13 \def (refl_equal nat (r (Bind 
+Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x 
+(CHead x0 (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in (ex_intro2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) 
+u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead 
+x0 (Bind Abbr) u) H14 t) H12)))))) H10)) c2 H8)))) H7)))) (\lambda (H5: 
+(csuba g (CHead c (Bind Abst) t) c2)).(\lambda (H6: (drop (r (Bind Abst) n) O 
+c (CHead d1 (Bind Abbr) u))).(let H7 \def (csuba_gen_abst g c c2 t H5) in 
+(or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda 
+(d2: C).(csuba g c 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 c d2)))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex2 C (\lambda (d2: C).(drop 
+(S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) 
+(\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) 
+(\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 
+(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (ex2 C (\lambda 
+(d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g 
+d1 d2))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x (Bind Abst) 
+t))).(\lambda (H10: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abst) t) 
+(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind 
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H11 \def (H c d1 u H6 g x 
+H10) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u))) 
+(\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O 
+(CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g 
+d1 d2))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr) 
+u))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def (refl_equal nat (r (Bind 
+Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x 
+(CHead x0 (Bind Abbr) u))) H12 (r (Bind Abbr) n) H14) in (ex_intro2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) 
+u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) n x (CHead 
+x0 (Bind Abbr) u) H15 t) H13)))))) H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3 
+C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc 
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))).(ex4_3_ind 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 c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 a)))) (ex2 C (\lambda (d2: C).(drop (S n) O 
+c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda 
+(x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 
+(Bind Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t 
+(asucc g x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind 
+Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H13 \def (H c d1 u 
+H6 g x0 H10) in (ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind 
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S 
+n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
+C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H14: (drop n O x0 (CHead x 
+(Bind Abbr) u))).(\lambda (H15: (csuba g d1 x)).(let H16 \def (refl_equal nat 
+(r (Bind Abbr) n)) in (let H17 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 
+O x0 (CHead x (Bind Abbr) u))) H14 (r (Bind Abbr) n) H16) in (ex_intro2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind 
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 
+(CHead x (Bind Abbr) u) H17 x1) H15)))))) H13)) c2 H9)))))))) H8)) H7)))) 
+(\lambda (H5: (csuba g (CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r 
+(Bind Void) n) O c (CHead d1 (Bind Abbr) u))).(let H7 \def (csuba_gen_void g 
+c c2 t H5) in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) t))) 
+(\lambda (d2: C).(csuba g c d2)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: 
+C).(\lambda (H8: (eq C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g c 
+x)).(eq_ind_r C (CHead x (Bind Void) t) (\lambda (c0: C).(ex2 C (\lambda (d2: 
+C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
+d2)))) (let H10 \def (H c d1 u H6 g x H9) in (ex2_ind C (\lambda (d2: 
+C).(drop n O x (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) 
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind 
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x0: C).(\lambda (H11: 
+(drop n O x (CHead x0 (Bind Abbr) u))).(\lambda (H12: (csuba g d1 x0)).(let 
+H13 \def (refl_equal nat (r (Bind Abbr) n)) in (let H14 \def (eq_ind nat n 
+(\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) u))) H11 (r (Bind Abbr) 
+n) H13) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) 
+t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop 
+(Bind Void) n x (CHead x0 (Bind Abbr) u) H14 t) H12)))))) H10)) c2 H8)))) 
+H7)))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g (CHead c (Flat f) 
+t) c2)).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) 
+u))).(let H5 \def (csuba_gen_flat g c c2 t f H3) in (ex2_2_ind C T (\lambda 
+(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g c d2))) (ex2 C (\lambda (d2: C).(drop (S n) O c2 
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) x1))).(\lambda 
+(H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c0: C).(ex2 
+C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
+C).(csuba g d1 d2)))) (let H8 \def (H0 d1 u H4 g x0 H7) in (ex2_ind C 
+(\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
+C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) 
+x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda 
+(x: C).(\lambda (H9: (drop (S n) O x0 (CHead x (Bind Abbr) u))).(\lambda 
+(H10: (csuba g d1 x)).(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 
+(Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x 
+(drop_drop (Flat f) n x0 (CHead x (Bind Abbr) u) H9 x1) H10)))) H8)) c2 
+H6))))) H5))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u) t n 
+H1)))))))))))) c1)))) i).
+
+theorem csuba_drop_abst:
+ \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i 
+O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba 
+g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop i O 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).(drop i O 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)))))))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: 
+C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: 
+G).(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop n 
+O 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).(drop n O 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 (c1: C).(\lambda (d1: C).(\lambda (u1: 
+T).(\lambda (H: (drop O O c1 (CHead d1 (Bind Abst) u1))).(\lambda (g: 
+G).(\lambda (c2: C).(\lambda (H0: (csuba g c1 c2)).(let H1 \def (eq_ind C c1 
+(\lambda (c: C).(csuba g c c2)) H0 (CHead d1 (Bind Abst) u1) (drop_gen_refl 
+c1 (CHead d1 (Bind Abst) u1) H)) in (let H2 \def (csuba_gen_abst g d1 c2 u1 
+H1) in (or_ind (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))))) (or (ex2 C 
+(\lambda (d2: C).(drop O O 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).(drop O O 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 (H3: (ex2 C (\lambda 
+(d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
+d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop O O 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).(drop O O 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 (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst) 
+u1))).(\lambda (H5: (csuba g d1 x)).(eq_ind_r C (CHead x (Bind Abst) u1) 
+(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O 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).(drop O O 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))))))) 
+(or_introl (ex2 C (\lambda (d2: C).(drop O O (CHead x (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).(drop O O (CHead x (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).(drop O O (CHead x 
+(Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) 
+x (drop_refl (CHead x (Bind Abst) u1)) H5)) c2 H4)))) H3)) (\lambda (H3: 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 
+(CHead d2 (Bind 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)))))).(ex4_3_ind 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)))) (or (ex2 C 
+(\lambda (d2: C).(drop O O 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).(drop O O 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 (x0: C).(\lambda (x1: 
+T).(\lambda (x2: A).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abbr) 
+x1))).(\lambda (H5: (csuba g d1 x0)).(\lambda (H6: (arity g d1 u1 (asucc g 
+x2))).(\lambda (H7: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) 
+(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O 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).(drop O O 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))))))) 
+(or_intror (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Abbr) x1) (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).(drop O O (CHead x0 (Bind Abbr) x1) 
+(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))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abbr) x1) (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)))) 
+x0 x1 x2 (drop_refl (CHead x0 (Bind Abbr) x1)) H5 H6 H7)) c2 H4)))))))) H3)) 
+H2)))))))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: 
+C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: 
+G).(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop n 
+O 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).(drop n O 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 (c1: C).(C_ind (\lambda (c: C).(\forall (d1: 
+C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abst) u1)) \to (\forall 
+(g: G).(\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda (d2: C).(drop 
+(S n) O 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).(drop (S n) O 
+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 (n0: nat).(\lambda (d1: 
+C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) (CHead d1 (Bind 
+Abst) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (csuba g (CSort n0) 
+c2)).(and3_ind (eq C (CHead d1 (Bind Abst) u1) (CSort n0)) (eq nat (S n) O) 
+(eq nat O O) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind 
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 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 (H2: (eq C (CHead d1 (Bind Abst) u1) (CSort n0))).(\lambda 
+(_: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (match H2 in eq 
+return (\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c (CSort n0)) \to (or 
+(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda 
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop (S n) O 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))))))))) with 
+[refl_equal \Rightarrow (\lambda (H5: (eq C (CHead d1 (Bind Abst) u1) (CSort 
+n0))).(let H6 \def (eq_ind C (CHead d1 (Bind Abst) u1) (\lambda (e: C).(match 
+e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | 
+(CHead _ _ _) \Rightarrow True])) I (CSort n0) H5) in (False_ind (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O 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)))))) H6)))]) in 
+(H5 (refl_equal C (CSort n0))))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind 
+Abst) u1) H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: 
+C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abst) u1)) \to (\forall 
+(g: G).(\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda (d2: C).(drop 
+(S n) O 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).(drop (S n) O 
+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 (k: K).(\lambda (t: 
+T).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H1: (drop (S n) O (CHead c k 
+t) (CHead d1 (Bind Abst) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: 
+(csuba g (CHead c k t) c2)).(K_ind (\lambda (k0: K).((csuba g (CHead c k0 t) 
+c2) \to ((drop (r k0 n) O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O 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).(drop (S n) O 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 
+(b: B).(\lambda (H3: (csuba g (CHead c (Bind b) t) c2)).(\lambda (H4: (drop 
+(r (Bind b) n) O c (CHead d1 (Bind Abst) u1))).(B_ind (\lambda (b0: 
+B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop (r (Bind b0) n) O c (CHead 
+d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O 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).(drop (S n) O 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 (H5: (csuba g (CHead c (Bind Abbr) t) c2)).(\lambda (H6: 
+(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abst) u1))).(let H7 \def 
+(csuba_gen_abbr g c c2 t H5) in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead 
+d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g c d2)) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O 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).(drop (S 
+n) O 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 (x: C).(\lambda (H8: 
+(eq C c2 (CHead x (Bind Abbr) t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C 
+(CHead x (Bind Abbr) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S 
+n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 
+C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 
+(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 H10 \def (H c d1 u1 H6 g x H9) in (or_ind 
+(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop n O x (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))))) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x (Bind Abbr) 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).(drop (S n) O (CHead x (Bind Abbr) 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)))))) (\lambda (H11: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind 
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: 
+C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) 
+(or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) 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).(drop (S n) O (CHead x (Bind Abbr) 
+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)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x 
+(CHead x0 (Bind Abst) u1))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def 
+(refl_equal nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda 
+(n0: nat).(drop n0 O x (CHead x0 (Bind Abst) u1))) H12 (r (Bind Abbr) n) H14) 
+in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) 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).(drop (S n) O (CHead x 
+(Bind Abbr) 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))))) (ex_intro2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda 
+(d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead x0 (Bind Abst) 
+u1) H15 t) H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind 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)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop n O x (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)))) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x (Bind Abbr) 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).(drop (S n) O (CHead x (Bind Abbr) 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)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: 
+(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H13: (csuba g d1 
+x0)).(\lambda (H14: (arity g d1 u1 (asucc g x2))).(\lambda (H15: (arity g x0 
+x1 x2)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def 
+(eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) x1))) H12 
+(r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O 
+(CHead x (Bind Abbr) 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).(drop 
+(S n) O (CHead x (Bind Abbr) 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))))) (ex4_3_intro C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
+(Bind Abbr) t) (CHead d2 (Bind 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)))) x0 x1 x2 (drop_drop (Bind Abbr) 
+n x (CHead x0 (Bind Abbr) x1) H17 t) H13 H14 H15))))))))))) H11)) H10)) c2 
+H8)))) H7)))) (\lambda (H5: (csuba g (CHead c (Bind Abst) t) c2)).(\lambda 
+(H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) u1))).(let H7 \def 
+(csuba_gen_abst g c c2 t H5) in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 
+(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c 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 c 
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O 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).(drop (S n) O 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 (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) 
+t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 
+(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O 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).(drop (S n) O 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 (x: 
+C).(\lambda (H9: (eq C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g c 
+x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda 
+(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba 
+g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop (S n) O c0 (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 H11 \def (H c d1 u1 H6 g 
+x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) 
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (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))))) (or 
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) 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).(drop (S n) O (CHead x (Bind Abst) 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)))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C 
+(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
+C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind 
+Abst) 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).(drop (S n) O 
+(CHead x (Bind Abst) 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)))))) (\lambda (x0: 
+C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u1))).(\lambda (H14: 
+(csuba g d1 x0)).(let H15 \def (refl_equal nat (r (Bind Abbr) n)) in (let H16 
+\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abst) 
+u1))) H13 (r (Bind Abbr) n) H15) in (or_introl (ex2 C (\lambda (d2: C).(drop 
+(S n) O (CHead x (Bind Abst) 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).(drop (S n) O (CHead x (Bind Abst) 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))))) 
+(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) 
+n x (CHead x0 (Bind Abst) u1) H16 t) H14))))))) H12)) (\lambda (H12: (ex4_3 C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 
+(Bind 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)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop n O x (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)))) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x (Bind Abst) 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).(drop (S n) O (CHead x (Bind Abst) 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)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: 
+(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H14: (csuba g d1 
+x0)).(\lambda (H15: (arity g d1 u1 (asucc g x2))).(\lambda (H16: (arity g x0 
+x1 x2)).(let H17 \def (refl_equal nat (r (Bind Abbr) n)) in (let H18 \def 
+(eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) x1))) H13 
+(r (Bind Abbr) n) H17) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O 
+(CHead x (Bind Abst) 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).(drop 
+(S n) O (CHead x (Bind Abst) 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))))) (ex4_3_intro C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
+(Bind Abst) t) (CHead d2 (Bind 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)))) x0 x1 x2 (drop_drop (Bind Abst) 
+n x (CHead x0 (Bind Abbr) x1) H18 t) H14 H15 H16))))))))))) H12)) H11)) c2 
+H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind 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 c d2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O 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).(drop (S n) O 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 (x0: 
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind 
+Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t (asucc g 
+x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) 
+(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O 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).(drop (S n) O c0 (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 H13 \def (H c d1 u1 H6 g x0 H10) in (or_ind (ex2 C (\lambda 
+(d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
+d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n 
+O x0 (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))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead 
+x0 (Bind Abbr) x1) (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).(drop (S 
+n) O (CHead x0 (Bind Abbr) x1) (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 
+(H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (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).(drop (S n) O (CHead x0 (Bind Abbr) x1) 
+(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 (x: C).(\lambda (H15: (drop n O x0 (CHead 
+x (Bind Abst) u1))).(\lambda (H16: (csuba g d1 x)).(let H17 \def (refl_equal 
+nat (r (Bind Abbr) n)) in (let H18 \def (eq_ind nat n (\lambda (n0: 
+nat).(drop n0 O x0 (CHead x (Bind Abst) u1))) H15 (r (Bind Abbr) n) H17) in 
+(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) 
+(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).(drop (S n) O (CHead x0 
+(Bind Abbr) x1) (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).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 (CHead x 
+(Bind Abst) u1) H18 x1) H16))))))) H14)) (\lambda (H14: (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind 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)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop n O x0 (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)))) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (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).(drop (S n) O (CHead x0 (Bind Abbr) x1) (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 (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H15: 
+(drop n O x0 (CHead x3 (Bind Abbr) x4))).(\lambda (H16: (csuba g d1 
+x3)).(\lambda (H17: (arity g d1 u1 (asucc g x5))).(\lambda (H18: (arity g x3 
+x4 x5)).(let H19 \def (refl_equal nat (r (Bind Abbr) n)) in (let H20 \def 
+(eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x3 (Bind Abbr) x4))) 
+H15 (r (Bind Abbr) n) H19) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) 
+O (CHead x0 (Bind Abbr) x1) (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).(drop (S n) O (CHead x0 (Bind Abbr) x1) (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))))) 
+(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
+n) O (CHead x0 (Bind Abbr) x1) (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)))) x3 x4 x5 
+(drop_drop (Bind Abbr) n x0 (CHead x3 (Bind Abbr) x4) H20 x1) H16 H17 
+H18))))))))))) H14)) H13)) c2 H9)))))))) H8)) H7)))) (\lambda (H5: (csuba g 
+(CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead 
+d1 (Bind Abst) u1))).(let H7 \def (csuba_gen_void g c c2 t H5) in (ex2_ind C 
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: C).(csuba 
+g c d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O 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).(drop (S n) O 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 (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Void) 
+t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Void) t) 
+(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind 
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (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 H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda (d2: 
+C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x 
+(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))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x 
+(Bind Void) 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).(drop (S n) O 
+(CHead x (Bind Void) 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)))))) (\lambda 
+(H11: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) 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).(drop (S n) O (CHead x (Bind Void) 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)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead 
+x0 (Bind Abst) u1))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def 
+(refl_equal nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda 
+(n0: nat).(drop n0 O x (CHead x0 (Bind Abst) u1))) H12 (r (Bind Abbr) n) H14) 
+in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) 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).(drop (S n) O (CHead x 
+(Bind Void) 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))))) (ex_intro2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda 
+(d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abst) 
+u1) H15 t) H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind 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)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop n O x (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)))) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x (Bind Void) 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).(drop (S n) O (CHead x (Bind Void) 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)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: 
+(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H13: (csuba g d1 
+x0)).(\lambda (H14: (arity g d1 u1 (asucc g x2))).(\lambda (H15: (arity g x0 
+x1 x2)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def 
+(eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) x1))) H12 
+(r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O 
+(CHead x (Bind Void) 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).(drop 
+(S n) O (CHead x (Bind Void) 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))))) (ex4_3_intro C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
+(Bind Void) t) (CHead d2 (Bind 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)))) x0 x1 x2 (drop_drop (Bind Void) 
+n x (CHead x0 (Bind Abbr) x1) H17 t) H13 H14 H15))))))))))) H11)) H10)) c2 
+H8)))) H7)))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g (CHead c 
+(Flat f) t) c2)).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abst) 
+u1))).(let H5 \def (csuba_gen_flat g c c2 t f H3) in (ex2_2_ind C T (\lambda 
+(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g c d2))) (or (ex2 C (\lambda (d2: C).(drop (S n) O 
+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).(drop (S n) O 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 (x0: C).(\lambda (x1: T).(\lambda (H6: (eq 
+C c2 (CHead x0 (Flat f) x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C 
+(CHead x0 (Flat f) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S 
+n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 
+C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 
+(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 H8 \def (H0 d1 u1 H4 g x0 H7) in (or_ind (ex2 
+C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O x0 (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))))) (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (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).(drop (S n) O (CHead x0 (Flat f) x1) 
+(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: (ex2 C (\lambda (d2: C).(drop (S n) O 
+x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C 
+(\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
+C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat 
+f) x1) (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).(drop (S n) O (CHead 
+x0 (Flat f) x1) (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 (x: C).(\lambda (H10: 
+(drop (S n) O x0 (CHead x (Bind Abst) u1))).(\lambda (H11: (csuba g d1 
+x)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) 
+(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).(drop (S n) O (CHead x0 
+(Flat f) x1) (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).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda 
+(d2: C).(csuba g d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u1) 
+H10 x1) H11))))) H9)) (\lambda (H9: (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind 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)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop (S n) O x0 (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)))) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Flat f) x1) (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).(drop (S n) O (CHead x0 (Flat f) x1) (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 (x2: C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H10: 
+(drop (S n) O x0 (CHead x2 (Bind Abbr) x3))).(\lambda (H11: (csuba g d1 
+x2)).(\lambda (H12: (arity g d1 u1 (asucc g x4))).(\lambda (H13: (arity g x2 
+x3 x4)).(or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) 
+x1) (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).(drop (S n) O (CHead x0 
+(Flat f) x1) (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))))) (ex4_3_intro C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) 
+(CHead d2 (Bind 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)))) x2 x3 x4 (drop_drop (Flat f) n x0 (CHead x2 (Bind 
+Abbr) x3) H10 x1) H11 H12 H13))))))))) H9)) H8)) c2 H6))))) H5))))) k H2 
+(drop_gen_drop k c (CHead d1 (Bind Abst) u1) t n H1)))))))))))) c1)))) i).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/fwd.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/fwd.ma
new file mode 100644
index 000000000..c5c65696a
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/fwd.ma
@@ -0,0 +1,411 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/fwd".
+
+include "csuba/defs.ma".
+
+theorem csuba_gen_abbr:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g 
+(CHead d1 (Bind Abbr) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 
+(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))))))).
+
+theorem csuba_gen_void:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g 
+(CHead d1 (Bind Void) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 
+(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))))))).
+
+theorem csuba_gen_abst:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g 
+(CHead d1 (Bind Abst) u1) c) \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))))))))))
+\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: 
+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) 
+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))))))) 
+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: 
+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 
+(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 
+(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))))))).
+
+theorem csuba_gen_flat:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall 
+(f: F).((csuba g (CHead d1 (Flat f) u1) 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)))))))))
+\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 
+(\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)))))))).
+
+theorem csuba_gen_bind:
+ \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall 
+(v1: T).((csuba g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(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 
+(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 
+(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 
+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)))))))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/getl.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/getl.ma
new file mode 100644
index 000000000..2bb976390
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/getl.ma
@@ -0,0 +1,463 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/getl".
+
+include "csuba/drop.ma".
+
+include "csuba/clear.ma".
+
+include "getl/clear.ma".
+
+theorem csuba_getl_abbr:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall 
+(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g 
+c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) 
+(\lambda (d2: C).(csuba g d1 d2))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda 
+(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u))).(let H0 \def 
+(getl_gen_all c1 (CHead d1 (Bind Abbr) u) i H) in (ex2_ind C (\lambda (e: 
+C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u))) 
+(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (x: 
+C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind 
+Abbr) u))).((match x in C return (\lambda (c: C).((drop i O c1 c) \to ((clear 
+c (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C 
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
+C).(csuba g d1 d2)))))))) with [(CSort n) \Rightarrow (\lambda (_: (drop i O 
+c1 (CSort n))).(\lambda (H4: (clear (CSort n) (CHead d1 (Bind Abbr) 
+u))).(clear_gen_sort (CHead d1 (Bind Abbr) u) n H4 (\forall (c2: C).((csuba g 
+c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) 
+(\lambda (d2: C).(csuba g d1 d2)))))))) | (CHead c k t) \Rightarrow (\lambda 
+(H3: (drop i O c1 (CHead c k t))).(\lambda (H4: (clear (CHead c k t) (CHead 
+d1 (Bind Abbr) u))).((match k in K return (\lambda (k0: K).((drop i O c1 
+(CHead c k0 t)) \to ((clear (CHead c k0 t) (CHead d1 (Bind Abbr) u)) \to 
+(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) with 
+[(Bind b) \Rightarrow (\lambda (H5: (drop i O c1 (CHead c (Bind b) 
+t))).(\lambda (H6: (clear (CHead c (Bind b) t) (CHead d1 (Bind Abbr) 
+u))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda 
+(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c0 _ _) \Rightarrow c0])) 
+(CHead d1 (Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 
+(Bind Abbr) u) t H6)) in ((let H8 \def (f_equal C B (\lambda (e: C).(match e 
+in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 
+_) \Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) 
+(CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abbr) u) t H6)) in 
+((let H9 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: 
+C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead 
+d1 (Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind 
+Abbr) u) t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 
+c)).(\lambda (c2: C).(\lambda (H12: (csuba g c1 c2)).(let H13 \def (eq_ind_r 
+T t (\lambda (t0: T).(drop i O c1 (CHead c (Bind b) t0))) H5 u H9) in (let 
+H14 \def (eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead c (Bind b0) u))) 
+H13 Abbr H10) in (let H15 \def (eq_ind_r C c (\lambda (c0: C).(drop i O c1 
+(CHead c0 (Bind Abbr) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abbr i c1 
+d1 u H15 g c2 H12) in (ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: 
+C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) 
+(\lambda (x0: C).(\lambda (H17: (drop i O c2 (CHead x0 (Bind Abbr) 
+u))).(\lambda (H18: (csuba g d1 x0)).(ex_intro2 C (\lambda (d2: C).(getl i c2 
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (getl_intro i 
+c2 (CHead x0 (Bind Abbr) u) (CHead x0 (Bind Abbr) u) H17 (clear_bind Abbr x0 
+u)) H18)))) H16)))))))))) H8)) H7)))) | (Flat f) \Rightarrow (\lambda (H5: 
+(drop i O c1 (CHead c (Flat f) t))).(\lambda (H6: (clear (CHead c (Flat f) t) 
+(CHead d1 (Bind Abbr) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c0: 
+C).((drop i O c0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g c0 c2) 
+\to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda 
+(d2: C).(csuba g d1 d2))))))) (nat_ind (\lambda (n: nat).(\forall (x0: 
+C).((drop n O x0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g x0 c2) 
+\to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda 
+(d2: C).(csuba g d1 d2)))))))) (\lambda (x0: C).(\lambda (H8: (drop O O x0 
+(CHead c (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g x0 c2)).(let 
+H10 \def (eq_ind C x0 (\lambda (c0: C).(csuba g c0 c2)) H9 (CHead c (Flat f) 
+t) (drop_gen_refl x0 (CHead c (Flat f) t) H8)) in (let H_y \def (clear_flat c 
+(CHead d1 (Bind Abbr) u) (clear_gen_flat f c (CHead d1 (Bind Abbr) u) t H6) f 
+t) in (let H11 \def (csuba_clear_conf g (CHead c (Flat f) t) c2 H10 (CHead d1 
+(Bind Abbr) u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 (Bind 
+Abbr) u) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(getl O 
+c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda 
+(x1: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abbr) u) x1)).(\lambda (H13: 
+(clear c2 x1)).(let H14 \def (csuba_gen_abbr g d1 x1 u H12) in (ex2_ind C 
+(\lambda (d2: C).(eq C x1 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba 
+g d1 d2)) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u))) 
+(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x2: C).(\lambda (H15: (eq C x1 
+(CHead x2 (Bind Abbr) u))).(\lambda (H16: (csuba g d1 x2)).(let H17 \def 
+(eq_ind C x1 (\lambda (c0: C).(clear c2 c0)) H13 (CHead x2 (Bind Abbr) u) 
+H15) in (ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u))) 
+(\lambda (d2: C).(csuba g d1 d2)) x2 (getl_intro O c2 (CHead x2 (Bind Abbr) 
+u) c2 (drop_refl c2) H17) H16))))) H14))))) H11)))))))) (\lambda (n: 
+nat).(\lambda (H8: ((\forall (x0: C).((drop n O x0 (CHead c (Flat f) t)) \to 
+(\forall (c2: C).((csuba g x0 c2) \to (ex2 C (\lambda (d2: C).(getl n c2 
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))).(\lambda 
+(x0: C).(\lambda (H9: (drop (S n) O x0 (CHead c (Flat f) t))).(\lambda (c2: 
+C).(\lambda (H10: (csuba g x0 c2)).(let H11 \def (drop_clear x0 (CHead c 
+(Flat f) t) n H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: 
+C).(\lambda (v: T).(clear x0 (CHead e (Bind b) v))))) (\lambda (_: 
+B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead c (Flat f) t))))) (ex2 
+C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
+C).(csuba g d1 d2))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: 
+T).(\lambda (H12: (clear x0 (CHead x2 (Bind x1) x3))).(\lambda (H13: (drop n 
+O x2 (CHead c (Flat f) t))).(let H14 \def (csuba_clear_conf g x0 c2 H10 
+(CHead x2 (Bind x1) x3) H12) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead 
+x2 (Bind x1) x3) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: 
+C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
+d2))) (\lambda (x4: C).(\lambda (H15: (csuba g (CHead x2 (Bind x1) x3) 
+x4)).(\lambda (H16: (clear c2 x4)).(let H17 \def (csuba_gen_bind g x1 x2 x4 
+x3 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
+T).(eq C x4 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csuba g x2 e2)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x5: 
+B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H18: (eq C x4 (CHead x6 (Bind 
+x5) x7))).(\lambda (H19: (csuba g x2 x6)).(let H20 \def (eq_ind C x4 (\lambda 
+(c0: C).(clear c2 c0)) H16 (CHead x6 (Bind x5) x7) H18) in (let H21 \def (H8 
+x2 H13 x6 H19) in (ex2_ind C (\lambda (d2: C).(getl n x6 (CHead d2 (Bind 
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S 
+n) c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda 
+(x8: C).(\lambda (H22: (getl n x6 (CHead x8 (Bind Abbr) u))).(\lambda (H23: 
+(csuba g d1 x8)).(ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x8 (getl_clear_bind x5 c2 x6 x7 
+H20 (CHead x8 (Bind Abbr) u) n H22) H23)))) H21)))))))) H17))))) H14))))))) 
+H11)))))))) i) H7))))]) H3 H4)))]) H1 H2)))) H0))))))).
+
+theorem csuba_getl_abst:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall 
+(i: nat).((getl i c1 (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba 
+g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i 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).(getl i 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)))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda 
+(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u1))).(let H0 \def 
+(getl_gen_all c1 (CHead d1 (Bind Abst) u1) i H) in (ex2_ind C (\lambda (e: 
+C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u1))) 
+(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i 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).(getl i 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 (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear 
+x (CHead d1 (Bind Abst) u1))).((match x in C return (\lambda (c: C).((drop i 
+O c1 c) \to ((clear c (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba 
+g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i 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).(getl i 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))))))))))) 
+with [(CSort n) \Rightarrow (\lambda (_: (drop i O c1 (CSort n))).(\lambda 
+(H4: (clear (CSort n) (CHead d1 (Bind Abst) u1))).(clear_gen_sort (CHead d1 
+(Bind Abst) u1) n H4 (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C 
+(\lambda (d2: C).(getl i 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).(getl i 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))))))))))) | (CHead c k t) 
+\Rightarrow (\lambda (H3: (drop i O c1 (CHead c k t))).(\lambda (H4: (clear 
+(CHead c k t) (CHead d1 (Bind Abst) u1))).((match k in K return (\lambda (k0: 
+K).((drop i O c1 (CHead c k0 t)) \to ((clear (CHead c k0 t) (CHead d1 (Bind 
+Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: 
+C).(getl i 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).(getl i 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))))))))))) with [(Bind b) \Rightarrow (\lambda (H5: (drop 
+i O c1 (CHead c (Bind b) t))).(\lambda (H6: (clear (CHead c (Bind b) t) 
+(CHead d1 (Bind Abst) u1))).(let H7 \def (f_equal C C (\lambda (e: C).(match 
+e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c0 _ 
+_) \Rightarrow c0])) (CHead d1 (Bind Abst) u1) (CHead c (Bind b) t) 
+(clear_gen_bind b c (CHead d1 (Bind Abst) u1) t H6)) in ((let H8 \def 
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with 
+[(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 in K 
+return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
+\Rightarrow Abst])])) (CHead d1 (Bind Abst) u1) (CHead c (Bind b) t) 
+(clear_gen_bind b c (CHead d1 (Bind Abst) u1) t H6)) in ((let H9 \def 
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
+[(CSort _) \Rightarrow u1 | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind 
+Abst) u1) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abst) u1) 
+t H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 c)).(\lambda 
+(c2: C).(\lambda (H12: (csuba g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda 
+(t0: T).(drop i O c1 (CHead c (Bind b) t0))) H5 u1 H9) in (let H14 \def 
+(eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead c (Bind b0) u1))) H13 Abst 
+H10) in (let H15 \def (eq_ind_r C c (\lambda (c0: C).(drop i O c1 (CHead c0 
+(Bind Abst) u1))) H14 d1 H11) in (let H16 \def (csuba_drop_abst i c1 d1 u1 
+H15 g c2 H12) in (or_ind (ex2 C (\lambda (d2: C).(drop i O 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).(drop i O 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))))) (or (ex2 C (\lambda (d2: C).(getl i 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).(getl i 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 
+(H17: (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C 
+(\lambda (d2: C).(getl i 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).(getl i 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 (x0: C).(\lambda 
+(H18: (drop i O c2 (CHead x0 (Bind Abst) u1))).(\lambda (H19: (csuba g d1 
+x0)).(or_introl (ex2 C (\lambda (d2: C).(getl i 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).(getl i 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))))) 
+(ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda 
+(d2: C).(csuba g d1 d2)) x0 (getl_intro i c2 (CHead x0 (Bind Abst) u1) (CHead 
+x0 (Bind Abst) u1) H18 (clear_bind Abst x0 u1)) H19))))) H17)) (\lambda (H17: 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 
+(CHead d2 (Bind 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)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop i O 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)))) (or (ex2 C 
+(\lambda (d2: C).(getl i 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).(getl i 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 (x0: C).(\lambda (x1: 
+T).(\lambda (x2: A).(\lambda (H18: (drop i O c2 (CHead x0 (Bind Abbr) 
+x1))).(\lambda (H19: (csuba g d1 x0)).(\lambda (H20: (arity g d1 u1 (asucc g 
+x2))).(\lambda (H21: (arity g x0 x1 x2)).(or_intror (ex2 C (\lambda (d2: 
+C).(getl i 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).(getl i 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))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(getl i 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)))) x0 x1 x2 
+(getl_intro i c2 (CHead x0 (Bind Abbr) x1) (CHead x0 (Bind Abbr) x1) H18 
+(clear_bind Abbr x0 x1)) H19 H20 H21))))))))) H17)) H16)))))))))) H8)) H7)))) 
+| (Flat f) \Rightarrow (\lambda (H5: (drop i O c1 (CHead c (Flat f) 
+t))).(\lambda (H6: (clear (CHead c (Flat f) t) (CHead d1 (Bind Abst) 
+u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c0: C).((drop i O c0 (CHead 
+c (Flat f) t)) \to (\forall (c2: C).((csuba g c0 c2) \to (or (ex2 C (\lambda 
+(d2: C).(getl i 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).(getl i 
+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)))))))))) (nat_ind (\lambda (n: nat).(\forall (x0: 
+C).((drop n O x0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g x0 c2) 
+\to (or (ex2 C (\lambda (d2: C).(getl n 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).(getl n 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 
+(x0: C).(\lambda (H8: (drop O O x0 (CHead c (Flat f) t))).(\lambda (c2: 
+C).(\lambda (H9: (csuba g x0 c2)).(let H10 \def (eq_ind C x0 (\lambda (c0: 
+C).(csuba g c0 c2)) H9 (CHead c (Flat f) t) (drop_gen_refl x0 (CHead c (Flat 
+f) t) H8)) in (let H_y \def (clear_flat c (CHead d1 (Bind Abst) u1) 
+(clear_gen_flat f c (CHead d1 (Bind Abst) u1) t H6) f t) in (let H11 \def 
+(csuba_clear_conf g (CHead c (Flat f) t) c2 H10 (CHead d1 (Bind Abst) u1) 
+H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 (Bind Abst) u1) e2)) 
+(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O 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).(getl O 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 (x1: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abst) u1) 
+x1)).(\lambda (H13: (clear c2 x1)).(let H14 \def (csuba_gen_abst g d1 x1 u1 
+H12) in (or_ind (ex2 C (\lambda (d2: C).(eq C x1 (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 x1 (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))))) (or (ex2 C 
+(\lambda (d2: C).(getl O 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).(getl O 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 (H15: (ex2 C (\lambda 
+(d2: C).(eq C x1 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
+d2)))).(ex2_ind C (\lambda (d2: C).(eq C x1 (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(getl O 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).(getl O 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 (x2: C).(\lambda (H16: (eq C x1 (CHead x2 (Bind Abst) 
+u1))).(\lambda (H17: (csuba g d1 x2)).(let H18 \def (eq_ind C x1 (\lambda 
+(c0: C).(clear c2 c0)) H13 (CHead x2 (Bind Abst) u1) H16) in (or_introl (ex2 
+C (\lambda (d2: C).(getl O 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).(getl O 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))))) (ex_intro2 C (\lambda (d2: 
+C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) 
+x2 (getl_intro O c2 (CHead x2 (Bind Abst) u1) c2 (drop_refl c2) H18) 
+H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(eq C x1 (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)))))).(ex4_3_ind C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x1 (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)))) (or (ex2 C (\lambda (d2: C).(getl O 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).(getl O 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 (x2: 
+C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H16: (eq C x1 (CHead x2 (Bind 
+Abbr) x3))).(\lambda (H17: (csuba g d1 x2)).(\lambda (H18: (arity g d1 u1 
+(asucc g x4))).(\lambda (H19: (arity g x2 x3 x4)).(let H20 \def (eq_ind C x1 
+(\lambda (c0: C).(clear c2 c0)) H13 (CHead x2 (Bind Abbr) x3) H16) in 
+(or_intror (ex2 C (\lambda (d2: C).(getl O 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).(getl O 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))))) (ex4_3_intro C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O 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)))) x2 x3 x4 (getl_intro O c2 (CHead x2 (Bind Abbr) x3) c2 (drop_refl 
+c2) H20) H17 H18 H19)))))))))) H15)) H14))))) H11)))))))) (\lambda (n: 
+nat).(\lambda (H8: ((\forall (x0: C).((drop n O x0 (CHead c (Flat f) t)) \to 
+(\forall (c2: C).((csuba g x0 c2) \to (or (ex2 C (\lambda (d2: C).(getl n 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).(getl n 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 (x0: C).(\lambda (H9: (drop (S n) O x0 (CHead c (Flat 
+f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x0 c2)).(let H11 \def 
+(drop_clear x0 (CHead c (Flat f) t) n H9) in (ex2_3_ind B C T (\lambda (b: 
+B).(\lambda (e: C).(\lambda (v: T).(clear x0 (CHead e (Bind b) v))))) 
+(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead c (Flat f) 
+t))))) (or (ex2 C (\lambda (d2: C).(getl (S n) 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).(getl (S n) 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 (x1: 
+B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (clear x0 (CHead x2 (Bind 
+x1) x3))).(\lambda (H13: (drop n O x2 (CHead c (Flat f) t))).(let H14 \def 
+(csuba_clear_conf g x0 c2 H10 (CHead x2 (Bind x1) x3) H12) in (ex2_ind C 
+(\lambda (e2: C).(csuba g (CHead x2 (Bind x1) x3) e2)) (\lambda (e2: 
+C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) 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).(getl (S n) 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 (x4: C).(\lambda (H15: (csuba g (CHead x2 (Bind x1) x3) 
+x4)).(\lambda (H16: (clear c2 x4)).(let H17 \def (csuba_gen_bind g x1 x2 x4 
+x3 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
+T).(eq C x4 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csuba g x2 e2)))) (or (ex2 C (\lambda (d2: C).(getl (S n) 
+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).(getl (S n) 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 (x5: B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H18: 
+(eq C x4 (CHead x6 (Bind x5) x7))).(\lambda (H19: (csuba g x2 x6)).(let H20 
+\def (eq_ind C x4 (\lambda (c0: C).(clear c2 c0)) H16 (CHead x6 (Bind x5) x7) 
+H18) in (let H21 \def (H8 x2 H13 x6 H19) in (or_ind (ex2 C (\lambda (d2: 
+C).(getl n x6 (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).(getl n x6 
+(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))))) (or (ex2 C (\lambda (d2: C).(getl (S n) 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).(getl (S n) 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 (H22: (ex2 C (\lambda (d2: C).(getl n x6 (CHead d2 (Bind 
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: 
+C).(getl n x6 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) 
+(or (ex2 C (\lambda (d2: C).(getl (S n) 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).(getl (S n) 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 (x8: 
+C).(\lambda (H23: (getl n x6 (CHead x8 (Bind Abst) u1))).(\lambda (H24: 
+(csuba g d1 x8)).(or_introl (ex2 C (\lambda (d2: C).(getl (S n) 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).(getl (S n) 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))))) (ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) 
+u1))) (\lambda (d2: C).(csuba g d1 d2)) x8 (getl_clear_bind x5 c2 x6 x7 H20 
+(CHead x8 (Bind Abst) u1) n H23) H24))))) H22)) (\lambda (H22: (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x6 (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)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(getl n x6 (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)))) (or (ex2 C (\lambda (d2: 
+C).(getl (S n) 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).(getl (S 
+n) 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 (x8: C).(\lambda (x9: T).(\lambda 
+(x10: A).(\lambda (H23: (getl n x6 (CHead x8 (Bind Abbr) x9))).(\lambda (H24: 
+(csuba g d1 x8)).(\lambda (H25: (arity g d1 u1 (asucc g x10))).(\lambda (H26: 
+(arity g x8 x9 x10)).(or_intror (ex2 C (\lambda (d2: C).(getl (S n) 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).(getl (S n) 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))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(getl (S n) 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)))) x8 x9 x10 (getl_clear_bind x5 c2 
+x6 x7 H20 (CHead x8 (Bind Abbr) x9) n H23) H24 H25 H26))))))))) H22)) 
+H21)))))))) H17))))) H14))))))) H11)))))))) i) H7))))]) H3 H4)))]) H1 H2)))) 
+H0))))))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/props.ma
new file mode 100644
index 000000000..b0e0b8080
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/props.ma
@@ -0,0 +1,27 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csuba/props".
+
+include "csuba/defs.ma".
+
+theorem csuba_refl:
+ \forall (g: G).(\forall (c: C).(csuba g c c))
+\def
+ \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csuba g c0 c0)) 
+(\lambda (n: nat).(csuba_sort g n)) (\lambda (c0: C).(\lambda (H: (csuba g c0 
+c0)).(\lambda (k: K).(\lambda (t: T).(csuba_head g c0 c0 H k t))))) c)).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/dec.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/dec.ma
new file mode 100644
index 000000000..9495a5ef8
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/dec.ma
@@ -0,0 +1,200 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/dec".
+
+include "nf2/defs.ma".
+
+include "pr2/clen.ma".
+
+include "pr2/fwd.ma".
+
+include "pr0/dec.ma".
+
+include "C/props.ma".
+
+theorem nf2_dec:
+ \forall (c: C).(\forall (t1: T).(or (nf2 c t1) (ex2 T (\lambda (t2: T).((eq 
+T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)))))
+\def
+ \lambda (c: C).(c_tail_ind (\lambda (c0: C).(\forall (t1: T).(or (\forall 
+(t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 
+t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))))) (\lambda 
+(n: nat).(\lambda (t1: T).(let H_x \def (nf0_dec t1) in (let H \def H_x in 
+(or_ind (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: 
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2))) 
+(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T 
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr2 (CSort n) t1 t2)))) (\lambda (H0: ((\forall (t2: T).((pr0 t1 t2) \to 
+(eq T t1 t2))))).(or_introl (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T 
+t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
+(\lambda (t2: T).(pr2 (CSort n) t1 t2))) (\lambda (t2: T).(\lambda (H1: (pr2 
+(CSort n) t1 t2)).(let H_y \def (pr2_gen_csort t1 t2 n H1) in (H0 t2 
+H_y)))))) (\lambda (H0: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall 
+(P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)))).(ex2_ind T (\lambda (t2: 
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)) 
+(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T 
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr2 (CSort n) t1 t2)))) (\lambda (x: T).(\lambda (H1: (((eq T t1 x) \to 
+(\forall (P: Prop).P)))).(\lambda (H2: (pr0 t1 x)).(or_intror (\forall (t2: 
+T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T 
+t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2))) 
+(ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
+(\lambda (t2: T).(pr2 (CSort n) t1 t2)) x H1 (pr2_free (CSort n) t1 x 
+H2)))))) H0)) H))))) (\lambda (c0: C).(\lambda (H: ((\forall (t1: T).(or 
+(\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: 
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 
+t2))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (t1: T).(let H_x \def (H 
+t1) in (let H0 \def H_x in (or_ind (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T 
+t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
+(\lambda (t2: T).(pr2 c0 t1 t2))) (or (\forall (t2: T).((pr2 (CTail k t c0) 
+t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall 
+(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (H1: 
+((\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))))).(match k in K return 
+(\lambda (k0: K).(or (\forall (t2: T).((pr2 (CTail k0 t c0) t1 t2) \to (eq T 
+t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
+(\lambda (t2: T).(pr2 (CTail k0 t c0) t1 t2))))) with [(Bind b) \Rightarrow 
+(match b in B return (\lambda (b0: B).(or (\forall (t2: T).((pr2 (CTail (Bind 
+b0) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to 
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind b0) t c0) t1 
+t2))))) with [Abbr \Rightarrow (let H_x0 \def (dnf_dec t t1 (clen c0)) in 
+(let H2 \def H_x0 in (ex_ind T (\lambda (v: T).(or (subst0 (clen c0) t t1 
+(lift (S O) (clen c0) v)) (eq T t1 (lift (S O) (clen c0) v)))) (or (\forall 
+(t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T 
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) (\lambda (x: T).(\lambda (H3: (or 
+(subst0 (clen c0) t t1 (lift (S O) (clen c0) x)) (eq T t1 (lift (S O) (clen 
+c0) x)))).(or_ind (subst0 (clen c0) t t1 (lift (S O) (clen c0) x)) (eq T t1 
+(lift (S O) (clen c0) x)) (or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) 
+t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall 
+(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) 
+(\lambda (H4: (subst0 (clen c0) t t1 (lift (S O) (clen c0) x))).(let H_x1 
+\def (getl_ctail_clen Abbr t c0) in (let H5 \def H_x1 in (ex_ind nat (\lambda 
+(n: nat).(getl (clen c0) (CTail (Bind Abbr) t c0) (CHead (CSort n) (Bind 
+Abbr) t))) (or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) \to (eq 
+T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
+(\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) (\lambda (x0: 
+nat).(\lambda (H6: (getl (clen c0) (CTail (Bind Abbr) t c0) (CHead (CSort x0) 
+(Bind Abbr) t))).(or_intror (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) 
+t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall 
+(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2))) 
+(ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
+(\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)) (lift (S O) (clen c0) 
+x) (\lambda (H7: (eq T t1 (lift (S O) (clen c0) x))).(\lambda (P: Prop).(let 
+H8 \def (eq_ind T t1 (\lambda (t0: T).(subst0 (clen c0) t t0 (lift (S O) 
+(clen c0) x))) H4 (lift (S O) (clen c0) x) H7) in (subst0_gen_lift_false x t 
+(lift (S O) (clen c0) x) (S O) (clen c0) (clen c0) (le_n (clen c0)) (eq_ind_r 
+nat (plus (S O) (clen c0)) (\lambda (n: nat).(lt (clen c0) n)) (le_n (plus (S 
+O) (clen c0))) (plus (clen c0) (S O)) (plus_comm (clen c0) (S O))) H8 P)))) 
+(pr2_delta (CTail (Bind Abbr) t c0) (CSort x0) t (clen c0) H6 t1 t1 (pr0_refl 
+t1) (lift (S O) (clen c0) x) H4))))) H5)))) (\lambda (H4: (eq T t1 (lift (S 
+O) (clen c0) x))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: 
+T).((pr2 c0 t0 t2) \to (eq T t0 t2)))) H1 (lift (S O) (clen c0) x) H4) in 
+(eq_ind_r T (lift (S O) (clen c0) x) (\lambda (t0: T).(or (\forall (t2: 
+T).((pr2 (CTail (Bind Abbr) t c0) t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda 
+(t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 
+(CTail (Bind Abbr) t c0) t0 t2))))) (or_introl (\forall (t2: T).((pr2 (CTail 
+(Bind Abbr) t c0) (lift (S O) (clen c0) x) t2) \to (eq T (lift (S O) (clen 
+c0) x) t2))) (ex2 T (\lambda (t2: T).((eq T (lift (S O) (clen c0) x) t2) \to 
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) (lift 
+(S O) (clen c0) x) t2))) (\lambda (t2: T).(\lambda (H6: (pr2 (CTail (Bind 
+Abbr) t c0) (lift (S O) (clen c0) x) t2)).(let H_x1 \def (pr2_gen_ctail (Bind 
+Abbr) c0 t (lift (S O) (clen c0) x) t2 H6) in (let H7 \def H_x1 in (or_ind 
+(pr2 c0 (lift (S O) (clen c0) x) t2) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) 
+(Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda 
+(t0: T).(subst0 (clen c0) t t0 t2))) (eq T (lift (S O) (clen c0) x) t2) 
+(\lambda (H8: (pr2 c0 (lift (S O) (clen c0) x) t2)).(H5 t2 H8)) (\lambda (H8: 
+(ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 
+(lift (S O) (clen c0) x) t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 
+t2)))).(ex3_ind T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda 
+(t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda (t0: T).(subst0 (clen c0) 
+t t0 t2)) (eq T (lift (S O) (clen c0) x) t2) (\lambda (x0: T).(\lambda (_: 
+(eq K (Bind Abbr) (Bind Abbr))).(\lambda (H10: (pr0 (lift (S O) (clen c0) x) 
+x0)).(\lambda (H11: (subst0 (clen c0) t x0 t2)).(ex2_ind T (\lambda (t3: 
+T).(eq T x0 (lift (S O) (clen c0) t3))) (\lambda (t3: T).(pr0 x t3)) (eq T 
+(lift (S O) (clen c0) x) t2) (\lambda (x1: T).(\lambda (H12: (eq T x0 (lift 
+(S O) (clen c0) x1))).(\lambda (_: (pr0 x x1)).(let H14 \def (eq_ind T x0 
+(\lambda (t0: T).(subst0 (clen c0) t t0 t2)) H11 (lift (S O) (clen c0) x1) 
+H12) in (subst0_gen_lift_false x1 t t2 (S O) (clen c0) (clen c0) (le_n (clen 
+c0)) (eq_ind_r nat (plus (S O) (clen c0)) (\lambda (n: nat).(lt (clen c0) n)) 
+(le_n (plus (S O) (clen c0))) (plus (clen c0) (S O)) (plus_comm (clen c0) (S 
+O))) H14 (eq T (lift (S O) (clen c0) x) t2)))))) (pr0_gen_lift x x0 (S O) 
+(clen c0) H10)))))) H8)) H7)))))) t1 H4))) H3))) H2))) | Abst \Rightarrow 
+(or_introl (\forall (t2: T).((pr2 (CTail (Bind Abst) t c0) t1 t2) \to (eq T 
+t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
+(\lambda (t2: T).(pr2 (CTail (Bind Abst) t c0) t1 t2))) (\lambda (t2: 
+T).(\lambda (H2: (pr2 (CTail (Bind Abst) t c0) t1 t2)).(let H_x0 \def 
+(pr2_gen_ctail (Bind Abst) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind 
+(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) 
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) 
+(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T 
+(\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) 
+(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq 
+K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
+T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: 
+(eq K (Bind Abst) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: 
+(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Abst) (\lambda (ee: 
+K).(match ee in K return (\lambda (_: K).Prop) with [(Bind b0) \Rightarrow 
+(match b0 in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | 
+Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow 
+False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3)))))) 
+| Void \Rightarrow (or_introl (\forall (t2: T).((pr2 (CTail (Bind Void) t c0) 
+t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall 
+(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Void) t c0) t1 t2))) 
+(\lambda (t2: T).(\lambda (H2: (pr2 (CTail (Bind Void) t c0) t1 t2)).(let 
+H_x0 \def (pr2_gen_ctail (Bind Void) c0 t t1 t2 H2) in (let H3 \def H_x0 in 
+(or_ind (pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) 
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) 
+(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T 
+(\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) 
+(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq 
+K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
+T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: 
+(eq K (Bind Void) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: 
+(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Void) (\lambda (ee: 
+K).(match ee 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 (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) 
+H3))))))]) | (Flat f) \Rightarrow (or_introl (\forall (t2: T).((pr2 (CTail 
+(Flat f) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 
+t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Flat f) t c0) 
+t1 t2))) (\lambda (t2: T).(\lambda (H2: (pr2 (CTail (Flat f) t c0) t1 
+t2)).(let H_x0 \def (pr2_gen_ctail (Flat f) c0 t t1 t2 H2) in (let H3 \def 
+H_x0 in (or_ind (pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Flat f) (Bind 
+Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 
+t2))) (eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: 
+(ex3 T (\lambda (_: T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 
+t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: 
+T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
+T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: 
+(eq K (Flat f) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: (subst0 
+(clen c0) t x0 t2)).(let H8 \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 Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) 
+H4)) H3))))))])) (\lambda (H1: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to 
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))).(ex2_ind T 
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr2 c0 t1 t2)) (or (\forall (t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T 
+t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
+(\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (x: T).(\lambda (H2: 
+(((eq T t1 x) \to (\forall (P: Prop).P)))).(\lambda (H3: (pr2 c0 t1 
+x)).(or_intror (\forall (t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 
+t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
+(\lambda (t2: T).(pr2 (CTail k t c0) t1 t2))) (ex_intro2 T (\lambda (t2: 
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t 
+c0) t1 t2)) x H2 (pr2_ctail c0 t1 x H3 k t)))))) H1)) H0)))))))) c).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/defs.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/defs.ma
new file mode 100644
index 000000000..134910436
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/defs.ma
@@ -0,0 +1,26 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/defs".
+
+include "pr2/defs.ma".
+
+definition nf2:
+ C \to (T \to Prop)
+\def
+ \lambda (c: C).(\lambda (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (eq T t1 
+t2)))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/fwd.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/fwd.ma
new file mode 100644
index 000000000..a9ad986e1
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/fwd.ma
@@ -0,0 +1,116 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/fwd".
+
+include "nf2/defs.ma".
+
+include "pr2/clen.ma".
+
+theorem nf2_gen_base__aux:
+ \forall (k: K).(\forall (t: T).(\forall (u: T).((eq T (THead k u t) t) \to 
+(\forall (P: Prop).P))))
+\def
+ \lambda (k: K).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (u: T).((eq 
+T (THead k u t0) t0) \to (\forall (P: Prop).P)))) (\lambda (n: nat).(\lambda 
+(u: T).(\lambda (H: (eq T (THead k u (TSort n)) (TSort n))).(\lambda (P: 
+Prop).(let H0 \def (eq_ind T (THead k u (TSort n)) (\lambda (ee: T).(match ee 
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef 
+_) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H) in 
+(False_ind P H0)))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (H: (eq T 
+(THead k u (TLRef n)) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind T 
+(THead k u (TLRef n)) (\lambda (ee: T).(match ee in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead _ _ _) \Rightarrow True])) I (TLRef n) H) in (False_ind P H0)))))) 
+(\lambda (k0: K).(\lambda (t0: T).(\lambda (_: ((\forall (u: T).((eq T (THead 
+k u t0) t0) \to (\forall (P: Prop).P))))).(\lambda (t1: T).(\lambda (H0: 
+((\forall (u: T).((eq T (THead k u t1) t1) \to (\forall (P: 
+Prop).P))))).(\lambda (u: T).(\lambda (H1: (eq T (THead k u (THead k0 t0 t1)) 
+(THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \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 u (THead 
+k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H3 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | 
+(TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2])) (THead k u (THead 
+k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H4 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead 
+k0 t0 t1) | (TLRef _) \Rightarrow (THead k0 t0 t1) | (THead _ _ t2) 
+\Rightarrow t2])) (THead k u (THead k0 t0 t1)) (THead k0 t0 t1) H1) in 
+(\lambda (_: (eq T u t0)).(\lambda (H6: (eq K k k0)).(let H7 \def (eq_ind K k 
+(\lambda (k1: K).(\forall (u0: T).((eq T (THead k1 u0 t1) t1) \to (\forall 
+(P0: Prop).P0)))) H0 k0 H6) in (H7 t0 H4 P))))) H3)) H2)))))))))) t)).
+
+theorem nf2_gen_lref:
+ \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abbr) u)) \to ((nf2 c (TLRef i)) \to (\forall (P: Prop).P))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H0: ((\forall (t2: T).((pr2 
+c (TLRef i) t2) \to (eq T (TLRef i) t2))))).(\lambda (P: 
+Prop).(lift_gen_lref_false (S i) O i (le_O_n i) (le_n (plus O (S i))) u (H0 
+(lift (S i) O u) (pr2_delta c d u i H (TLRef i) (TLRef i) (pr0_refl (TLRef 
+i)) (lift (S i) O u) (subst0_lref u i))) P))))))).
+
+theorem nf2_gen_abst:
+ \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abst) u 
+t)) \to (land (nf2 c u) (nf2 (CHead c (Bind Abst) u) t)))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: 
+T).((pr2 c (THead (Bind Abst) u t) t2) \to (eq T (THead (Bind Abst) u t) 
+t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall (t2: 
+T).((pr2 (CHead c (Bind Abst) u) t t2) \to (eq T t t2))) (\lambda (t2: 
+T).(\lambda (H0: (pr2 c u t2)).(let H1 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | 
+(TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst) 
+u t) (THead (Bind Abst) t2 t) (H (THead (Bind Abst) t2 t) (pr2_head_1 c u t2 
+H0 (Bind Abst) t))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 c u 
+t0)) H0 u H1) in (eq_ind T u (\lambda (t0: T).(eq T u t0)) (refl_equal T u) 
+t2 H1))))) (\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind Abst) u) t 
+t2)).(let H1 \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 (Bind Abst) u t) (THead (Bind Abst) u t2) (H 
+(THead (Bind Abst) u t2) (let H_y \def (pr2_gen_cbind Abst c u t t2 H0) in 
+H_y))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 (CHead c (Bind 
+Abst) u) t t0)) H0 t H1) in (eq_ind T t (\lambda (t0: T).(eq T t t0)) 
+(refl_equal T t) t2 H1))))))))).
+
+theorem nf2_gen_cast:
+ \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Flat Cast) u 
+t)) \to (\forall (P: Prop).P))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (nf2 c (THead 
+(Flat Cast) u t))).(\lambda (P: Prop).(nf2_gen_base__aux (Flat Cast) t u (H t 
+(pr2_free c (THead (Flat Cast) u t) t (pr0_epsilon t t (pr0_refl t) u))) 
+P))))).
+
+theorem nf2_gen_flat:
+ \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c 
+(THead (Flat f) u t)) \to (land (nf2 c u) (nf2 c t))))))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: 
+((\forall (t2: T).((pr2 c (THead (Flat f) u t) t2) \to (eq T (THead (Flat f) 
+u t) t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall 
+(t2: T).((pr2 c t t2) \to (eq T t t2))) (\lambda (t2: T).(\lambda (H0: (pr2 c 
+u t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return 
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | 
+(THead _ t0 _) \Rightarrow t0])) (THead (Flat f) u t) (THead (Flat f) t2 t) 
+(H (THead (Flat f) t2 t) (pr2_head_1 c u t2 H0 (Flat f) t))) in H1))) 
+(\lambda (t2: T).(\lambda (H0: (pr2 c t t2)).(let H1 \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 (Flat f) u t) (THead (Flat f) u t2) (H (THead (Flat f) u t2) 
+(pr2_head_2 c u t t2 (Flat f) (pr2_cflat c t t2 H0 f u)))) in H1)))))))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/iso.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/iso.ma
new file mode 100644
index 000000000..c95449423
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/iso.ma
@@ -0,0 +1,129 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/iso".
+
+include "nf2/pr3.ma".
+
+include "pr3/fwd.ma".
+
+include "iso/fwd.ma".
+
+theorem nf2_iso_appls_lref:
+ \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: 
+TList).(\forall (u: T).((pr3 c (THeads (Flat Appl) vs (TLRef i)) u) \to (iso 
+(THeads (Flat Appl) vs (TLRef i)) u))))))
+\def
+ \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda 
+(vs: TList).(TList_ind (\lambda (t: TList).(\forall (u: T).((pr3 c (THeads 
+(Flat Appl) t (TLRef i)) u) \to (iso (THeads (Flat Appl) t (TLRef i)) u)))) 
+(\lambda (u: T).(\lambda (H0: (pr3 c (TLRef i) u)).(let H_y \def 
+(nf2_pr3_unfold c (TLRef i) u H0 H) in (let H1 \def (eq_ind_r T u (\lambda 
+(t: T).(pr3 c (TLRef i) t)) H0 (TLRef i) H_y) in (eq_ind T (TLRef i) (\lambda 
+(t: T).(iso (TLRef i) t)) (iso_lref i i) u H_y))))) (\lambda (t: T).(\lambda 
+(t0: TList).(\lambda (H0: ((\forall (u: T).((pr3 c (THeads (Flat Appl) t0 
+(TLRef i)) u) \to (iso (THeads (Flat Appl) t0 (TLRef i)) u))))).(\lambda (u: 
+T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef 
+i))) u)).(let H2 \def (pr3_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) u 
+H1) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T u (THead 
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
+t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat 
+Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: 
+T).(pr3 (CHead c (Bind b) u0) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat 
+Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: 
+T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2)))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef 
+i))) u) (\lambda (H3: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T u 
+(THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T u (THead (Flat 
+Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))) (iso 
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H4: (eq T u (THead (Flat Appl) x0 
+x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 
+(TLRef i)) x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(iso 
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (iso_head (Flat 
+Appl) t x0 (THeads (Flat Appl) t0 (TLRef i)) x1) u H4)))))) H3)) (\lambda 
+(H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat 
+Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: 
+T).(pr3 (CHead c (Bind b) u0) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind 
+Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c t u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(t2: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) u0) z1 
+t2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda 
+(_: (pr3 c (THead (Bind Abbr) x2 x3) u)).(\lambda (_: (pr3 c t x2)).(\lambda 
+(H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 
+x1))).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) 
+u0) x1 x3))))).(let H_y \def (H0 (THead (Bind Abst) x0 x1) H6) in 
+(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H_y (iso (THead (Flat Appl) t 
+(THeads (Flat Appl) t0 (TLRef i))) u))))))))))) H3)) (\lambda (H3: (ex6_6 B T 
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(S O) O u2) z2)) u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
+(CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat 
+Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: 
+T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) 
+u) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 
+Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
+x0) x1 x2))).(\lambda (_: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift 
+(S O) O x4) x3)) u)).(\lambda (_: (pr3 c t x4)).(\lambda (_: (pr3 c x1 
+x5)).(\lambda (_: (pr3 (CHead c (Bind x0) x5) x2 x3)).(let H_y \def (H0 
+(THead (Bind x0) x1 x2) H5) in (iso_flats_lref_bind_false Appl x0 i x1 x2 t0 
+H_y (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) 
+u))))))))))))))) H3)) H2))))))) vs)))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/lift1.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/lift1.ma
new file mode 100644
index 000000000..f939e56fa
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/lift1.ma
@@ -0,0 +1,84 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/lift1".
+
+include "nf2/props.ma".
+
+include "drop1/defs.ma".
+
+theorem nf2_lift1:
+ \forall (e: C).(\forall (hds: PList).(\forall (c: C).(\forall (t: T).((drop1 
+hds c e) \to ((nf2 e t) \to (nf2 c (lift1 hds t)))))))
+\def
+ \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
+(c: C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p 
+t))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c 
+e)).(\lambda (H0: (nf2 e t)).(let H1 \def (match H in drop1 return (\lambda 
+(p: PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (drop1 p c0 
+c1)).((eq PList p PNil) \to ((eq C c0 c) \to ((eq C c1 e) \to (nf2 c 
+t)))))))) with [(drop1_nil c0) \Rightarrow (\lambda (_: (eq PList PNil 
+PNil)).(\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq C c0 e)).(eq_ind C c 
+(\lambda (c1: C).((eq C c1 e) \to (nf2 c t))) (\lambda (H4: (eq C c 
+e)).(eq_ind C e (\lambda (c1: C).(nf2 c1 t)) H0 c (sym_eq C c e H4))) c0 
+(sym_eq C c0 c H2) H3)))) | (drop1_cons c1 c2 h d H1 c3 hds0 H2) \Rightarrow 
+(\lambda (H3: (eq PList (PCons h d hds0) PNil)).(\lambda (H4: (eq C c1 
+c)).(\lambda (H5: (eq C c3 e)).((let H6 \def (eq_ind PList (PCons h d hds0) 
+(\lambda (e0: PList).(match e0 in PList return (\lambda (_: PList).Prop) with 
+[PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in 
+(False_ind ((eq C c1 c) \to ((eq C c3 e) \to ((drop h d c1 c2) \to ((drop1 
+hds0 c2 c3) \to (nf2 c t))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList 
+PNil) (refl_equal C c) (refl_equal C e))))))) (\lambda (n: nat).(\lambda (n0: 
+nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).(\forall (t: 
+T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p t)))))))).(\lambda (c: 
+C).(\lambda (t: T).(\lambda (H0: (drop1 (PCons n n0 p) c e)).(\lambda (H1: 
+(nf2 e t)).(let H2 \def (match H0 in drop1 return (\lambda (p0: 
+PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (drop1 p0 c0 c1)).((eq 
+PList p0 (PCons n n0 p)) \to ((eq C c0 c) \to ((eq C c1 e) \to (nf2 c (lift n 
+n0 (lift1 p t)))))))))) with [(drop1_nil c0) \Rightarrow (\lambda (H2: (eq 
+PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq C c0 
+e)).((let H5 \def (eq_ind PList PNil (\lambda (e0: PList).(match e0 in PList 
+return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) 
+\Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c0 c) \to ((eq 
+C c0 e) \to (nf2 c (lift n n0 (lift1 p t))))) H5)) H3 H4)))) | (drop1_cons c1 
+c2 h d H2 c3 hds0 H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds0) 
+(PCons n n0 p))).(\lambda (H5: (eq C c1 c)).(\lambda (H6: (eq C c3 e)).((let 
+H7 \def (f_equal PList PList (\lambda (e0: PList).(match e0 in PList return 
+(\lambda (_: PList).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p0) 
+\Rightarrow p0])) (PCons h d hds0) (PCons n n0 p) H4) in ((let H8 \def 
+(f_equal PList nat (\lambda (e0: PList).(match e0 in PList return (\lambda 
+(_: PList).nat) with [PNil \Rightarrow d | (PCons _ n1 _) \Rightarrow n1])) 
+(PCons h d hds0) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat 
+(\lambda (e0: PList).(match e0 in PList return (\lambda (_: PList).nat) with 
+[PNil \Rightarrow h | (PCons n1 _ _) \Rightarrow n1])) (PCons h d hds0) 
+(PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to 
+((eq PList hds0 p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n1 d c1 c2) 
+\to ((drop1 hds0 c2 c3) \to (nf2 c (lift n n0 (lift1 p t)))))))))) (\lambda 
+(H10: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds0 p) \to 
+((eq C c1 c) \to ((eq C c3 e) \to ((drop n n1 c1 c2) \to ((drop1 hds0 c2 c3) 
+\to (nf2 c (lift n n0 (lift1 p t))))))))) (\lambda (H11: (eq PList hds0 
+p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c) \to ((eq C c3 e) \to 
+((drop n n0 c1 c2) \to ((drop1 p0 c2 c3) \to (nf2 c (lift n n0 (lift1 p 
+t)))))))) (\lambda (H12: (eq C c1 c)).(eq_ind C c (\lambda (c0: C).((eq C c3 
+e) \to ((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to (nf2 c (lift n n0 (lift1 p 
+t))))))) (\lambda (H13: (eq C c3 e)).(eq_ind C e (\lambda (c0: C).((drop n n0 
+c c2) \to ((drop1 p c2 c0) \to (nf2 c (lift n n0 (lift1 p t)))))) (\lambda 
+(H14: (drop n n0 c c2)).(\lambda (H15: (drop1 p c2 e)).(nf2_lift c2 (lift1 p 
+t) (H c2 t H15 H1) c n n0 H14))) c3 (sym_eq C c3 e H13))) c1 (sym_eq C c1 c 
+H12))) hds0 (sym_eq PList hds0 p H11))) d (sym_eq nat d n0 H10))) h (sym_eq 
+nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n 
+n0 p)) (refl_equal C c) (refl_equal C e))))))))))) hds)).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/pr3.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/pr3.ma
new file mode 100644
index 000000000..9c6dc2913
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/pr3.ma
@@ -0,0 +1,52 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/pr3".
+
+include "nf2/defs.ma".
+
+include "pr3/pr3.ma".
+
+theorem nf2_pr3_unfold:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to ((nf2 c 
+t1) \to (eq T t1 t2)))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((nf2 c t) \to (eq T t 
+t0)))) (\lambda (t: T).(\lambda (H0: (nf2 c t)).(H0 t (pr2_free c t t 
+(pr0_refl t))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 
+t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: (((nf2 c t0) 
+\to (eq T t0 t4)))).(\lambda (H3: (nf2 c t3)).(let H4 \def H3 in (let H5 \def 
+(eq_ind T t3 (\lambda (t: T).(nf2 c t)) H3 t0 (H4 t0 H0)) in (let H6 \def 
+(eq_ind T t3 (\lambda (t: T).(pr2 c t t0)) H0 t0 (H4 t0 H0)) in (eq_ind_r T 
+t0 (\lambda (t: T).(eq T t t4)) (H2 H5) t3 (H4 t0 H0)))))))))))) t1 t2 H)))).
+
+theorem nf2_pr3_confluence:
+ \forall (c: C).(\forall (t1: T).((nf2 c t1) \to (\forall (t2: T).((nf2 c t2) 
+\to (\forall (t: T).((pr3 c t t1) \to ((pr3 c t t2) \to (eq T t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (H: (nf2 c t1)).(\lambda (t2: 
+T).(\lambda (H0: (nf2 c t2)).(\lambda (t: T).(\lambda (H1: (pr3 c t 
+t1)).(\lambda (H2: (pr3 c t t2)).(ex2_ind T (\lambda (t0: T).(pr3 c t2 t0)) 
+(\lambda (t0: T).(pr3 c t1 t0)) (eq T t1 t2) (\lambda (x: T).(\lambda (H3: 
+(pr3 c t2 x)).(\lambda (H4: (pr3 c t1 x)).(let H_y \def (nf2_pr3_unfold c t1 
+x H4 H) in (let H5 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t1 t0)) H4 t1 
+H_y) in (let H6 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t2 t0)) H3 t1 H_y) 
+in (let H_y0 \def (nf2_pr3_unfold c t2 t1 H6 H0) in (let H7 \def (eq_ind T t2 
+(\lambda (t0: T).(pr3 c t0 t1)) H6 t1 H_y0) in (eq_ind_r T t1 (\lambda (t0: 
+T).(eq T t1 t0)) (refl_equal T t1) t2 H_y0))))))))) (pr3_confluence c t t2 H2 
+t1 H1))))))))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/props.ma
new file mode 100644
index 000000000..89f766ee4
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/props.ma
@@ -0,0 +1,199 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/props".
+
+include "nf2/defs.ma".
+
+include "pr2/fwd.ma".
+
+theorem nf2_sort:
+ \forall (c: C).(\forall (n: nat).(nf2 c (TSort n)))
+\def
+ \lambda (c: C).(\lambda (n: nat).(\lambda (t2: T).(\lambda (H: (pr2 c (TSort 
+n) t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal 
+T (TSort n)) t2 (pr2_gen_sort c t2 n H))))).
+
+theorem nf2_abst:
+ \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (b: B).(\forall (v: 
+T).(\forall (t: T).((nf2 (CHead c (Bind b) v) t) \to (nf2 c (THead (Bind 
+Abst) u t))))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2) 
+\to (eq T u t2))))).(\lambda (b: B).(\lambda (v: T).(\lambda (t: T).(\lambda 
+(H0: ((\forall (t2: T).((pr2 (CHead c (Bind b) v) t t2) \to (eq T t 
+t2))))).(\lambda (t2: T).(\lambda (H1: (pr2 c (THead (Bind Abst) u t) 
+t2)).(let H2 \def (pr2_gen_abst c u t t2 H1) 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).(pr2 c u u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
+(b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t t3))))) (eq T (THead 
+(Bind Abst) u t) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T t2 
+(THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2 c u x0)).(\lambda (H5: 
+((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t 
+x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T (THead 
+(Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) u x0 t 
+x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 b v))) t2 H3)))))) 
+H2)))))))))).
+
+theorem nf2_appl_lref:
+ \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (i: nat).((nf2 c 
+(TLRef i)) \to (nf2 c (THead (Flat Appl) u (TLRef i)))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2) 
+\to (eq T u t2))))).(\lambda (i: nat).(\lambda (H0: ((\forall (t2: T).((pr2 c 
+(TLRef i) t2) \to (eq T (TLRef i) t2))))).(\lambda (t2: T).(\lambda (H1: (pr2 
+c (THead (Flat Appl) u (TLRef i)) t2)).(let H2 \def (pr2_gen_appl c u (TLRef 
+i) t2 H1) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
+(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u 
+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 (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq 
+T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c u 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 T (THead (Flat Appl) u (TLRef i)) t2) (\lambda (H3: (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
+(TLRef i) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
+(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))) (eq T (THead (Flat 
+Appl) u (TLRef i)) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T 
+t2 (THead (Flat Appl) x0 x1))).(\lambda (H5: (pr2 c u x0)).(\lambda (H6: (pr2 
+c (TLRef i) x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t: T).(eq T 
+(THead (Flat Appl) u (TLRef i)) t)) (let H7 \def (eq_ind_r T x1 (\lambda (t: 
+T).(pr2 c (TLRef i) t)) H6 (TLRef i) (H0 x1 H6)) in (eq_ind T (TLRef i) 
+(\lambda (t: T).(eq T (THead (Flat Appl) u (TLRef i)) (THead (Flat Appl) x0 
+t))) (let H8 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c u t)) H5 u (H x0 H5)) 
+in (eq_ind T u (\lambda (t: T).(eq T (THead (Flat Appl) u (TLRef i)) (THead 
+(Flat Appl) t (TLRef i)))) (refl_equal T (THead (Flat Appl) u (TLRef i))) x0 
+(H x0 H5))) x1 (H0 x1 H6))) t2 H4)))))) H3)) (\lambda (H3: (ex4_4 T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u 
+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))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c u 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))))))) (eq T (THead (Flat Appl) u (TLRef i)) t2) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (eq T 
+(TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H5: (eq T t2 (THead (Bind 
+Abbr) x2 x3))).(\lambda (_: (pr2 c u x2)).(\lambda (_: ((\forall (b: 
+B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) x1 x3))))).(eq_ind_r T (THead 
+(Bind Abbr) x2 x3) (\lambda (t: T).(eq T (THead (Flat Appl) u (TLRef i)) t)) 
+(let H8 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 
+x1) H4) in (False_ind (eq T (THead (Flat Appl) u (TLRef i)) (THead (Bind 
+Abbr) x2 x3)) H8)) t2 H5))))))))) H3)) (\lambda (H3: (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c u u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind b) y1 
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat 
+Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u 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 T (THead (Flat Appl) u (TLRef i)) t2) 
+(\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda 
+(x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H5: (eq 
+T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda (H6: (eq T t2 (THead (Bind x0) 
+x5 (THead (Flat Appl) (lift (S O) O x4) x3)))).(\lambda (_: (pr2 c u 
+x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: (pr2 (CHead c (Bind x0) x5) x2 
+x3)).(eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) 
+x3)) (\lambda (t: T).(eq T (THead (Flat Appl) u (TLRef i)) t)) (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 (Bind x0) x1 x2) H5) in 
+(False_ind (eq T (THead (Flat Appl) u (TLRef i)) (THead (Bind x0) x5 (THead 
+(Flat Appl) (lift (S O) O x4) x3))) H10)) t2 H6))))))))))))) H3)) H2)))))))).
+
+theorem nf2_lref_abst:
+ \forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c 
+(CHead e (Bind Abst) u)) \to (nf2 c (TLRef i))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H: (getl i c (CHead e (Bind Abst) u))).(\lambda (t2: T).(\lambda (H0: (pr2 c 
+(TLRef i) t2)).(let H1 \def (pr2_gen_lref c t2 i H0) in (or_ind (eq T t2 
+(TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c (CHead d 
+(Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift (S i) O 
+u0))))) (eq T (TLRef i) t2) (\lambda (H2: (eq T t2 (TLRef i))).(eq_ind_r T 
+(TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2 
+H2)) (\lambda (H2: (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c 
+(CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift 
+(S i) O u0)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u0: T).(getl i c 
+(CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift 
+(S i) O u0)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
+(H3: (getl i c (CHead x0 (Bind Abbr) x1))).(\lambda (H4: (eq T t2 (lift (S i) 
+O x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T (TLRef i) t)) 
+(let H5 \def (eq_ind C (CHead e (Bind Abst) u) (\lambda (c0: C).(getl i c 
+c0)) H (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H 
+(CHead x0 (Bind Abbr) x1) H3)) in (let H6 \def (eq_ind C (CHead e (Bind Abst) 
+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 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 x0 (Bind 
+Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H (CHead x0 (Bind Abbr) x1) 
+H3)) in (False_ind (eq T (TLRef i) (lift (S i) O x1)) H6))) t2 H4))))) H2)) 
+H1)))))))).
+
+theorem nf2_lift:
+ \forall (d: C).(\forall (t: T).((nf2 d t) \to (\forall (c: C).(\forall (h: 
+nat).(\forall (i: nat).((drop h i c d) \to (nf2 c (lift h i t))))))))
+\def
+ \lambda (d: C).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).((pr2 d t t2) 
+\to (eq T t t2))))).(\lambda (c: C).(\lambda (h: nat).(\lambda (i: 
+nat).(\lambda (H0: (drop h i c d)).(\lambda (t2: T).(\lambda (H1: (pr2 c 
+(lift h i t) t2)).(let H2 \def (pr2_gen_lift c t t2 h i H1 d H0) in (ex2_ind 
+T (\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t t3)) 
+(eq T (lift h i t) t2) (\lambda (x: T).(\lambda (H3: (eq T t2 (lift h i 
+x))).(\lambda (H4: (pr2 d t x)).(eq_ind_r T (lift h i x) (\lambda (t0: T).(eq 
+T (lift h i t) t0)) (let H_y \def (H x H4) in (let H5 \def (eq_ind_r T x 
+(\lambda (t0: T).(pr2 d t t0)) H4 t H_y) in (eq_ind T t (\lambda (t0: T).(eq 
+T (lift h i t) (lift h i t0))) (refl_equal T (lift h i t)) x H_y))) t2 H3)))) 
+H2)))))))))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc1/defs.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc1/defs.ma
new file mode 100644
index 000000000..ac831f160
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc1/defs.ma
@@ -0,0 +1,26 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pc1/defs".
+
+include "pr1/defs.ma".
+
+definition pc1:
+ T \to (T \to Prop)
+\def
+ \lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda 
+(t: T).(pr1 t2 t)))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc1/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc1/props.ma
new file mode 100644
index 000000000..f670f0eaf
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc1/props.ma
@@ -0,0 +1,118 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pc1/props".
+
+include "pc1/defs.ma".
+
+include "pr1/pr1.ma".
+
+theorem pc1_pr0_r:
+ \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pc1 t1 t2)))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(ex_intro2 T 
+(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t2 (pr1_pr0 t1 t2 H) 
+(pr1_r t2)))).
+
+theorem pc1_pr0_x:
+ \forall (t1: T).(\forall (t2: T).((pr0 t2 t1) \to (pc1 t1 t2)))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t2 t1)).(ex_intro2 T 
+(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t1 (pr1_r t1) 
+(pr1_pr0 t2 t1 H)))).
+
+theorem pc1_pr0_u:
+ \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: T).((pc1 t2 
+t3) \to (pc1 t1 t3)))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr0 t1 t2)).(\lambda (t3: 
+T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t: 
+T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x: 
+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_u 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_r t) (pr1_r t)).
+
+theorem pc1_s:
+ \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (pc1 t2 t1)))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(let H0 \def H in 
+(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t2 
+t1) (\lambda (x: T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2 
+x)).(ex_intro2 T (\lambda (t: T).(pr1 t2 t)) (\lambda (t: T).(pr1 t1 t)) x H2 
+H1)))) H0)))).
+
+theorem pc1_head_1:
+ \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t: T).(\forall 
+(k: K).(pc1 (THead k u1 t) (THead k u2 t))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t: 
+T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t0: T).(pr1 u1 t0)) 
+(\lambda (t0: T).(pr1 u2 t0)) (pc1 (THead k u1 t) (THead k u2 t)) (\lambda 
+(x: T).(\lambda (H1: (pr1 u1 x)).(\lambda (H2: (pr1 u2 x)).(ex_intro2 T 
+(\lambda (t0: T).(pr1 (THead k u1 t) t0)) (\lambda (t0: T).(pr1 (THead k u2 
+t) t0)) (THead k x t) (pr1_head_1 u1 x H1 t k) (pr1_head_1 u2 x H2 t k))))) 
+H0)))))).
+
+theorem pc1_head_2:
+ \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (u: T).(\forall 
+(k: K).(pc1 (THead k u t1) (THead k u t2))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (u: 
+T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t)) 
+(\lambda (t: T).(pr1 t2 t)) (pc1 (THead k u t1) (THead k u t2)) (\lambda (x: 
+T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda 
+(t: T).(pr1 (THead k u t1) t)) (\lambda (t: T).(pr1 (THead k u t2) t)) (THead 
+k u x) (pr1_head_2 t1 x H1 u k) (pr1_head_2 t2 x H2 u k))))) H0)))))).
+
+theorem pc1_t:
+ \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (\forall (t3: T).((pc1 t2 
+t3) \to (pc1 t1 t3)))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(\lambda (t3: 
+T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t: 
+T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x: 
+T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(let H4 \def H in 
+(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t1 
+t3) (\lambda (x0: T).(\lambda (H5: (pr1 t1 x0)).(\lambda (H6: (pr1 t2 
+x0)).(ex2_ind T (\lambda (t: T).(pr1 x0 t)) (\lambda (t: T).(pr1 x t)) (pc1 
+t1 t3) (\lambda (x1: T).(\lambda (H7: (pr1 x0 x1)).(\lambda (H8: (pr1 x 
+x1)).(ex_intro2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x1 
+(pr1_t x0 t1 H5 x1 H7) (pr1_t x t3 H3 x1 H8))))) (pr1_confluence t2 x0 H6 x 
+H2))))) H4))))) H1)))))).
+
+theorem pc1_pr0_u2:
+ \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pc1 t0 
+t2) \to (pc1 t1 t2)))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr0 t0 t1)).(\lambda (t2: 
+T).(\lambda (H0: (pc1 t0 t2)).(pc1_t t0 t1 (pc1_pr0_x t1 t0 H) t2 H0))))).
+
+theorem pc1_head:
+ \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t1: T).(\forall 
+(t2: T).((pc1 t1 t2) \to (\forall (k: K).(pc1 (THead k u1 t1) (THead k u2 
+t2))))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (H0: (pc1 t1 t2)).(\lambda (k: K).(pc1_t (THead 
+k u2 t1) (THead k u1 t1) (pc1_head_1 u1 u2 H t1 k) (THead k u2 t2) 
+(pc1_head_2 t1 t2 H0 u2 k)))))))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/defs.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/defs.ma
new file mode 100644
index 000000000..6a652e41b
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/defs.ma
@@ -0,0 +1,33 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/defs".
+
+include "pr3/defs.ma".
+
+definition pc3:
+ C \to (T \to (T \to Prop))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr3 
+c t1 t)) (\lambda (t: T).(pr3 c t2 t))))).
+
+inductive pc3_left (c:C): T \to (T \to Prop) \def
+| pc3_left_r: \forall (t: T).(pc3_left c t t)
+| pc3_left_ur: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
+(t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3)))))
+| pc3_left_ux: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
+(t3: T).((pc3_left c t1 t3) \to (pc3_left c t2 t3))))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/props.ma
new file mode 100644
index 000000000..3819b5d36
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/props.ma
@@ -0,0 +1,440 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/props".
+
+include "pc3/defs.ma".
+
+include "pr3/pr3.ma".
+
+theorem clear_pc3_trans:
+ \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pc3 c2 t1 t2) \to 
+(\forall (c1: C).((clear c1 c2) \to (pc3 c1 t1 t2))))))
+\def
+ \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c2 t1 
+t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def H in (ex2_ind 
+T (\lambda (t: T).(pr3 c2 t1 t)) (\lambda (t: T).(pr3 c2 t2 t)) (pc3 c1 t1 
+t2) (\lambda (x: T).(\lambda (H2: (pr3 c2 t1 x)).(\lambda (H3: (pr3 c2 t2 
+x)).(ex_intro2 T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2 
+t)) x (clear_pr3_trans c2 t1 x H2 c1 H0) (clear_pr3_trans c2 t2 x H3 c1 
+H0))))) H1))))))).
+
+theorem pc3_pr2_r:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pc3 c 
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
+t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) 
+t2 (pr3_pr2 c t1 t2 H) (pr3_refl c t2))))).
+
+theorem pc3_pr2_x:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t2 t1) \to (pc3 c 
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t2 
+t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) 
+t1 (pr3_refl c t1) (pr3_pr2 c t2 t1 H))))).
+
+theorem pc3_pr3_r:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (pc3 c 
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
+t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) 
+t2 H (pr3_refl c t2))))).
+
+theorem pc3_pr3_x:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t2 t1) \to (pc3 c 
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t2 
+t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) 
+t1 (pr3_refl c t1) H)))).
+
+theorem pc3_pr3_t:
+ \forall (c: C).(\forall (t1: T).(\forall (t0: T).((pr3 c t1 t0) \to (\forall 
+(t2: T).((pr3 c t2 t0) \to (pc3 c t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H: (pr3 c t1 
+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
+ \lambda (c: C).(\lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr3 c t t0)) 
+(\lambda (t0: T).(pr3 c t t0)) t (pr3_refl c t) (pr3_refl c t))).
+
+theorem pc3_s:
+ \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pc3 c t1 t2) \to (pc3 c 
+t2 t1))))
+\def
+ \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc3 c t1 
+t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: 
+T).(pr3 c t2 t)) (pc3 c t2 t1) (\lambda (x: T).(\lambda (H1: (pr3 c t1 
+x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t2 t)) 
+(\lambda (t: T).(pr3 c t1 t)) x H2 H1)))) H0))))).
+
+theorem pc3_thin_dx:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall 
+(u: T).(\forall (f: F).(pc3 c (THead (Flat f) u t1) (THead (Flat f) u 
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 
+t2)).(\lambda (u: T).(\lambda (f: F).(let H0 \def H in (ex2_ind T (\lambda 
+(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 c (THead (Flat f) u 
+t1) (THead (Flat f) u t2)) (\lambda (x: T).(\lambda (H1: (pr3 c t1 
+x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c (THead 
+(Flat f) u t1) t)) (\lambda (t: T).(pr3 c (THead (Flat f) u t2) t)) (THead 
+(Flat f) u x) (pr3_thin_dx c t1 x H1 u f) (pr3_thin_dx c t2 x H2 u f))))) 
+H0))))))).
+
+theorem pc3_head_1:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall 
+(k: K).(\forall (t: T).(pc3 c (THead k u1 t) (THead k u2 t)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 
+u2)).(\lambda (k: K).(\lambda (t: T).(let H0 \def H in (ex2_ind T (\lambda 
+(t0: T).(pr3 c u1 t0)) (\lambda (t0: T).(pr3 c u2 t0)) (pc3 c (THead k u1 t) 
+(THead k u2 t)) (\lambda (x: T).(\lambda (H1: (pr3 c u1 x)).(\lambda (H2: 
+(pr3 c u2 x)).(ex_intro2 T (\lambda (t0: T).(pr3 c (THead k u1 t) t0)) 
+(\lambda (t0: T).(pr3 c (THead k u2 t) t0)) (THead k x t) (pr3_head_12 c u1 x 
+H1 k t t (pr3_refl (CHead c k x) t)) (pr3_head_12 c u2 x H2 k t t (pr3_refl 
+(CHead c k x) t)))))) H0))))))).
+
+theorem pc3_head_2:
+ \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall 
+(k: K).((pc3 (CHead c k u) t1 t2) \to (pc3 c (THead k u t1) (THead k u 
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(k: K).(\lambda (H: (pc3 (CHead c k u) t1 t2)).(let H0 \def H in (ex2_ind T 
+(\lambda (t: T).(pr3 (CHead c k u) t1 t)) (\lambda (t: T).(pr3 (CHead c k u) 
+t2 t)) (pc3 c (THead k u t1) (THead k u t2)) (\lambda (x: T).(\lambda (H1: 
+(pr3 (CHead c k u) t1 x)).(\lambda (H2: (pr3 (CHead c k u) t2 x)).(ex_intro2 
+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_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))))))
+\def
+ \lambda (t2: T).(\lambda (c: C).(\lambda (t1: T).(\lambda (H: (pc3 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)).(let H4 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: 
+T).(pr3 c t2 t)) (pc3 c t1 t3) (\lambda (x0: T).(\lambda (H5: (pr3 c t1 
+x0)).(\lambda (H6: (pr3 c t2 x0)).(ex2_ind T (\lambda (t: T).(pr3 c x0 t)) 
+(\lambda (t: T).(pr3 c x t)) (pc3 c t1 t3) (\lambda (x1: T).(\lambda (H7: 
+(pr3 c x0 x1)).(\lambda (H8: (pr3 c x x1)).(pc3_pr3_t c t1 x1 (pr3_t x0 t1 c 
+H5 x1 H7) t3 (pr3_t x t3 c H3 x1 H8))))) (pr3_confluence c t2 x0 H6 x H2))))) 
+H4))))) H1))))))).
+
+theorem pc3_pr2_u2:
+ \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall 
+(t2: T).((pc3 c t0 t2) \to (pc3 c t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0 
+t1)).(\lambda (t2: T).(\lambda (H0: (pc3 c t0 t2)).(pc3_t t0 c t1 (pc3_pr2_x 
+c t1 t0 H) t2 H0)))))).
+
+theorem pc3_head_12:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall 
+(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u2) t1 t2) \to (pc3 
+c (THead k u1 t1) (THead k u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 
+u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 
+(CHead c k u2) t1 t2)).(pc3_t (THead k u2 t1) c (THead k u1 t1) (pc3_head_1 c 
+u1 u2 H k t1) (THead k u2 t2) (pc3_head_2 c u2 t1 t2 k H0))))))))).
+
+theorem pc3_head_21:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall 
+(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u1) t1 t2) \to (pc3 
+c (THead k u1 t1) (THead k u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 
+u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 
+(CHead c k u1) t1 t2)).(pc3_t (THead k u1 t2) c (THead k u1 t1) (pc3_head_2 c 
+u1 t1 t2 k H0) (THead k u2 t2) (pc3_head_1 c u1 u2 H k t2))))))))).
+
+theorem pc3_pr0_pr2_t:
+ \forall (u1: T).(\forall (u2: T).((pr0 u2 u1) \to (\forall (c: C).(\forall 
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 
+(CHead c k u1) t1 t2))))))))
+\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)))))))))).
+
+theorem pc3_pr2_pr2_t:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u2 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))))))))
+\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)).((match k in 
+K return (\lambda (k0: K).((clear (CHead c k0 u2) (CHead d0 (Bind Abbr) u0)) 
+\to (pc3 (CHead c k0 u1) t0 t3))) with [(Bind b) \Rightarrow (\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 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 (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))) | (Flat f) \Rightarrow 
+(\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)))]) (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)))))).
+
+theorem pc3_pr2_pr3_t:
+ \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall 
+(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u2 u1) \to 
+(pc3 (CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2) 
+(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u2 u1) \to (pc3 
+(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c 
+u2 u1)).(pc3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_: 
+(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u2 u1) 
+\to (pc3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u2 
+u1)).(pc3_t t0 (CHead c k u1) t3 (pc3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2 
+u1 H3)))))))))) t1 t2 H)))))).
+
+theorem pc3_pr3_pc3_t:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u2 u1) \to (\forall 
+(t1: T).(\forall (t2: T).(\forall (k: K).((pc3 (CHead c k u2) t1 t2) \to (pc3 
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u2 
+u1)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall 
+(t2: T).(\forall (k: K).((pc3 (CHead c k t) t1 t2) \to (pc3 (CHead c k t0) t1 
+t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: 
+K).(\lambda (H0: (pc3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda 
+(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 
+t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pc3 
+(CHead c k t2) t4 t5) \to (pc3 (CHead c k t3) t4 t5))))))).(\lambda (t0: 
+T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pc3 (CHead c k t1) t0 
+t4)).(H2 t0 t4 k (let H4 \def H3 in (ex2_ind T (\lambda (t: T).(pr3 (CHead c 
+k t1) t0 t)) (\lambda (t: T).(pr3 (CHead c k t1) t4 t)) (pc3 (CHead c k t2) 
+t0 t4) (\lambda (x: T).(\lambda (H5: (pr3 (CHead c k t1) t0 x)).(\lambda (H6: 
+(pr3 (CHead c k t1) t4 x)).(pc3_t x (CHead c k t2) t0 (pc3_pr2_pr3_t c t1 t0 
+x k H5 t2 H0) t4 (pc3_s (CHead c k t2) x t4 (pc3_pr2_pr3_t c t1 t4 x k H6 t2 
+H0)))))) H4))))))))))))) u2 u1 H)))).
+
+theorem pc3_lift:
+ \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h 
+d c e) \to (\forall (t1: T).(\forall (t2: T).((pc3 e t1 t2) \to (pc3 c (lift 
+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: (pc3 e t1 
+t2)).(let H1 \def H0 in (ex2_ind T (\lambda (t: T).(pr3 e t1 t)) (\lambda (t: 
+T).(pr3 e t2 t)) (pc3 c (lift h d t1) (lift h d t2)) (\lambda (x: T).(\lambda 
+(H2: (pr3 e t1 x)).(\lambda (H3: (pr3 e t2 x)).(pc3_pr3_t c (lift h d t1) 
+(lift h d x) (pr3_lift c e h d H t1 x H2) (lift h d t2) (pr3_lift c e h d H 
+t2 x H3))))) H1))))))))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/arity.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/arity.ma
new file mode 100644
index 000000000..800e6a00e
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/arity.ma
@@ -0,0 +1,82 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/arity".
+
+include "ceqc/props.ma".
+
+theorem sc3_arity:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
+a) \to (sc3 g a c t)))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
+(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0: 
+A).(sc3 g a0 c0 t0)))) (\lambda (c0: C).(\lambda (n: nat).(conj (arity g c0 
+(TSort n) (ASort O n)) (sn3 c0 (TSort n)) (arity_sort g c0 n) (sn3_nf2 c0 
+(TSort n) (nf2_sort c0 n))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) 
+u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (H2: (sc3 g a0 
+d u)).(let H_y \def (sc3_abbr g a0 TNil) in (H_y i d u c0 (sc3_lift g a0 d u 
+H2 c0 (S i) O (getl_drop Abbr c0 d u i H0)) H0)))))))))) (\lambda (c0: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 
+(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc 
+g a0))).(\lambda (_: (sc3 g (asucc g a0) d u)).(let H3 \def (sc3_abst g a0 
+TNil) in (H3 c0 i (arity_abst g c0 d u i H0 a0 H1) (nf2_lref_abst c0 d u i 
+H0) I)))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda 
+(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u 
+a1)).(\lambda (H2: (sc3 g a1 c0 u)).(\lambda (t0: T).(\lambda (a2: 
+A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: (sc3 g 
+a2 (CHead c0 (Bind b) u) t0)).(let H_y \def (sc3_bind g b H0 a1 a2 TNil) in 
+(H_y c0 u t0 H4 H2))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda 
+(a1: A).(\lambda (H0: (arity g c0 u (asucc g a1))).(\lambda (H1: (sc3 g 
+(asucc g a1) c0 u)).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H2: (arity g 
+(CHead c0 (Bind Abst) u) t0 a2)).(\lambda (H3: (sc3 g a2 (CHead c0 (Bind 
+Abst) u) t0)).(conj (arity g c0 (THead (Bind Abst) u t0) (AHead a1 a2)) 
+(\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is: 
+PList).((drop1 is d c0) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is (THead 
+(Bind Abst) u t0))))))))) (arity_head g c0 u a1 H0 t0 a2 H2) (\lambda (d: 
+C).(\lambda (w: T).(\lambda (H4: (sc3 g a1 d w)).(\lambda (is: 
+PList).(\lambda (H5: (drop1 is d c0)).(let H6 \def (sc3_appl g a1 a2 TNil) in 
+(eq_ind_r T (THead (Bind Abst) (lift1 is u) (lift1 (Ss is) t0)) (\lambda (t1: 
+T).(sc3 g a2 d (THead (Flat Appl) w t1))) (H6 d w (lift1 (Ss is) t0) (let H_y 
+\def (sc3_bind g Abbr (\lambda (H7: (eq B Abbr Abst)).(not_abbr_abst H7)) a1 
+a2 TNil) in (H_y d w (lift1 (Ss is) t0) (let H7 \def (sc3_ceqc_trans g a2 
+TNil) in (H7 (CHead d (Bind Abst) (lift1 is u)) (lift1 (Ss is) t0) (sc3_lift1 
+g (CHead c0 (Bind Abst) u) a2 (Ss is) (CHead d (Bind Abst) (lift1 is u)) t0 
+H3 (drop1_skip_bind Abst c0 is d u H5)) (CHead d (Bind Abbr) w) (or_intror 
+(csubc g (CHead d (Bind Abbr) w) (CHead d (Bind Abst) (lift1 is u))) (csubc g 
+(CHead d (Bind Abst) (lift1 is u)) (CHead d (Bind Abbr) w)) (csubc_abst g d d 
+(csubc_refl g d) (lift1 is u) a1 (sc3_lift1 g c0 (asucc g a1) is d u H1 H5) w 
+H4)))) H4)) H4 (lift1 is u) (sc3_lift1 g c0 (asucc g a1) is d u H1 H5)) 
+(lift1 is (THead (Bind Abst) u t0)) (lift1_bind Abst is u 
+t0)))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: 
+A).(\lambda (_: (arity g c0 u a1)).(\lambda (H1: (sc3 g a1 c0 u)).(\lambda 
+(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda 
+(H3: (sc3 g (AHead a1 a2) c0 t0)).(let H4 \def H3 in (and_ind (arity g c0 t0 
+(AHead a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall 
+(is: PList).((drop1 is d c0) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is 
+t0)))))))) (sc3 g a2 c0 (THead (Flat Appl) u t0)) (\lambda (_: (arity g c0 t0 
+(AHead a1 a2))).(\lambda (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a1 d 
+w) \to (\forall (is: PList).((drop1 is d c0) \to (sc3 g a2 d (THead (Flat 
+Appl) w (lift1 is t0)))))))))).(let H_y \def (H6 c0 u H1 PNil) in (H_y 
+(drop1_nil c0))))) H4))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda 
+(a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda (H1: (sc3 g (asucc 
+g a0) c0 u)).(\lambda (t0: T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: 
+(sc3 g a0 c0 t0)).(let H_y \def (sc3_cast g a0 TNil) in (H_y c0 u H1 t0 
+H3)))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: 
+(arity g c0 t0 a1)).(\lambda (H1: (sc3 g a1 c0 t0)).(\lambda (a2: A).(\lambda 
+(H2: (leq g a1 a2)).(sc3_repl g a1 c0 t0 H1 a2 H2)))))))) c t a H))))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/defs.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/defs.ma
new file mode 100644
index 000000000..5942dd4f8
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/defs.ma
@@ -0,0 +1,34 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/defs".
+
+include "sn3/defs.ma".
+
+include "arity/defs.ma".
+
+include "drop1/defs.ma".
+
+definition sc3:
+ G \to (A \to (C \to (T \to Prop)))
+\def
+ let rec sc3 (g: G) (a: A) on a: (C \to (T \to Prop)) \def (\lambda (c: 
+C).(\lambda (t: T).(match a with [(ASort h n) \Rightarrow (land (arity g c t 
+(ASort h n)) (sn3 c t)) | (AHead a1 a2) \Rightarrow (land (arity g c t (AHead 
+a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is: 
+PList).((drop1 is d c) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is 
+t)))))))))]))) in sc3.
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/props.ma
new file mode 100644
index 000000000..86a674344
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/props.ma
@@ -0,0 +1,520 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/props".
+
+include "sc3/defs.ma".
+
+include "sn3/lift1.ma".
+
+include "nf2/lift1.ma".
+
+include "arity/lift1.ma".
+
+include "arity/aprem.ma".
+
+include "llt/props.ma".
+
+include "drop1/props.ma".
+
+theorem sc3_arity_gen:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((sc3 g a c 
+t) \to (arity g c t a)))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(A_ind 
+(\lambda (a0: A).((sc3 g a0 c t) \to (arity g c t a0))) (\lambda (n: 
+nat).(\lambda (n0: nat).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c 
+t))).(let H0 \def H in (and_ind (arity g c t (ASort n n0)) (sn3 c t) (arity g 
+c t (ASort n n0)) (\lambda (H1: (arity g c t (ASort n n0))).(\lambda (_: (sn3 
+c t)).H1)) H0))))) (\lambda (a0: A).(\lambda (_: (((sc3 g a0 c t) \to (arity 
+g c t a0)))).(\lambda (a1: A).(\lambda (_: (((sc3 g a1 c t) \to (arity g c t 
+a1)))).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: 
+C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) 
+\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H1 in 
+(and_ind (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g 
+a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat 
+Appl) w (lift1 is t)))))))) (arity g c t (AHead a0 a1)) (\lambda (H3: (arity 
+g c t (AHead a0 a1))).(\lambda (_: ((\forall (d: C).(\forall (w: T).((sc3 g 
+a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat 
+Appl) w (lift1 is t)))))))))).H3)) H2))))))) a)))).
+
+theorem sc3_repl:
+ \forall (g: G).(\forall (a1: A).(\forall (c: C).(\forall (t: T).((sc3 g a1 c 
+t) \to (\forall (a2: A).((leq g a1 a2) \to (sc3 g a2 c t)))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(llt_wf_ind (\lambda (a: A).(\forall (c: 
+C).(\forall (t: T).((sc3 g a c t) \to (\forall (a2: A).((leq g a a2) \to (sc3 
+g a2 c t))))))) (\lambda (a2: A).(A_ind (\lambda (a: A).(((\forall (a3: 
+A).((llt a3 a) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 c t) \to 
+(\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t))))))))) \to (\forall (c: 
+C).(\forall (t: T).((sc3 g a c t) \to (\forall (a3: A).((leq g a a3) \to (sc3 
+g a3 c t)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (_: ((\forall 
+(a3: A).((llt a3 (ASort n n0)) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 
+c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t)))))))))).(\lambda 
+(c: C).(\lambda (t: T).(\lambda (H0: (land (arity g c t (ASort n n0)) (sn3 c 
+t))).(\lambda (a3: A).(\lambda (H1: (leq g (ASort n n0) a3)).(let H2 \def H0 
+in (and_ind (arity g c t (ASort n n0)) (sn3 c t) (sc3 g a3 c t) (\lambda (H3: 
+(arity g c t (ASort n n0))).(\lambda (H4: (sn3 c t)).(let H_y \def 
+(arity_repl g c t (ASort n n0) H3 a3 H1) in (let H_x \def (leq_gen_sort g n 
+n0 a3 H1) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: 
+nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a3 (ASort h2 n2))))) (\lambda 
+(n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort n n0) k) 
+(aplus g (ASort h2 n2) k))))) (sc3 g a3 c t) (\lambda (x0: nat).(\lambda (x1: 
+nat).(\lambda (x2: nat).(\lambda (H6: (eq A a3 (ASort x1 x0))).(\lambda (_: 
+(eq A (aplus g (ASort n n0) x2) (aplus g (ASort x1 x0) x2))).(let H8 \def 
+(eq_ind A a3 (\lambda (a: A).(arity g c t a)) H_y (ASort x1 x0) H6) in 
+(eq_ind_r A (ASort x1 x0) (\lambda (a: A).(sc3 g a c t)) (conj (arity g c t 
+(ASort x1 x0)) (sn3 c t) H8 H4) a3 H6))))))) H5)))))) H2)))))))))) (\lambda 
+(a: A).(\lambda (_: ((((\forall (a3: A).((llt a3 a) \to (\forall (c: 
+C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to 
+(sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a c t) \to 
+(\forall (a3: A).((leq g a a3) \to (sc3 g a3 c t))))))))).(\lambda (a0: 
+A).(\lambda (H0: ((((\forall (a3: A).((llt a3 a0) \to (\forall (c: 
+C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to 
+(sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a0 c t) 
+\to (\forall (a3: A).((leq g a0 a3) \to (sc3 g a3 c t))))))))).(\lambda (H1: 
+((\forall (a3: A).((llt a3 (AHead a a0)) \to (\forall (c: C).(\forall (t: 
+T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c 
+t)))))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H2: (land (arity g c t 
+(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall 
+(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is 
+t)))))))))).(\lambda (a3: A).(\lambda (H3: (leq g (AHead a a0) a3)).(let H4 
+\def H2 in (and_ind (arity g c t (AHead a a0)) (\forall (d: C).(\forall (w: 
+T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d 
+(THead (Flat Appl) w (lift1 is t)))))))) (sc3 g a3 c t) (\lambda (H5: (arity 
+g c t (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a 
+d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat 
+Appl) w (lift1 is t)))))))))).(let H_x \def (leq_gen_head g a a0 a3 H3) in 
+(let H7 \def H_x in (ex3_2_ind A A (\lambda (a4: A).(\lambda (a5: A).(eq A a3 
+(AHead a4 a5)))) (\lambda (a4: A).(\lambda (_: A).(leq g a a4))) (\lambda (_: 
+A).(\lambda (a5: A).(leq g a0 a5))) (sc3 g a3 c t) (\lambda (x0: A).(\lambda 
+(x1: A).(\lambda (H8: (eq A a3 (AHead x0 x1))).(\lambda (H9: (leq g a 
+x0)).(\lambda (H10: (leq g a0 x1)).(eq_ind_r A (AHead x0 x1) (\lambda (a4: 
+A).(sc3 g a4 c t)) (conj (arity g c t (AHead x0 x1)) (\forall (d: C).(\forall 
+(w: T).((sc3 g x0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g x1 
+d (THead (Flat Appl) w (lift1 is t)))))))) (arity_repl g c t (AHead a a0) H5 
+(AHead x0 x1) (leq_head g a x0 H9 a0 x1 H10)) (\lambda (d: C).(\lambda (w: 
+T).(\lambda (H11: (sc3 g x0 d w)).(\lambda (is: PList).(\lambda (H12: (drop1 
+is d c)).(H0 (\lambda (a4: A).(\lambda (H13: (llt a4 a0)).(\lambda (c0: 
+C).(\lambda (t0: T).(\lambda (H14: (sc3 g a4 c0 t0)).(\lambda (a5: 
+A).(\lambda (H15: (leq g a4 a5)).(H1 a4 (llt_trans a4 a0 (AHead a a0) H13 
+(llt_head_dx a a0)) c0 t0 H14 a5 H15)))))))) d (THead (Flat Appl) w (lift1 is 
+t)) (H6 d w (H1 x0 (llt_repl g a x0 H9 (AHead a a0) (llt_head_sx a a0)) d w 
+H11 a (leq_sym g a x0 H9)) is H12) x1 H10))))))) a3 H8)))))) H7))))) 
+H4)))))))))))) a2)) a1)).
+
+theorem sc3_lift:
+ \forall (g: G).(\forall (a: A).(\forall (e: C).(\forall (t: T).((sc3 g a e 
+t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) 
+\to (sc3 g a c (lift h d t))))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (e: 
+C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h: 
+nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d t)))))))))) 
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (e: C).(\lambda (t: T).(\lambda 
+(H: (land (arity g e t (ASort n n0)) (sn3 e t))).(\lambda (c: C).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H0: (drop h d c e)).(let H1 \def H in 
+(and_ind (arity g e t (ASort n n0)) (sn3 e t) (land (arity g c (lift h d t) 
+(ASort n n0)) (sn3 c (lift h d t))) (\lambda (H2: (arity g e t (ASort n 
+n0))).(\lambda (H3: (sn3 e t)).(conj (arity g c (lift h d t) (ASort n n0)) 
+(sn3 c (lift h d t)) (arity_lift g e t (ASort n n0) H2 c h d H0) (sn3_lift e 
+t H3 c h d H0)))) H1))))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (e: 
+C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h: 
+nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d 
+t))))))))))).(\lambda (a1: A).(\lambda (_: ((\forall (e: C).(\forall (t: 
+T).((sc3 g a1 e t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: 
+nat).((drop h d c e) \to (sc3 g a1 c (lift h d t))))))))))).(\lambda (e: 
+C).(\lambda (t: T).(\lambda (H1: (land (arity g e t (AHead a0 a1)) (\forall 
+(d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d 
+e) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(\lambda (c: 
+C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h d c e)).(let H3 
+\def H1 in (and_ind (arity g e t (AHead a0 a1)) (\forall (d0: C).(\forall (w: 
+T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 e) \to (sc3 g a1 
+d0 (THead (Flat Appl) w (lift1 is t)))))))) (land (arity g c (lift h d t) 
+(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall 
+(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is 
+(lift h d t)))))))))) (\lambda (H4: (arity g e t (AHead a0 a1))).(\lambda 
+(H5: ((\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: 
+PList).((drop1 is d0 e) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is 
+t)))))))))).(conj (arity g c (lift h d t) (AHead a0 a1)) (\forall (d0: 
+C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) 
+\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (lift h d t))))))))) 
+(arity_lift g e t (AHead a0 a1) H4 c h d H2) (\lambda (d0: C).(\lambda (w: 
+T).(\lambda (H6: (sc3 g a0 d0 w)).(\lambda (is: PList).(\lambda (H7: (drop1 
+is d0 c)).(let H_y \def (H5 d0 w H6 (PConsTail is h d)) in (eq_ind T (lift1 
+(PConsTail is h d) t) (\lambda (t0: T).(sc3 g a1 d0 (THead (Flat Appl) w 
+t0))) (H_y (drop1_cons_tail c e h d H2 is d0 H7)) (lift1 is (lift h d t)) 
+(lift1_cons_tail t h d is))))))))))) H3))))))))))))) a)).
+
+theorem sc3_lift1:
+ \forall (g: G).(\forall (e: C).(\forall (a: A).(\forall (hds: 
+PList).(\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 hds c e) 
+\to (sc3 g a c (lift1 hds t)))))))))
+\def
+ \lambda (g: G).(\lambda (e: C).(\lambda (a: A).(\lambda (hds: 
+PList).(PList_ind (\lambda (p: PList).(\forall (c: C).(\forall (t: T).((sc3 g 
+a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t))))))) (\lambda (c: 
+C).(\lambda (t: T).(\lambda (H: (sc3 g a e t)).(\lambda (H0: (drop1 PNil c 
+e)).(let H1 \def (match H0 in drop1 return (\lambda (p: PList).(\lambda (c0: 
+C).(\lambda (c1: C).(\lambda (_: (drop1 p c0 c1)).((eq PList p PNil) \to ((eq 
+C c0 c) \to ((eq C c1 e) \to (sc3 g a c t)))))))) with [(drop1_nil c0) 
+\Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c0 
+c)).(\lambda (H3: (eq C c0 e)).(eq_ind C c (\lambda (c1: C).((eq C c1 e) \to 
+(sc3 g a c t))) (\lambda (H4: (eq C c e)).(eq_ind C e (\lambda (c1: C).(sc3 g 
+a c1 t)) H c (sym_eq C c e H4))) c0 (sym_eq C c0 c H2) H3)))) | (drop1_cons 
+c1 c2 h d H1 c3 hds0 H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds0) 
+PNil)).(\lambda (H4: (eq C c1 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def 
+(eq_ind PList (PCons h d hds0) (\lambda (e0: PList).(match e0 in PList return 
+(\lambda (_: PList).Prop) with [PNil \Rightarrow False | (PCons _ _ _) 
+\Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c) \to ((eq C c3 e) 
+\to ((drop h d c1 c2) \to ((drop1 hds0 c2 c3) \to (sc3 g a c t))))) H6)) H4 
+H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c) (refl_equal C 
+e))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda 
+(H: ((\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 p c e) \to 
+(sc3 g a c (lift1 p t)))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: 
+(sc3 g a e t)).(\lambda (H1: (drop1 (PCons n n0 p) c e)).(let H2 \def (match 
+H1 in drop1 return (\lambda (p0: PList).(\lambda (c0: C).(\lambda (c1: 
+C).(\lambda (_: (drop1 p0 c0 c1)).((eq PList p0 (PCons n n0 p)) \to ((eq C c0 
+c) \to ((eq C c1 e) \to (sc3 g a c (lift n n0 (lift1 p t)))))))))) with 
+[(drop1_nil c0) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0 
+p))).(\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq C c0 e)).((let H5 \def 
+(eq_ind PList PNil (\lambda (e0: PList).(match e0 in PList return (\lambda 
+(_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow 
+False])) I (PCons n n0 p) H2) in (False_ind ((eq C c0 c) \to ((eq C c0 e) \to 
+(sc3 g a c (lift n n0 (lift1 p t))))) H5)) H3 H4)))) | (drop1_cons c1 c2 h d 
+H2 c3 hds0 H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds0) (PCons n 
+n0 p))).(\lambda (H5: (eq C c1 c)).(\lambda (H6: (eq C c3 e)).((let H7 \def 
+(f_equal PList PList (\lambda (e0: PList).(match e0 in PList return (\lambda 
+(_: PList).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p0) \Rightarrow 
+p0])) (PCons h d hds0) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat 
+(\lambda (e0: PList).(match e0 in PList return (\lambda (_: PList).nat) with 
+[PNil \Rightarrow d | (PCons _ n1 _) \Rightarrow n1])) (PCons h d hds0) 
+(PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e0: 
+PList).(match e0 in PList return (\lambda (_: PList).nat) with [PNil 
+\Rightarrow h | (PCons n1 _ _) \Rightarrow n1])) (PCons h d hds0) (PCons n n0 
+p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds0 
+p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n1 d c1 c2) \to ((drop1 hds0 
+c2 c3) \to (sc3 g a c (lift n n0 (lift1 p t)))))))))) (\lambda (H10: (eq nat 
+d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds0 p) \to ((eq C c1 c) 
+\to ((eq C c3 e) \to ((drop n n1 c1 c2) \to ((drop1 hds0 c2 c3) \to (sc3 g a 
+c (lift n n0 (lift1 p t))))))))) (\lambda (H11: (eq PList hds0 p)).(eq_ind 
+PList p (\lambda (p0: PList).((eq C c1 c) \to ((eq C c3 e) \to ((drop n n0 c1 
+c2) \to ((drop1 p0 c2 c3) \to (sc3 g a c (lift n n0 (lift1 p t)))))))) 
+(\lambda (H12: (eq C c1 c)).(eq_ind C c (\lambda (c0: C).((eq C c3 e) \to 
+((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to (sc3 g a c (lift n n0 (lift1 p 
+t))))))) (\lambda (H13: (eq C c3 e)).(eq_ind C e (\lambda (c0: C).((drop n n0 
+c c2) \to ((drop1 p c2 c0) \to (sc3 g a c (lift n n0 (lift1 p t)))))) 
+(\lambda (H14: (drop n n0 c c2)).(\lambda (H15: (drop1 p c2 e)).(sc3_lift g a 
+c2 (lift1 p t) (H c2 t H0 H15) c n n0 H14))) c3 (sym_eq C c3 e H13))) c1 
+(sym_eq C c1 c H12))) hds0 (sym_eq PList hds0 p H11))) d (sym_eq nat d n0 
+H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal 
+PList (PCons n n0 p)) (refl_equal C c) (refl_equal C e))))))))))) hds)))).
+
+axiom sc3_abbr:
+ \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (i: 
+nat).(\forall (d: C).(\forall (v: T).(\forall (c: C).((sc3 g a c (THeads 
+(Flat Appl) vs (lift (S i) O v))) \to ((getl i c (CHead d (Bind Abbr) v)) \to 
+(sc3 g a c (THeads (Flat Appl) vs (TLRef i)))))))))))
+.
+
+theorem sc3_cast:
+ \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall 
+(u: T).((sc3 g (asucc g a) c (THeads (Flat Appl) vs u)) \to (\forall (t: 
+T).((sc3 g a c (THeads (Flat Appl) vs t)) \to (sc3 g a c (THeads (Flat Appl) 
+vs (THead (Flat Cast) u t))))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs: 
+TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat 
+Appl) vs u)) \to (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to 
+(sc3 g a0 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (\lambda 
+(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: 
+T).(\lambda (H: (sc3 g (match n with [O \Rightarrow (ASort O (next g n0)) | 
+(S h) \Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u))).(\lambda (t: 
+T).(\lambda (H0: (land (arity g c (THeads (Flat Appl) vs t) (ASort n n0)) 
+(sn3 c (THeads (Flat Appl) vs t)))).((match n in nat return (\lambda (n1: 
+nat).((sc3 g (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) 
+\Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u)) \to ((land (arity g c 
+(THeads (Flat Appl) vs t) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) 
+\to (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort 
+n1 n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))) with [O 
+\Rightarrow (\lambda (H1: (sc3 g (ASort O (next g n0)) c (THeads (Flat Appl) 
+vs u))).(\lambda (H2: (land (arity g c (THeads (Flat Appl) vs t) (ASort O 
+n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let H3 \def H1 in (and_ind (arity g 
+c (THeads (Flat Appl) vs u) (ASort O (next g n0))) (sn3 c (THeads (Flat Appl) 
+vs u)) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) 
+(ASort O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))) 
+(\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (ASort O (next g 
+n0)))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def H2 in 
+(and_ind (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads 
+(Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) 
+u t)) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))) 
+(\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort O n0))).(\lambda 
+(H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g c (THeads (Flat Appl) 
+vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs 
+(THead (Flat Cast) u t))) (arity_appls_cast g c u t vs (ASort O n0) H4 H7) 
+(sn3_appls_cast c vs u H5 t H8)))) H6)))) H3)))) | (S n1) \Rightarrow 
+(\lambda (H1: (sc3 g (ASort n1 n0) c (THeads (Flat Appl) vs u))).(\lambda 
+(H2: (land (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c 
+(THeads (Flat Appl) vs t)))).(let H3 \def H1 in (and_ind (arity g c (THeads 
+(Flat Appl) vs u) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs u)) (land 
+(arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) 
+(sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity 
+g c (THeads (Flat Appl) vs u) (ASort n1 n0))).(\lambda (H5: (sn3 c (THeads 
+(Flat Appl) vs u))).(let H6 \def H2 in (and_ind (arity g c (THeads (Flat 
+Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity 
+g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c 
+(THeads (Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c 
+(THeads (Flat Appl) vs t) (ASort (S n1) n0))).(\lambda (H8: (sn3 c (THeads 
+(Flat Appl) vs t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat 
+Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat 
+Cast) u t))) (arity_appls_cast g c u t vs (ASort (S n1) n0) H4 H7) 
+(sn3_appls_cast c vs u H5 t H8)))) H6)))) H3))))]) H H0))))))))) (\lambda 
+(a0: A).(\lambda (_: ((\forall (vs: TList).(\forall (c: C).(\forall (u: 
+T).((sc3 g (asucc g a0) c (THeads (Flat Appl) vs u)) \to (\forall (t: 
+T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to (sc3 g a0 c (THeads (Flat 
+Appl) vs (THead (Flat Cast) u t))))))))))).(\lambda (a1: A).(\lambda (H0: 
+((\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a1) c 
+(THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a1 c (THeads (Flat 
+Appl) vs t)) \to (sc3 g a1 c (THeads (Flat Appl) vs (THead (Flat Cast) u 
+t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: T).(\lambda 
+(H1: (land (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g a1))) 
+(\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: 
+PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 
+is (THeads (Flat Appl) vs u))))))))))).(\lambda (t: T).(\lambda (H2: (land 
+(arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: C).(\forall 
+(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 
+d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))))).(let H3 
+\def H1 in (and_ind (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g 
+a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: 
+PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 
+is (THeads (Flat Appl) vs u))))))))) (land (arity g c (THeads (Flat Appl) vs 
+(THead (Flat Cast) u t)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 
+g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead 
+(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u 
+t))))))))))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (AHead a0 
+(asucc g a1)))).(\lambda (H5: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d 
+w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead 
+(Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))))).(let H6 \def H2 
+in (and_ind (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: 
+C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) 
+\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs 
+t))))))))) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) 
+(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall 
+(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is 
+(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))) (\lambda (H7: (arity 
+g c (THeads (Flat Appl) vs t) (AHead a0 a1))).(\lambda (H8: ((\forall (d: 
+C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) 
+\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs 
+t))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) 
+(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall 
+(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is 
+(THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (arity_appls_cast g c 
+u t vs (AHead a0 a1) H4 H7) (\lambda (d: C).(\lambda (w: T).(\lambda (H9: 
+(sc3 g a0 d w)).(\lambda (is: PList).(\lambda (H10: (drop1 is d c)).(let H_y 
+\def (H0 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 
+is vs) (lift1 is (THead (Flat Cast) u t))) (\lambda (t0: T).(sc3 g a1 d 
+(THead (Flat Appl) w t0))) (eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1 
+is t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat Appl) 
+(lifts1 is vs) t0)))) (H_y d (lift1 is u) (eq_ind T (lift1 is (THeads (Flat 
+Appl) vs u)) (\lambda (t0: T).(sc3 g (asucc g a1) d (THead (Flat Appl) w 
+t0))) (H5 d w H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is u)) 
+(lifts1_flat Appl is u vs)) (lift1 is t) (eq_ind T (lift1 is (THeads (Flat 
+Appl) vs t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w t0))) (H8 d w 
+H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is t)) (lifts1_flat Appl 
+is t vs))) (lift1 is (THead (Flat Cast) u t)) (lift1_flat Cast is u t)) 
+(lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (lifts1_flat Appl 
+is (THead (Flat Cast) u t) vs))))))))))) H6)))) H3)))))))))))) a)).
+
+axiom sc3_bind:
+ \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (a1: 
+A).(\forall (a2: A).(\forall (vs: TList).(\forall (c: C).(\forall (v: 
+T).(\forall (t: T).((sc3 g a2 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts 
+(S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a2 c (THeads (Flat Appl) vs 
+(THead (Bind b) v t)))))))))))))
+.
+
+axiom sc3_appl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (vs: 
+TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 c (THeads 
+(Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: 
+T).((sc3 g (asucc g a1) c w) \to (sc3 g a2 c (THeads (Flat Appl) vs (THead 
+(Flat Appl) v (THead (Bind Abst) w t))))))))))))))
+.
+
+theorem sc3_props__sc3_sn3_abst:
+ \forall (g: G).(\forall (a: A).(land (\forall (c: C).(\forall (t: T).((sc3 g 
+a c t) \to (sn3 c t)))) (\forall (vs: TList).(\forall (i: nat).(let t \def 
+(THeads (Flat Appl) vs (TLRef i)) in (\forall (c: C).((arity g c t a) \to 
+((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a c t))))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(land (\forall (c: 
+C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall (vs: 
+TList).(\forall (i: nat).(let t \def (THeads (Flat Appl) vs (TLRef i)) in 
+(\forall (c: C).((arity g c t a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to 
+(sc3 g a0 c t)))))))))) (\lambda (n: nat).(\lambda (n0: nat).(conj (\forall 
+(c: C).(\forall (t: T).((land (arity g c t (ASort n n0)) (sn3 c t)) \to (sn3 
+c t)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c 
+(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) \to ((nf2 c (TLRef i)) \to 
+((sns3 c vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n 
+n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))) (\lambda (c: 
+C).(\lambda (t: T).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c 
+t))).(let H0 \def H in (and_ind (arity g c t (ASort n n0)) (sn3 c t) (sn3 c 
+t) (\lambda (_: (arity g c t (ASort n n0))).(\lambda (H2: (sn3 c t)).H2)) 
+H0))))) (\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H: 
+(arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n n0))).(\lambda (H0: 
+(nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(conj (arity g c (THeads (Flat 
+Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i))) H 
+(sn3_appls_lref c i H0 vs H1))))))))))) (\lambda (a0: A).(\lambda (H: (land 
+(\forall (c: C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall 
+(vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads (Flat Appl) 
+vs (TLRef i)) a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a0 c 
+(THeads (Flat Appl) vs (TLRef i))))))))))).(\lambda (a1: A).(\lambda (H0: 
+(land (\forall (c: C).(\forall (t: T).((sc3 g a1 c t) \to (sn3 c t)))) 
+(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads 
+(Flat Appl) vs (TLRef i)) a1) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to 
+(sc3 g a1 c (THeads (Flat Appl) vs (TLRef i))))))))))).(conj (\forall (c: 
+C).(\forall (t: T).((land (arity g c t (AHead a0 a1)) (\forall (d: 
+C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) 
+\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t))))))))) \to (sn3 c t)))) 
+(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads 
+(Flat Appl) vs (TLRef i)) (AHead a0 a1)) \to ((nf2 c (TLRef i)) \to ((sns3 c 
+vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1)) 
+(\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: 
+PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads 
+(Flat Appl) vs (TLRef i))))))))))))))))) (\lambda (c: C).(\lambda (t: 
+T).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall 
+(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 
+d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H in (and_ind 
+(\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 t0)))) 
+(\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads 
+(Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to 
+(sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) (\lambda (_: 
+((\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 
+t0)))))).(\lambda (H4: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0: 
+C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) 
+\to ((sns3 c0 vs) \to (sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef 
+i))))))))))).(let H5 \def H0 in (and_ind (\forall (c0: C).(\forall (t0: 
+T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))) (\forall (vs: TList).(\forall (i: 
+nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a1) \to 
+((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 (THeads (Flat Appl) vs 
+(TLRef i))))))))) (sn3 c t) (\lambda (H6: ((\forall (c0: C).(\forall (t0: 
+T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))))).(\lambda (_: ((\forall (vs: 
+TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs 
+(TLRef i)) a1) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 
+(THeads (Flat Appl) vs (TLRef i))))))))))).(let H8 \def H1 in (and_ind (arity 
+g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to 
+(\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w 
+(lift1 is t)))))))) (sn3 c t) (\lambda (H9: (arity g c t (AHead a0 
+a1))).(\lambda (H10: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to 
+(\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w 
+(lift1 is t)))))))))).(let H_y \def (arity_aprem g c t (AHead a0 a1) H9 O a0) 
+in (let H11 \def (H_y (aprem_zero a0 a1)) in (ex2_3_ind C T nat (\lambda (d: 
+C).(\lambda (_: T).(\lambda (j: nat).(drop j O d c)))) (\lambda (d: 
+C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g a0))))) (sn3 c t) 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H12: (drop x2 
+O x0 c)).(\lambda (H13: (arity g x0 x1 (asucc g a0))).(let H_y0 \def (H10 
+(CHead x0 (Bind Abst) x1) (TLRef O) (H4 TNil O (CHead x0 (Bind Abst) x1) 
+(arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1) a0 
+H13) (nf2_lref_abst (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1)) 
+I) (PCons (S x2) O PNil)) in (let H_y1 \def (H6 (CHead x0 (Bind Abst) x1) 
+(THead (Flat Appl) (TLRef O) (lift (S x2) O t)) (H_y0 (drop1_cons (CHead x0 
+(Bind Abst) x1) c (S x2) O (drop_drop (Bind Abst) x2 x0 c H12 x1) c PNil 
+(drop1_nil c)))) in (let H14 \def (sn3_gen_flat Appl (CHead x0 (Bind Abst) 
+x1) (TLRef O) (lift (S x2) O t) H_y1) in (and_ind (sn3 (CHead x0 (Bind Abst) 
+x1) (TLRef O)) (sn3 (CHead x0 (Bind Abst) x1) (lift (S x2) O t)) (sn3 c t) 
+(\lambda (_: (sn3 (CHead x0 (Bind Abst) x1) (TLRef O))).(\lambda (H16: (sn3 
+(CHead x0 (Bind Abst) x1) (lift (S x2) O t))).(sn3_gen_lift (CHead x0 (Bind 
+Abst) x1) t (S x2) O H16 c (drop_drop (Bind Abst) x2 x0 c H12 x1)))) 
+H14))))))))) H11))))) H8)))) H5)))) H2))))) (\lambda (vs: TList).(\lambda (i: 
+nat).(\lambda (c: C).(\lambda (H1: (arity g c (THeads (Flat Appl) vs (TLRef 
+i)) (AHead a0 a1))).(\lambda (H2: (nf2 c (TLRef i))).(\lambda (H3: (sns3 c 
+vs)).(conj (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1)) 
+(\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: 
+PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads 
+(Flat Appl) vs (TLRef i)))))))))) H1 (\lambda (d: C).(\lambda (w: T).(\lambda 
+(H4: (sc3 g a0 d w)).(\lambda (is: PList).(\lambda (H5: (drop1 is d c)).(let 
+H6 \def H in (and_ind (\forall (c0: C).(\forall (t: T).((sc3 g a0 c0 t) \to 
+(sn3 c0 t)))) (\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: 
+C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef 
+i0)) \to ((sns3 c0 vs0) \to (sc3 g a0 c0 (THeads (Flat Appl) vs0 (TLRef 
+i0))))))))) (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs 
+(TLRef i))))) (\lambda (H7: ((\forall (c0: C).(\forall (t: T).((sc3 g a0 c0 
+t) \to (sn3 c0 t)))))).(\lambda (_: ((\forall (vs0: TList).(\forall (i0: 
+nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a0) 
+\to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a0 c0 (THeads (Flat 
+Appl) vs0 (TLRef i0))))))))))).(let H9 \def H0 in (and_ind (\forall (c0: 
+C).(\forall (t: T).((sc3 g a1 c0 t) \to (sn3 c0 t)))) (\forall (vs0: 
+TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) 
+vs0 (TLRef i0)) a1) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 
+c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a1 d (THead (Flat Appl) 
+w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (_: ((\forall (c0: 
+C).(\forall (t: T).((sc3 g a1 c0 t) \to (sn3 c0 t)))))).(\lambda (H11: 
+((\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 
+(THeads (Flat Appl) vs0 (TLRef i0)) a1) \to ((nf2 c0 (TLRef i0)) \to ((sns3 
+c0 vs0) \to (sc3 g a1 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))))).(let 
+H_y \def (H11 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) 
+(lifts1 is vs) (lift1 is (TLRef i))) (\lambda (t: T).(sc3 g a1 d (THead (Flat 
+Appl) w t))) (eq_ind_r T (TLRef (trans is i)) (\lambda (t: T).(sc3 g a1 d 
+(THead (Flat Appl) w (THeads (Flat Appl) (lifts1 is vs) t)))) (H_y (trans is 
+i) d (eq_ind T (lift1 is (TLRef i)) (\lambda (t: T).(arity g d (THead (Flat 
+Appl) w (THeads (Flat Appl) (lifts1 is vs) t)) a1)) (eq_ind T (lift1 is 
+(THeads (Flat Appl) vs (TLRef i))) (\lambda (t: T).(arity g d (THead (Flat 
+Appl) w t) a1)) (arity_appl g d w a0 (sc3_arity_gen g d w a0 H4) (lift1 is 
+(THeads (Flat Appl) vs (TLRef i))) a1 (arity_lift1 g (AHead a0 a1) c is d 
+(THeads (Flat Appl) vs (TLRef i)) H5 H1)) (THeads (Flat Appl) (lifts1 is vs) 
+(lift1 is (TLRef i))) (lifts1_flat Appl is (TLRef i) vs)) (TLRef (trans is 
+i)) (lift1_lref is i)) (eq_ind T (lift1 is (TLRef i)) (\lambda (t: T).(nf2 d 
+t)) (nf2_lift1 c is d (TLRef i) H5 H2) (TLRef (trans is i)) (lift1_lref is 
+i)) (conj (sn3 d w) (sns3 d (lifts1 is vs)) (H7 d w H4) (sns3_lifts1 c is d 
+H5 vs H3))) (lift1 is (TLRef i)) (lift1_lref is i)) (lift1 is (THeads (Flat 
+Appl) vs (TLRef i))) (lifts1_flat Appl is (TLRef i) vs))))) H9)))) 
+H6))))))))))))))))))) a)).
+
+theorem sc3_sn3:
+ \forall (g: G).(\forall (a: A).(\forall (c: C).(\forall (t: T).((sc3 g a c 
+t) \to (sn3 c t)))))
+\def
+ \lambda (g: G).(\lambda (a: A).(\lambda (c: C).(\lambda (t: T).(\lambda (H: 
+(sc3 g a c t)).(let H_x \def (sc3_props__sc3_sn3_abst g a) in (let H0 \def 
+H_x in (and_ind (\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 
+c0 t0)))) (\forall (vs: TList).(\forall (i: nat).(let t0 \def (THeads (Flat 
+Appl) vs (TLRef i)) in (\forall (c0: C).((arity g c0 t0 a) \to ((nf2 c0 
+(TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a c0 t0)))))))) (sn3 c t) (\lambda 
+(H1: ((\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 c0 
+t0)))))).(\lambda (_: ((\forall (vs: TList).(\forall (i: nat).(let t0 \def 
+(THeads (Flat Appl) vs (TLRef i)) in (\forall (c0: C).((arity g c0 t0 a) \to 
+((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a c0 t0)))))))))).(H1 c t 
+H))) H0))))))).
+
+theorem sc3_abst:
+ \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall 
+(i: nat).((arity g c (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c (TLRef 
+i)) \to ((sns3 c vs) \to (sc3 g a c (THeads (Flat Appl) vs (TLRef i))))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(\lambda (vs: TList).(\lambda (c: C).(\lambda 
+(i: nat).(\lambda (H: (arity g c (THeads (Flat Appl) vs (TLRef i)) 
+a)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(let H_x \def 
+(sc3_props__sc3_sn3_abst g a) in (let H2 \def H_x in (and_ind (\forall (c0: 
+C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0 t)))) (\forall (vs0: 
+TList).(\forall (i0: nat).(let t \def (THeads (Flat Appl) vs0 (TLRef i0)) in 
+(\forall (c0: C).((arity g c0 t a) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 
+vs0) \to (sc3 g a c0 t)))))))) (sc3 g a c (THeads (Flat Appl) vs (TLRef i))) 
+(\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0 
+t)))))).(\lambda (H4: ((\forall (vs0: TList).(\forall (i0: nat).(let t \def 
+(THeads (Flat Appl) vs0 (TLRef i0)) in (\forall (c0: C).((arity g c0 t a) \to 
+((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a c0 t)))))))))).(H4 vs i 
+c H H0 H1))) H2)))))))))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/defs.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/defs.ma
new file mode 100644
index 000000000..46e2fda80
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/defs.ma
@@ -0,0 +1,31 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/defs".
+
+include "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 
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)).
+
+definition sns3:
+ C \to (TList \to Prop)
+\def
+ let rec sns3 (c: C) (ts: TList) on ts: Prop \def (match ts with [TNil 
+\Rightarrow True | (TCons t ts0) \Rightarrow (land (sn3 c t) (sns3 c ts0))]) 
+in sns3.
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/fwd.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/fwd.ma
new file mode 100644
index 000000000..9b751cedf
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/fwd.ma
@@ -0,0 +1,101 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/fwd".
+
+include "sn3/defs.ma".
+
+include "pr3/props.ma".
+
+theorem sn3_gen_flat:
+ \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
+(THead (Flat f) u t)) \to (land (sn3 c u) (sn3 c t))))))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: 
+(sn3 c (THead (Flat f) u t))).(insert_eq T (THead (Flat f) u t) (\lambda (t0: 
+T).(sn3 c t0)) (land (sn3 c u) (sn3 c t)) (\lambda (y: T).(\lambda (H0: (sn3 
+c y)).(unintro T t (\lambda (t0: T).((eq T y (THead (Flat f) u t0)) \to (land 
+(sn3 c u) (sn3 c t0)))) (unintro T u (\lambda (t0: T).(\forall (x: T).((eq T 
+y (THead (Flat f) t0 x)) \to (land (sn3 c t0) (sn3 c x))))) (sn3_ind c 
+(\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat f) x 
+x0)) \to (land (sn3 c x) (sn3 c x0)))))) (\lambda (t1: T).(\lambda (H1: 
+((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 
+t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to 
+(\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall (x0: 
+T).((eq T t2 (THead (Flat f) x x0)) \to (land (sn3 c x) (sn3 c 
+x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1 (THead 
+(Flat f) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: 
+T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to 
+(\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Flat f) x1 x2)) \to (land 
+(sn3 c x1) (sn3 c x2))))))))) H2 (THead (Flat f) x x0) H3) in (let H5 \def 
+(eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall 
+(P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead (Flat f) x x0) 
+H3) in (conj (sn3 c x) (sn3 c x0) (sn3_sing c x (\lambda (t2: T).(\lambda 
+(H6: (((eq T x t2) \to (\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x 
+t2)).(let H8 \def (H4 (THead (Flat f) t2 x0) (\lambda (H8: (eq T (THead (Flat 
+f) x x0) (THead (Flat f) t2 x0))).(\lambda (P: Prop).(let H9 \def (f_equal T 
+T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) 
+(THead (Flat f) x x0) (THead (Flat f) t2 x0) H8) in (let H10 \def (eq_ind_r T 
+t2 (\lambda (t0: T).(pr3 c x t0)) H7 x H9) in (let H11 \def (eq_ind_r T t2 
+(\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H6 x H9) in (H11 
+(refl_equal T x) P)))))) (pr3_head_12 c x t2 H7 (Flat f) x0 x0 (pr3_refl 
+(CHead c (Flat f) t2) x0)) t2 x0 (refl_equal T (THead (Flat f) t2 x0))) in 
+(and_ind (sn3 c t2) (sn3 c x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda 
+(_: (sn3 c x0)).H9)) H8)))))) (sn3_sing c x0 (\lambda (t2: T).(\lambda (H6: 
+(((eq T x0 t2) \to (\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x0 t2)).(let 
+H8 \def (H4 (THead (Flat f) x t2) (\lambda (H8: (eq T (THead (Flat f) x x0) 
+(THead (Flat f) x t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda 
+(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 
+| (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat f) 
+x x0) (THead (Flat f) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: 
+T).(pr3 c x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: 
+T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in (H11 (refl_equal 
+T x0) P)))))) (pr3_thin_dx c x0 t2 H7 x f) x t2 (refl_equal T (THead (Flat f) 
+x t2))) in (and_ind (sn3 c x) (sn3 c t2) (sn3 c t2) (\lambda (_: (sn3 c 
+x)).(\lambda (H10: (sn3 c t2)).H10)) H8))))))))))))))) y H0))))) H))))).
+
+theorem sn3_gen_lift:
+ \forall (c1: C).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((sn3 c1 
+(lift h d t)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t)))))))
+\def
+ \lambda (c1: C).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
+(H: (sn3 c1 (lift h d t))).(insert_eq T (lift h d t) (\lambda (t0: T).(sn3 c1 
+t0)) (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t))) (\lambda (y: 
+T).(\lambda (H0: (sn3 c1 y)).(unintro T t (\lambda (t0: T).((eq T y (lift h d 
+t0)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t0))))) (sn3_ind c1 
+(\lambda (t0: T).(\forall (x: T).((eq T t0 (lift h d x)) \to (\forall (c2: 
+C).((drop h d c1 c2) \to (sn3 c2 x)))))) (\lambda (t1: T).(\lambda (H1: 
+((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t1 
+t2) \to (sn3 c1 t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to 
+(\forall (P: Prop).P))) \to ((pr3 c1 t1 t2) \to (\forall (x: T).((eq T t2 
+(lift h d x)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 
+x)))))))))).(\lambda (x: T).(\lambda (H3: (eq T t1 (lift h d x))).(\lambda 
+(c2: C).(\lambda (H4: (drop h d c1 c2)).(let H5 \def (eq_ind T t1 (\lambda 
+(t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to 
+((pr3 c1 t0 t2) \to (\forall (x0: T).((eq T t2 (lift h d x0)) \to (\forall 
+(c3: C).((drop h d c1 c3) \to (sn3 c3 x0))))))))) H2 (lift h d x) H3) in (let 
+H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to 
+(\forall (P: Prop).P))) \to ((pr3 c1 t0 t2) \to (sn3 c1 t2))))) H1 (lift h d 
+x) H3) in (sn3_sing c2 x (\lambda (t2: T).(\lambda (H7: (((eq T x t2) \to 
+(\forall (P: Prop).P)))).(\lambda (H8: (pr3 c2 x t2)).(H5 (lift h d t2) 
+(\lambda (H9: (eq T (lift h d x) (lift h d t2))).(\lambda (P: Prop).(let H10 
+\def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c2 x t0)) H8 x (lift_inj x t2 h d 
+H9)) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to 
+(\forall (P0: Prop).P0))) H7 x (lift_inj x t2 h d H9)) in (H11 (refl_equal T 
+x) P))))) (pr3_lift c1 c2 h d H4 x t2 H8) t2 (refl_equal T (lift h d t2)) c2 
+H4)))))))))))))) y H0)))) H))))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/lift1.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/lift1.ma
new file mode 100644
index 000000000..3533a3989
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/lift1.ma
@@ -0,0 +1,90 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/lift1".
+
+include "sn3/props.ma".
+
+include "drop1/defs.ma".
+
+include "lift1/fwd.ma".
+
+theorem sns3_lifts1:
+ \forall (e: C).(\forall (hds: PList).(\forall (c: C).((drop1 hds c e) \to 
+(\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 hds ts)))))))
+\def
+ \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
+(c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to (sns3 c 
+(lifts1 p ts))))))) (\lambda (c: C).(\lambda (H: (drop1 PNil c e)).(\lambda 
+(ts: TList).(\lambda (H0: (sns3 e ts)).(let H1 \def (match H in drop1 return 
+(\lambda (p: PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (drop1 p 
+c0 c1)).((eq PList p PNil) \to ((eq C c0 c) \to ((eq C c1 e) \to (sns3 c 
+(lifts1 PNil ts))))))))) with [(drop1_nil c0) \Rightarrow (\lambda (_: (eq 
+PList PNil PNil)).(\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq C c0 
+e)).(eq_ind C c (\lambda (c1: C).((eq C c1 e) \to (sns3 c (lifts1 PNil ts)))) 
+(\lambda (H4: (eq C c e)).(eq_ind C e (\lambda (c1: C).(sns3 c1 (lifts1 PNil 
+ts))) (eq_ind_r TList ts (\lambda (t: TList).(sns3 e t)) H0 (lifts1 PNil ts) 
+(lifts1_nil ts)) c (sym_eq C c e H4))) c0 (sym_eq C c0 c H2) H3)))) | 
+(drop1_cons c1 c2 h d H1 c3 hds0 H2) \Rightarrow (\lambda (H3: (eq PList 
+(PCons h d hds0) PNil)).(\lambda (H4: (eq C c1 c)).(\lambda (H5: (eq C c3 
+e)).((let H6 \def (eq_ind PList (PCons h d hds0) (\lambda (e0: PList).(match 
+e0 in PList return (\lambda (_: PList).Prop) with [PNil \Rightarrow False | 
+(PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c) \to 
+((eq C c3 e) \to ((drop h d c1 c2) \to ((drop1 hds0 c2 c3) \to (sns3 c 
+(lifts1 PNil ts)))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) 
+(refl_equal C c) (refl_equal C e))))))) (\lambda (n: nat).(\lambda (n0: 
+nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).((drop1 p c e) \to 
+(\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 p ts)))))))).(\lambda 
+(c: C).(\lambda (H0: (drop1 (PCons n n0 p) c e)).(\lambda (ts: 
+TList).(\lambda (H1: (sns3 e ts)).(let H2 \def (match H0 in drop1 return 
+(\lambda (p0: PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (drop1 p0 
+c0 c1)).((eq PList p0 (PCons n n0 p)) \to ((eq C c0 c) \to ((eq C c1 e) \to 
+(sns3 c (lifts1 (PCons n n0 p) ts))))))))) with [(drop1_nil c0) \Rightarrow 
+(\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c0 
+c)).(\lambda (H4: (eq C c0 e)).((let H5 \def (eq_ind PList PNil (\lambda (e0: 
+PList).(match e0 in PList return (\lambda (_: PList).Prop) with [PNil 
+\Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in 
+(False_ind ((eq C c0 c) \to ((eq C c0 e) \to (sns3 c (lifts1 (PCons n n0 p) 
+ts)))) H5)) H3 H4)))) | (drop1_cons c1 c2 h d H2 c3 hds0 H3) \Rightarrow 
+(\lambda (H4: (eq PList (PCons h d hds0) (PCons n n0 p))).(\lambda (H5: (eq C 
+c1 c)).(\lambda (H6: (eq C c3 e)).((let H7 \def (f_equal PList PList (\lambda 
+(e0: PList).(match e0 in PList return (\lambda (_: PList).PList) with [PNil 
+\Rightarrow hds0 | (PCons _ _ p0) \Rightarrow p0])) (PCons h d hds0) (PCons n 
+n0 p) H4) in ((let H8 \def (f_equal PList nat (\lambda (e0: PList).(match e0 
+in PList return (\lambda (_: PList).nat) with [PNil \Rightarrow d | (PCons _ 
+n1 _) \Rightarrow n1])) (PCons h d hds0) (PCons n n0 p) H4) in ((let H9 \def 
+(f_equal PList nat (\lambda (e0: PList).(match e0 in PList return (\lambda 
+(_: PList).nat) with [PNil \Rightarrow h | (PCons n1 _ _) \Rightarrow n1])) 
+(PCons h d hds0) (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq 
+nat d n0) \to ((eq PList hds0 p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop 
+n1 d c1 c2) \to ((drop1 hds0 c2 c3) \to (sns3 c (lifts1 (PCons n n0 p) 
+ts))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1: 
+nat).((eq PList hds0 p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n n1 c1 
+c2) \to ((drop1 hds0 c2 c3) \to (sns3 c (lifts1 (PCons n n0 p) ts)))))))) 
+(\lambda (H11: (eq PList hds0 p)).(eq_ind PList p (\lambda (p0: PList).((eq C 
+c1 c) \to ((eq C c3 e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2 c3) \to (sns3 
+c (lifts1 (PCons n n0 p) ts))))))) (\lambda (H12: (eq C c1 c)).(eq_ind C c 
+(\lambda (c0: C).((eq C c3 e) \to ((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to 
+(sns3 c (lifts1 (PCons n n0 p) ts)))))) (\lambda (H13: (eq C c3 e)).(eq_ind C 
+e (\lambda (c0: C).((drop n n0 c c2) \to ((drop1 p c2 c0) \to (sns3 c (lifts1 
+(PCons n n0 p) ts))))) (\lambda (H14: (drop n n0 c c2)).(\lambda (H15: (drop1 
+p c2 e)).(eq_ind_r TList (lifts n n0 (lifts1 p ts)) (\lambda (t: TList).(sns3 
+c t)) (sns3_lifts c c2 n n0 H14 (lifts1 p ts) (H c2 H15 ts H1)) (lifts1 
+(PCons n n0 p) ts) (lifts1_cons n n0 p ts)))) c3 (sym_eq C c3 e H13))) c1 
+(sym_eq C c1 c H12))) hds0 (sym_eq PList hds0 p H11))) d (sym_eq nat d n0 
+H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal 
+PList (PCons n n0 p)) (refl_equal C c) (refl_equal C e))))))))))) hds)).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/nf2.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/nf2.ma
new file mode 100644
index 000000000..1279cc30b
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/nf2.ma
@@ -0,0 +1,62 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/nf2".
+
+include "sn3/defs.ma".
+
+include "nf2/dec.ma".
+
+include "nf2/pr3.ma".
+
+theorem sn3_nf2:
+ \forall (c: C).(\forall (t: T).((nf2 c t) \to (sn3 c t)))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (H: (nf2 c t)).(sn3_sing c t 
+(\lambda (t2: T).(\lambda (H0: (((eq T t t2) \to (\forall (P: 
+Prop).P)))).(\lambda (H1: (pr3 c t t2)).(let H_y \def (nf2_pr3_unfold c t t2 
+H1 H) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c t t0)) H1 t H_y) 
+in (let H3 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T t t0) \to (\forall (P: 
+Prop).P))) H0 t H_y) in (eq_ind T t (\lambda (t0: T).(sn3 c t0)) (H3 
+(refl_equal T t) (sn3 c t)) t2 H_y)))))))))).
+
+theorem nf2_sn3:
+ \forall (c: C).(\forall (t: T).((sn3 c t) \to (ex2 T (\lambda (u: T).(pr3 c 
+t u)) (\lambda (u: T).(nf2 c u)))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(sn3_ind c (\lambda 
+(t0: T).(ex2 T (\lambda (u: T).(pr3 c t0 u)) (\lambda (u: T).(nf2 c u)))) 
+(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall 
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
+(ex2 T (\lambda (u: T).(pr3 c t2 u)) (\lambda (u: T).(nf2 c u)))))))).(let 
+H_x \def (nf2_dec c t1) in (let H2 \def H_x in (or_ind (nf2 c t1) (ex2 T 
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr2 c t1 t2))) (ex2 T (\lambda (u: T).(pr3 c t1 u)) (\lambda (u: T).(nf2 
+c u))) (\lambda (H3: (nf2 c t1)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u)) 
+(\lambda (u: T).(nf2 c u)) t1 (pr3_refl c t1) H3)) (\lambda (H3: (ex2 T 
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr2 c t1 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t1 t2) \to (\forall 
+(P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)) (ex2 T (\lambda (u: T).(pr3 c 
+t1 u)) (\lambda (u: T).(nf2 c u))) (\lambda (x: T).(\lambda (H4: (((eq T t1 
+x) \to (\forall (P: Prop).P)))).(\lambda (H5: (pr2 c t1 x)).(let H_y \def (H1 
+x H4) in (let H6 \def (H_y (pr3_pr2 c t1 x H5)) in (ex2_ind T (\lambda (u: 
+T).(pr3 c x u)) (\lambda (u: T).(nf2 c u)) (ex2 T (\lambda (u: T).(pr3 c t1 
+u)) (\lambda (u: T).(nf2 c u))) (\lambda (x0: T).(\lambda (H7: (pr3 c x 
+x0)).(\lambda (H8: (nf2 c x0)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u)) 
+(\lambda (u: T).(nf2 c u)) x0 (pr3_sing c x t1 H5 x0 H7) H8)))) H6)))))) H3)) 
+H2)))))) t H))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/props.ma
new file mode 100644
index 000000000..dd51aca1d
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/props.ma
@@ -0,0 +1,1174 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/sn3/props".
+
+include "sn3/nf2.ma".
+
+include "sn3/fwd.ma".
+
+include "nf2/iso.ma".
+
+include "pr3/iso.ma".
+
+include "iso/props.ma".
+
+theorem sn3_pr3_trans:
+ \forall (c: C).(\forall (t1: T).((sn3 c t1) \to (\forall (t2: T).((pr3 c t1 
+t2) \to (sn3 c t2)))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (H: (sn3 c t1)).(sn3_ind c (\lambda 
+(t: T).(\forall (t2: T).((pr3 c t t2) \to (sn3 c t2)))) (\lambda (t2: 
+T).(\lambda (H0: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: 
+Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H1: ((\forall 
+(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to 
+(\forall (t4: T).((pr3 c t3 t4) \to (sn3 c t4)))))))).(\lambda (t3: 
+T).(\lambda (H2: (pr3 c t2 t3)).(sn3_sing c t3 (\lambda (t0: T).(\lambda (H3: 
+(((eq T t3 t0) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 c t3 t0)).(let 
+H_x \def (term_dec t2 t3) in (let H5 \def H_x in (or_ind (eq T t2 t3) ((eq T 
+t2 t3) \to (\forall (P: Prop).P)) (sn3 c t0) (\lambda (H6: (eq T t2 t3)).(let 
+H7 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t t0)) H4 t2 H6) in (let H8 
+\def (eq_ind_r T t3 (\lambda (t: T).((eq T t t0) \to (\forall (P: Prop).P))) 
+H3 t2 H6) in (let H9 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t2 t)) H2 t2 
+H6) in (H0 t0 H8 H7))))) (\lambda (H6: (((eq T t2 t3) \to (\forall (P: 
+Prop).P)))).(H1 t3 H6 H2 t0 H4)) H5)))))))))))) t1 H))).
+
+theorem sn3_pr2_intro:
+ \forall (c: C).(\forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to 
+(\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (H: ((\forall (t2: T).((((eq T t1 
+t2) \to (\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c 
+t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H0: (((eq T t1 t2) \to 
+(\forall (P: Prop).P)))).(\lambda (H1: (pr3 c t1 t2)).(let H2 \def H0 in 
+((let H3 \def H in (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(((\forall 
+(t3: T).((((eq T t t3) \to (\forall (P: Prop).P))) \to ((pr2 c t t3) \to (sn3 
+c t3))))) \to ((((eq T t t0) \to (\forall (P: Prop).P))) \to (sn3 c t0))))) 
+(\lambda (t: T).(\lambda (H4: ((\forall (t3: T).((((eq T t t3) \to (\forall 
+(P: Prop).P))) \to ((pr2 c t t3) \to (sn3 c t3)))))).(\lambda (H5: (((eq T t 
+t) \to (\forall (P: Prop).P)))).(H4 t H5 (pr2_free c t t (pr0_refl t)))))) 
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (H4: (pr2 c t4 t3)).(\lambda (t5: 
+T).(\lambda (H5: (pr3 c t3 t5)).(\lambda (H6: ((((\forall (t6: T).((((eq T t3 
+t6) \to (\forall (P: Prop).P))) \to ((pr2 c t3 t6) \to (sn3 c t6))))) \to 
+((((eq T t3 t5) \to (\forall (P: Prop).P))) \to (sn3 c t5))))).(\lambda (H7: 
+((\forall (t6: T).((((eq T t4 t6) \to (\forall (P: Prop).P))) \to ((pr2 c t4 
+t6) \to (sn3 c t6)))))).(\lambda (H8: (((eq T t4 t5) \to (\forall (P: 
+Prop).P)))).(let H_x \def (term_dec t4 t3) in (let H9 \def H_x in (or_ind (eq 
+T t4 t3) ((eq T t4 t3) \to (\forall (P: Prop).P)) (sn3 c t5) (\lambda (H10: 
+(eq T t4 t3)).(let H11 \def (eq_ind T t4 (\lambda (t: T).((eq T t t5) \to 
+(\forall (P: Prop).P))) H8 t3 H10) in (let H12 \def (eq_ind T t4 (\lambda (t: 
+T).(\forall (t6: T).((((eq T t t6) \to (\forall (P: Prop).P))) \to ((pr2 c t 
+t6) \to (sn3 c t6))))) H7 t3 H10) in (let H13 \def (eq_ind T t4 (\lambda (t: 
+T).(pr2 c t t3)) H4 t3 H10) in (H6 H12 H11))))) (\lambda (H10: (((eq T t4 t3) 
+\to (\forall (P: Prop).P)))).(sn3_pr3_trans c t3 (H7 t3 H10 H4) t5 H5)) 
+H9))))))))))) t1 t2 H1 H3)) H2)))))))).
+
+theorem sn3_cast:
+ \forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: T).((sn3 c t) \to 
+(sn3 c (THead (Flat Cast) u t))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c u)).(sn3_ind c (\lambda 
+(t: T).(\forall (t0: T).((sn3 c t0) \to (sn3 c (THead (Flat Cast) t t0))))) 
+(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall 
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
+(\forall (t: T).((sn3 c t) \to (sn3 c (THead (Flat Cast) t2 
+t))))))))).(\lambda (t: T).(\lambda (H2: (sn3 c t)).(sn3_ind c (\lambda (t0: 
+T).(sn3 c (THead (Flat Cast) t1 t0))) (\lambda (t0: T).(\lambda (H3: 
+((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 
+t2) \to (sn3 c t2)))))).(\lambda (H4: ((\forall (t2: T).((((eq T t0 t2) \to 
+(\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c (THead (Flat Cast) t1 
+t2))))))).(sn3_pr2_intro c (THead (Flat Cast) t1 t0) (\lambda (t2: 
+T).(\lambda (H5: (((eq T (THead (Flat Cast) t1 t0) t2) \to (\forall (P: 
+Prop).P)))).(\lambda (H6: (pr2 c (THead (Flat Cast) t1 t0) t2)).(let H7 \def 
+(pr2_gen_cast c t1 t0 t2 H6) 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).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3)))) (pr2 c 
+t0 t2) (sn3 c t2) (\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 
+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).(pr2 c t1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3))) (sn3 c t2) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead (Flat Cast) x0 
+x1))).(\lambda (H10: (pr2 c t1 x0)).(\lambda (H11: (pr2 c t0 x1)).(let H12 
+\def (eq_ind T t2 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) t3) \to 
+(\forall (P: Prop).P))) H5 (THead (Flat Cast) x0 x1) H9) in (eq_ind_r T 
+(THead (Flat Cast) x0 x1) (\lambda (t3: T).(sn3 c t3)) (let H_x \def 
+(term_dec x0 t1) in (let H13 \def H_x in (or_ind (eq T x0 t1) ((eq T x0 t1) 
+\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) x0 x1)) (\lambda (H14: 
+(eq T x0 t1)).(let H15 \def (eq_ind T x0 (\lambda (t3: T).((eq T (THead (Flat 
+Cast) t1 t0) (THead (Flat Cast) t3 x1)) \to (\forall (P: Prop).P))) H12 t1 
+H14) in (let H16 \def (eq_ind T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1 
+H14) in (eq_ind_r T t1 (\lambda (t3: T).(sn3 c (THead (Flat Cast) t3 x1))) 
+(let H_x0 \def (term_dec t0 x1) in (let H17 \def H_x0 in (or_ind (eq T t0 x1) 
+((eq T t0 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) t1 x1)) 
+(\lambda (H18: (eq T t0 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t3: 
+T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat Cast) t1 t3)) \to (\forall 
+(P: Prop).P))) H15 t0 H18) in (let H20 \def (eq_ind_r T x1 (\lambda (t3: 
+T).(pr2 c t0 t3)) H11 t0 H18) in (eq_ind T t0 (\lambda (t3: T).(sn3 c (THead 
+(Flat Cast) t1 t3))) (H19 (refl_equal T (THead (Flat Cast) t1 t0)) (sn3 c 
+(THead (Flat Cast) t1 t0))) x1 H18)))) (\lambda (H18: (((eq T t0 x1) \to 
+(\forall (P: Prop).P)))).(H4 x1 H18 (pr3_pr2 c t0 x1 H11))) H17))) x0 H14)))) 
+(\lambda (H14: (((eq T x0 t1) \to (\forall (P: Prop).P)))).(H1 x0 (\lambda 
+(H15: (eq T t1 x0)).(\lambda (P: Prop).(let H16 \def (eq_ind_r T x0 (\lambda 
+(t3: T).((eq T t3 t1) \to (\forall (P0: Prop).P0))) H14 t1 H15) in (let H17 
+\def (eq_ind_r T x0 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead 
+(Flat Cast) t3 x1)) \to (\forall (P0: Prop).P0))) H12 t1 H15) in (let H18 
+\def (eq_ind_r T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1 H15) in (H16 
+(refl_equal T t1) P)))))) (pr3_pr2 c t1 x0 H10) x1 (let H_x0 \def (term_dec 
+t0 x1) in (let H15 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to 
+(\forall (P: Prop).P)) (sn3 c x1) (\lambda (H16: (eq T t0 x1)).(let H17 \def 
+(eq_ind_r T x1 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat 
+Cast) x0 t3)) \to (\forall (P: Prop).P))) H12 t0 H16) in (let H18 \def 
+(eq_ind_r T x1 (\lambda (t3: T).(pr2 c t0 t3)) H11 t0 H16) in (eq_ind T t0 
+(\lambda (t3: T).(sn3 c t3)) (sn3_sing c t0 H3) x1 H16)))) (\lambda (H16: 
+(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H3 x1 H16 (pr3_pr2 c t0 x1 
+H11))) H15))))) H13))) t2 H9))))))) H8)) (\lambda (H8: (pr2 c t0 
+t2)).(sn3_pr3_trans c t0 (sn3_sing c t0 H3) t2 (pr3_pr2 c t0 t2 H8))) 
+H7))))))))) t H2)))))) u H))).
+
+theorem sn3_appl_lref:
+ \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (v: 
+T).((sn3 c v) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))
+\def
+ \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda 
+(v: T).(\lambda (H0: (sn3 c v)).(sn3_ind c (\lambda (t: T).(sn3 c (THead 
+(Flat Appl) t (TLRef i)))) (\lambda (t1: T).(\lambda (_: ((\forall (t2: 
+T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c 
+t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
+Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c (THead (Flat Appl) t2 (TLRef 
+i)))))))).(sn3_pr2_intro c (THead (Flat Appl) t1 (TLRef i)) (\lambda (t2: 
+T).(\lambda (H3: (((eq T (THead (Flat Appl) t1 (TLRef i)) t2) \to (\forall 
+(P: Prop).P)))).(\lambda (H4: (pr2 c (THead (Flat Appl) t1 (TLRef i)) 
+t2)).(let H5 \def (pr2_gen_appl c t1 (TLRef i) t2 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).(pr2 c t1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) 
+(sn3 c t2) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T 
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr2 c t1 
+x0)).(\lambda (H9: (pr2 c (TLRef i) x1)).(let H10 \def (eq_ind T t2 (\lambda 
+(t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to (\forall (P: Prop).P))) 
+H3 (THead (Flat Appl) x0 x1) H7) in (eq_ind_r T (THead (Flat Appl) x0 x1) 
+(\lambda (t: T).(sn3 c t)) (let H11 \def (eq_ind_r T x1 (\lambda (t: T).((eq 
+T (THead (Flat Appl) t1 (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P: 
+Prop).P))) H10 (TLRef i) (H x1 H9)) in (let H12 \def (eq_ind_r T x1 (\lambda 
+(t: T).(pr2 c (TLRef i) t)) H9 (TLRef i) (H x1 H9)) in (eq_ind T (TLRef i) 
+(\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H_x \def (term_dec t1 
+x0) in (let H13 \def H_x in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall 
+(P: Prop).P)) (sn3 c (THead (Flat Appl) x0 (TLRef i))) (\lambda (H14: (eq T 
+t1 x0)).(let H15 \def (eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat 
+Appl) t1 (TLRef i)) (THead (Flat Appl) t (TLRef i))) \to (\forall (P: 
+Prop).P))) H11 t1 H14) in (let H16 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c 
+t1 t)) H8 t1 H14) in (eq_ind T t1 (\lambda (t: T).(sn3 c (THead (Flat Appl) t 
+(TLRef i)))) (H15 (refl_equal T (THead (Flat Appl) t1 (TLRef i))) (sn3 c 
+(THead (Flat Appl) t1 (TLRef i)))) x0 H14)))) (\lambda (H14: (((eq T t1 x0) 
+\to (\forall (P: Prop).P)))).(H2 x0 H14 (pr3_pr2 c t1 x0 H8))) H13))) x1 (H 
+x1 H9)))) t2 H7))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T 
+T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))) 
+(sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H7: (eq T (TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H8: 
+(eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c t1 x2)).(\lambda (_: 
+((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let 
+H11 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) 
+t) \to (\forall (P: Prop).P))) H3 (THead (Bind Abbr) x2 x3) H8) in (eq_ind_r 
+T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) (let H12 \def (eq_ind 
+T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
+\Rightarrow False])) I (THead (Bind Abst) x0 x1) H7) in (False_ind (sn3 c 
+(THead (Bind Abbr) x2 x3)) H12)) t2 H8)))))))))) 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 (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq 
+T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind b) y1 
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat 
+Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2) (\lambda (x0: B).(\lambda (x1: 
+T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: 
+T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H8: (eq T (TLRef i) (THead 
+(Bind x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x5 (THead (Flat 
+Appl) (lift (S O) O x4) x3)))).(\lambda (_: (pr2 c t1 x4)).(\lambda (_: (pr2 
+c x1 x5)).(\lambda (_: (pr2 (CHead c (Bind x0) x5) x2 x3)).(let H13 \def 
+(eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to 
+(\forall (P: Prop).P))) H3 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) 
+O x4) x3)) H9) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S 
+O) O x4) x3)) (\lambda (t: T).(sn3 c t)) (let H14 \def (eq_ind T (TLRef i) 
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
+False])) I (THead (Bind x0) x1 x2) H8) in (False_ind (sn3 c (THead (Bind x0) 
+x5 (THead (Flat Appl) (lift (S O) O x4) x3))) H14)) t2 H9)))))))))))))) H6)) 
+H5))))))))) v H0))))).
+
+theorem sn3_appl_cast:
+ \forall (c: C).(\forall (v: T).(\forall (u: T).((sn3 c (THead (Flat Appl) v 
+u)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) v t)) \to (sn3 c (THead 
+(Flat Appl) v (THead (Flat Cast) u t))))))))
+\def
+ \lambda (c: C).(\lambda (v: T).(\lambda (u: T).(\lambda (H: (sn3 c (THead 
+(Flat Appl) v u))).(insert_eq T (THead (Flat Appl) v u) (\lambda (t: T).(sn3 
+c t)) (\forall (t: T).((sn3 c (THead (Flat Appl) v t)) \to (sn3 c (THead 
+(Flat Appl) v (THead (Flat Cast) u t))))) (\lambda (y: T).(\lambda (H0: (sn3 
+c y)).(unintro T u (\lambda (t: T).((eq T y (THead (Flat Appl) v t)) \to 
+(\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to (sn3 c (THead (Flat 
+Appl) v (THead (Flat Cast) t t0))))))) (unintro T v (\lambda (t: T).(\forall 
+(x: T).((eq T y (THead (Flat Appl) t x)) \to (\forall (t0: T).((sn3 c (THead 
+(Flat Appl) t t0)) \to (sn3 c (THead (Flat Appl) t (THead (Flat Cast) x 
+t0)))))))) (sn3_ind c (\lambda (t: T).(\forall (x: T).(\forall (x0: T).((eq T 
+t (THead (Flat Appl) x x0)) \to (\forall (t0: T).((sn3 c (THead (Flat Appl) x 
+t0)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 t0))))))))) 
+(\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall 
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
+(\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat Appl) x x0)) \to 
+(\forall (t: T).((sn3 c (THead (Flat Appl) x t)) \to (sn3 c (THead (Flat 
+Appl) x (THead (Flat Cast) x0 t))))))))))))).(\lambda (x: T).(\lambda (x0: 
+T).(\lambda (H3: (eq T t1 (THead (Flat Appl) x x0))).(\lambda (t: T).(\lambda 
+(H4: (sn3 c (THead (Flat Appl) x t))).(insert_eq T (THead (Flat Appl) x t) 
+(\lambda (t0: T).(sn3 c t0)) (sn3 c (THead (Flat Appl) x (THead (Flat Cast) 
+x0 t))) (\lambda (y0: T).(\lambda (H5: (sn3 c y0)).(unintro T t (\lambda (t0: 
+T).((eq T y0 (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead 
+(Flat Cast) x0 t0))))) (sn3_ind c (\lambda (t0: T).(\forall (x1: T).((eq T t0 
+(THead (Flat Appl) x x1)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) 
+x0 x1)))))) (\lambda (t0: T).(\lambda (H6: ((\forall (t2: T).((((eq T t0 t2) 
+\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2)))))).(\lambda 
+(H7: ((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 
+c t0 t2) \to (\forall (x1: T).((eq T t2 (THead (Flat Appl) x x1)) \to (sn3 c 
+(THead (Flat Appl) x (THead (Flat Cast) x0 x1)))))))))).(\lambda (x1: 
+T).(\lambda (H8: (eq T t0 (THead (Flat Appl) x x1))).(let H9 \def (eq_ind T 
+t0 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: 
+Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).((eq T t3 (THead (Flat 
+Appl) x x2)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 
+x2))))))))) H7 (THead (Flat Appl) x x1) H8) in (let H10 \def (eq_ind T t0 
+(\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) 
+\to ((pr3 c t2 t3) \to (sn3 c t3))))) H6 (THead (Flat Appl) x x1) H8) in (let 
+H11 \def (eq_ind T t1 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to 
+(\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).(\forall (x3: 
+T).((eq T t3 (THead (Flat Appl) x2 x3)) \to (\forall (t4: T).((sn3 c (THead 
+(Flat Appl) x2 t4)) \to (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x3 
+t4)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H12 \def (eq_ind T t1 
+(\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) 
+\to ((pr3 c t2 t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in 
+(sn3_pr2_intro c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (\lambda 
+(t2: T).(\lambda (H13: (((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 
+x1)) t2) \to (\forall (P: Prop).P)))).(\lambda (H14: (pr2 c (THead (Flat 
+Appl) x (THead (Flat Cast) x0 x1)) t2)).(let H15 \def (pr2_gen_appl c x 
+(THead (Flat Cast) x0 x1) t2 H14) in (or3_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
+(THead (Flat Cast) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: 
+B).(\forall (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 (Flat Cast) 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 t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) 
+y2) z1 z2)))))))) (sn3 c t2) (\lambda (H16: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
+(THead (Flat Cast) x0 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).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Flat Cast) 
+x0 x1) t3))) (sn3 c t2) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H17: (eq 
+T t2 (THead (Flat Appl) x2 x3))).(\lambda (H18: (pr2 c x x2)).(\lambda (H19: 
+(pr2 c (THead (Flat Cast) x0 x1) x3)).(let H20 \def (eq_ind T t2 (\lambda 
+(t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to 
+(\forall (P: Prop).P))) H13 (THead (Flat Appl) x2 x3) H17) in (eq_ind_r T 
+(THead (Flat Appl) x2 x3) (\lambda (t3: T).(sn3 c t3)) (let H21 \def 
+(pr2_gen_cast c x0 x1 x3 H19) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3)))) (pr2 c 
+x1 x3) (sn3 c (THead (Flat Appl) x2 x3)) (\lambda (H22: (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x3 (THead 
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3))) (sn3 c (THead (Flat Appl) x2 
+x3)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H23: (eq T x3 (THead (Flat 
+Cast) x4 x5))).(\lambda (H24: (pr2 c x0 x4)).(\lambda (H25: (pr2 c x1 
+x5)).(let H26 \def (eq_ind T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x 
+(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P: 
+Prop).P))) H20 (THead (Flat Cast) x4 x5) H23) in (eq_ind_r T (THead (Flat 
+Cast) x4 x5) (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 t3))) (let H_x 
+\def (term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) in (let 
+H27 \def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 
+x4)) ((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) \to (\forall 
+(P: Prop).P)) (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x4 x5))) 
+(\lambda (H28: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 
+x4))).(let H29 \def (f_equal T T (\lambda (e: T).(match e in T return 
+(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | 
+(THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x0) (THead (Flat Appl) 
+x2 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T 
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) 
+\Rightarrow x0 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x0) 
+(THead (Flat Appl) x2 x4) H28) in (\lambda (H31: (eq T x x2)).(let H32 \def 
+(eq_ind_r T x4 (\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat 
+Cast) x0 x1)) (THead (Flat Appl) x2 (THead (Flat Cast) t3 x5))) \to (\forall 
+(P: Prop).P))) H26 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t3: 
+T).(pr2 c x0 t3)) H24 x0 H30) in (eq_ind T x0 (\lambda (t3: T).(sn3 c (THead 
+(Flat Appl) x2 (THead (Flat Cast) t3 x5)))) (let H34 \def (eq_ind_r T x2 
+(\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) 
+(THead (Flat Appl) t3 (THead (Flat Cast) x0 x5))) \to (\forall (P: Prop).P))) 
+H32 x H31) in (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18 
+x H31) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead 
+(Flat Cast) x0 x5)))) (let H_x0 \def (term_dec (THead (Flat Appl) x x1) 
+(THead (Flat Appl) x x5)) in (let H36 \def H_x0 in (or_ind (eq T (THead (Flat 
+Appl) x x1) (THead (Flat Appl) x x5)) ((eq T (THead (Flat Appl) x x1) (THead 
+(Flat Appl) x x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x 
+(THead (Flat Cast) x0 x5))) (\lambda (H37: (eq T (THead (Flat Appl) x x1) 
+(THead (Flat Appl) x x5))).(let H38 \def (f_equal T T (\lambda (e: T).(match 
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) 
+\Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x1) 
+(THead (Flat Appl) x x5) H37) in (let H39 \def (eq_ind_r T x5 (\lambda (t3: 
+T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) 
+x (THead (Flat Cast) x0 t3))) \to (\forall (P: Prop).P))) H34 x1 H38) in (let 
+H40 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H38) in 
+(eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) 
+x0 t3)))) (H39 (refl_equal T (THead (Flat Appl) x (THead (Flat Cast) x0 x1))) 
+(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)))) x5 H38))))) (\lambda 
+(H37: (((eq T (THead (Flat Appl) x x1) (THead (Flat Appl) x x5)) \to (\forall 
+(P: Prop).P)))).(H9 (THead (Flat Appl) x x5) H37 (pr3_pr2 c (THead (Flat 
+Appl) x x1) (THead (Flat Appl) x x5) (pr2_thin_dx c x1 x5 H25 x Appl)) x5 
+(refl_equal T (THead (Flat Appl) x x5)))) H36))) x2 H31))) x4 H30))))) H29))) 
+(\lambda (H28: (((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) 
+\to (\forall (P: Prop).P)))).(let H_x0 \def (term_dec (THead (Flat Appl) x 
+x1) (THead (Flat Appl) x2 x5)) in (let H29 \def H_x0 in (or_ind (eq T (THead 
+(Flat Appl) x x1) (THead (Flat Appl) x2 x5)) ((eq T (THead (Flat Appl) x x1) 
+(THead (Flat Appl) x2 x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat 
+Appl) x2 (THead (Flat Cast) x4 x5))) (\lambda (H30: (eq T (THead (Flat Appl) 
+x x1) (THead (Flat Appl) x2 x5))).(let H31 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | 
+(TLRef _) \Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) 
+x x1) (THead (Flat Appl) x2 x5) H30) in ((let H32 \def (f_equal T T (\lambda 
+(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 
+| (TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat 
+Appl) x x1) (THead (Flat Appl) x2 x5) H30) in (\lambda (H33: (eq T x 
+x2)).(let H34 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H32) 
+in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 (THead (Flat 
+Cast) x4 t3)))) (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead 
+(Flat Appl) x x0) (THead (Flat Appl) t3 x4)) \to (\forall (P: Prop).P))) H28 
+x H33) in (let H36 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18 x 
+H33) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead 
+(Flat Cast) x4 x1)))) (H11 (THead (Flat Appl) x x4) H35 (pr3_pr2 c (THead 
+(Flat Appl) x x0) (THead (Flat Appl) x x4) (pr2_thin_dx c x0 x4 H24 x Appl)) 
+x x4 (refl_equal T (THead (Flat Appl) x x4)) x1 (sn3_sing c (THead (Flat 
+Appl) x x1) H10)) x2 H33))) x5 H32)))) H31))) (\lambda (H30: (((eq T (THead 
+(Flat Appl) x x1) (THead (Flat Appl) x2 x5)) \to (\forall (P: 
+Prop).P)))).(H11 (THead (Flat Appl) x2 x4) H28 (pr3_head_12 c x x2 (pr3_pr2 c 
+x x2 H18) (Flat Appl) x0 x4 (pr3_pr2 (CHead c (Flat Appl) x2) x0 x4 
+(pr2_cflat c x0 x4 H24 Appl x2))) x2 x4 (refl_equal T (THead (Flat Appl) x2 
+x4)) x5 (H10 (THead (Flat Appl) x2 x5) H30 (pr3_head_12 c x x2 (pr3_pr2 c x 
+x2 H18) (Flat Appl) x1 x5 (pr3_pr2 (CHead c (Flat Appl) x2) x1 x5 (pr2_cflat 
+c x1 x5 H25 Appl x2)))))) H29)))) H27))) x3 H23))))))) H22)) (\lambda (H22: 
+(pr2 c x1 x3)).(let H_x \def (term_dec (THead (Flat Appl) x x1) (THead (Flat 
+Appl) x2 x3)) in (let H23 \def H_x in (or_ind (eq T (THead (Flat Appl) x x1) 
+(THead (Flat Appl) x2 x3)) ((eq T (THead (Flat Appl) x x1) (THead (Flat Appl) 
+x2 x3)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x2 x3)) (\lambda 
+(H24: (eq T (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3))).(let H25 
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
+with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t3 _) 
+\Rightarrow t3])) (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3) H24) in 
+((let H26 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
+T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ 
+t3) \Rightarrow t3])) (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3) H24) 
+in (\lambda (H27: (eq T x x2)).(let H28 \def (eq_ind_r T x3 (\lambda (t3: 
+T).(pr2 c x1 t3)) H22 x1 H26) in (let H29 \def (eq_ind_r T x3 (\lambda (t3: 
+T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) 
+x2 t3)) \to (\forall (P: Prop).P))) H20 x1 H26) in (eq_ind T x1 (\lambda (t3: 
+T).(sn3 c (THead (Flat Appl) x2 t3))) (let H30 \def (eq_ind_r T x2 (\lambda 
+(t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat 
+Appl) t3 x1)) \to (\forall (P: Prop).P))) H29 x H27) in (let H31 \def 
+(eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18 x H27) in (eq_ind T x 
+(\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 x1))) (sn3_sing c (THead (Flat 
+Appl) x x1) H10) x2 H27))) x3 H26))))) H25))) (\lambda (H24: (((eq T (THead 
+(Flat Appl) x x1) (THead (Flat Appl) x2 x3)) \to (\forall (P: 
+Prop).P)))).(H10 (THead (Flat Appl) x2 x3) H24 (pr3_head_12 c x x2 (pr3_pr2 c 
+x x2 H18) (Flat Appl) x1 x3 (pr3_pr2 (CHead c (Flat Appl) x2) x1 x3 
+(pr2_cflat c x1 x3 H22 Appl x2))))) H23)))) H21)) t2 H17))))))) H16)) 
+(\lambda (H16: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: 
+T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) 
+x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x 
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))) 
+(sn3 c t2) (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: 
+T).(\lambda (H17: (eq T (THead (Flat Cast) x0 x1) (THead (Bind Abst) x2 
+x3))).(\lambda (H18: (eq T t2 (THead (Bind Abbr) x4 x5))).(\lambda (_: (pr2 c 
+x x4)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+u0) x3 x5))))).(let H21 \def (eq_ind T t2 (\lambda (t3: T).((eq T (THead 
+(Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P: Prop).P))) H13 
+(THead (Bind Abbr) x4 x5) H18) in (eq_ind_r T (THead (Bind Abbr) x4 x5) 
+(\lambda (t3: T).(sn3 c t3)) (let H22 \def (eq_ind T (THead (Flat Cast) x0 
+x1) (\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) x2 
+x3) H17) in (False_ind (sn3 c (THead (Bind Abbr) x4 x5)) H22)) t2 
+H18)))))))))) H16)) (\lambda (H16: (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 (Flat 
+Cast) 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 
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) 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 t2 (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2) 
+(\lambda (x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda 
+(x6: T).(\lambda (x7: T).(\lambda (_: (not (eq B x2 Abst))).(\lambda (H18: 
+(eq T (THead (Flat Cast) x0 x1) (THead (Bind x2) x3 x4))).(\lambda (H19: (eq 
+T t2 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda 
+(_: (pr2 c x x6)).(\lambda (_: (pr2 c x3 x7)).(\lambda (_: (pr2 (CHead c 
+(Bind x2) x7) x4 x5)).(let H23 \def (eq_ind T t2 (\lambda (t3: T).((eq T 
+(THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P: 
+Prop).P))) H13 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) 
+H19) in (eq_ind_r T (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) 
+x5)) (\lambda (t3: T).(sn3 c t3)) (let H24 \def (eq_ind T (THead (Flat Cast) 
+x0 x1) (\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 x2) x3 x4) 
+H18) in (False_ind (sn3 c (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) 
+O x6) x5))) H24)) t2 H19)))))))))))))) H16)) H15))))))))))))))) y0 H5)))) 
+H4))))))))) y H0))))) H)))).
+
+theorem sn3_appl_appl:
+ \forall (v1: T).(\forall (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in 
+(\forall (c: C).((sn3 c u1) \to (\forall (v2: T).((sn3 c v2) \to (((\forall 
+(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to 
+(sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 
+u1)))))))))
+\def
+ \lambda (v1: T).(\lambda (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in 
+(\lambda (c: C).(\lambda (H: (sn3 c (THead (Flat Appl) v1 t1))).(insert_eq T 
+(THead (Flat Appl) v1 t1) (\lambda (t: T).(sn3 c t)) (\forall (v2: T).((sn3 c 
+v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) v1 t1) u2) \to ((((iso 
+(THead (Flat Appl) v1 t1) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead 
+(Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) 
+v1 t1)))))) (\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t1 (\lambda 
+(t: T).((eq T y (THead (Flat Appl) v1 t)) \to (\forall (v2: T).((sn3 c v2) 
+\to (((\forall (u2: T).((pr3 c (THead (Flat Appl) v1 t) u2) \to ((((iso 
+(THead (Flat Appl) v1 t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead 
+(Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) 
+v1 t)))))))) (unintro T v1 (\lambda (t: T).(\forall (x: T).((eq T y (THead 
+(Flat Appl) t x)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: 
+T).((pr3 c (THead (Flat Appl) t x) u2) \to ((((iso (THead (Flat Appl) t x) 
+u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to 
+(sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) t x))))))))) (sn3_ind c 
+(\lambda (t: T).(\forall (x: T).(\forall (x0: T).((eq T t (THead (Flat Appl) 
+x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c (THead 
+(Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall 
+(P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead 
+(Flat Appl) v2 (THead (Flat Appl) x x0)))))))))) (\lambda (t2: T).(\lambda 
+(H1: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 
+c t2 t3) \to (sn3 c t3)))))).(\lambda (H2: ((\forall (t3: T).((((eq T t2 t3) 
+\to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x: T).(\forall 
+(x0: T).((eq T t3 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) 
+\to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso 
+(THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead 
+(Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) x 
+x0)))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t2 
+(THead (Flat Appl) x x0))).(\lambda (v2: T).(\lambda (H4: (sn3 c 
+v2)).(sn3_ind c (\lambda (t: T).(((\forall (u2: T).((pr3 c (THead (Flat Appl) 
+x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: 
+Prop).P))) \to (sn3 c (THead (Flat Appl) t u2)))))) \to (sn3 c (THead (Flat 
+Appl) t (THead (Flat Appl) x x0))))) (\lambda (t0: T).(\lambda (H5: ((\forall 
+(t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to 
+(sn3 c t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t0 t3) \to (\forall 
+(P: Prop).P))) \to ((pr3 c t0 t3) \to (((\forall (u2: T).((pr3 c (THead (Flat 
+Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: 
+Prop).P))) \to (sn3 c (THead (Flat Appl) t3 u2)))))) \to (sn3 c (THead (Flat 
+Appl) t3 (THead (Flat Appl) x x0))))))))).(\lambda (H7: ((\forall (u2: 
+T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) 
+u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 
+u2))))))).(let H8 \def (eq_ind T t2 (\lambda (t: T).(\forall (t3: T).((((eq T 
+t t3) \to (\forall (P: Prop).P))) \to ((pr3 c t t3) \to (\forall (x1: 
+T).(\forall (x2: T).((eq T t3 (THead (Flat Appl) x1 x2)) \to (\forall (v3: 
+T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x1 x2) u2) 
+\to ((((iso (THead (Flat Appl) x1 x2) u2) \to (\forall (P: Prop).P))) \to 
+(sn3 c (THead (Flat Appl) v3 u2)))))) \to (sn3 c (THead (Flat Appl) v3 (THead 
+(Flat Appl) x1 x2))))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H9 
+\def (eq_ind T t2 (\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to 
+(\forall (P: Prop).P))) \to ((pr3 c t t3) \to (sn3 c t3))))) H1 (THead (Flat 
+Appl) x x0) H3) in (sn3_pr2_intro c (THead (Flat Appl) t0 (THead (Flat Appl) 
+x x0)) (\lambda (t3: T).(\lambda (H10: (((eq T (THead (Flat Appl) t0 (THead 
+(Flat Appl) x x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H11: (pr2 c 
+(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t3)).(let H12 \def 
+(pr2_gen_appl c t0 (THead (Flat Appl) x x0) t3 H11) in (or3_ind (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda 
+(t4: T).(pr2 c (THead (Flat Appl) x x0) t4)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) 
+x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq 
+T (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) 
+y2) z1 z2)))))))) (sn3 c t3) (\lambda (H13: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c 
+(THead (Flat Appl) x x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda 
+(t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Flat Appl) 
+x x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H14: (eq T 
+t3 (THead (Flat Appl) x1 x2))).(\lambda (H15: (pr2 c t0 x1)).(\lambda (H16: 
+(pr2 c (THead (Flat Appl) x x0) x2)).(let H17 \def (eq_ind T t3 (\lambda (t: 
+T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: 
+Prop).P))) H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat 
+Appl) x1 x2) (\lambda (t: T).(sn3 c t)) (let H18 \def (pr2_gen_appl c x x0 x2 
+H16) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead 
+(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
+(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4)))) (ex4_4 T T T T (\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 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 x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c 
+(THead (Flat Appl) x1 x2)) (\lambda (H19: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t4: T).(eq T x2 (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c x0 
+t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead 
+(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
+(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4))) (sn3 c (THead (Flat Appl) x1 
+x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H20: (eq T x2 (THead (Flat 
+Appl) x3 x4))).(\lambda (H21: (pr2 c x x3)).(\lambda (H22: (pr2 c x0 
+x4)).(let H23 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 
+(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: 
+Prop).P))) H17 (THead (Flat Appl) x3 x4) H20) in (eq_ind_r T (THead (Flat 
+Appl) x3 x4) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H_x \def 
+(term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) in (let H24 
+\def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) 
+((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) \to (\forall (P: 
+Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 x4))) (\lambda 
+(H25: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))).(let H26 
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
+with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) 
+\Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H25) in 
+((let H27 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
+T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ 
+t) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H25) 
+in (\lambda (H28: (eq T x x3)).(let H29 \def (eq_ind_r T x4 (\lambda (t: 
+T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) 
+x1 (THead (Flat Appl) x3 t))) \to (\forall (P: Prop).P))) H23 x0 H27) in (let 
+H30 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) H22 x0 H27) in (eq_ind 
+T x0 (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 t)))) 
+(let H31 \def (eq_ind_r T x3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 
+(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 (THead (Flat Appl) t x0))) 
+\to (\forall (P: Prop).P))) H29 x H28) in (let H32 \def (eq_ind_r T x3 
+(\lambda (t: T).(pr2 c x t)) H21 x H28) in (eq_ind T x (\lambda (t: T).(sn3 c 
+(THead (Flat Appl) x1 (THead (Flat Appl) t x0)))) (let H_x0 \def (term_dec t0 
+x1) in (let H33 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to (\forall 
+(P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x x0))) 
+(\lambda (H34: (eq T t0 x1)).(let H35 \def (eq_ind_r T x1 (\lambda (t: 
+T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) 
+t (THead (Flat Appl) x x0))) \to (\forall (P: Prop).P))) H31 t0 H34) in (let 
+H36 \def (eq_ind_r T x1 (\lambda (t: T).(pr2 c t0 t)) H15 t0 H34) in (eq_ind 
+T t0 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (THead (Flat Appl) x x0)))) 
+(H35 (refl_equal T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))) (sn3 c 
+(THead (Flat Appl) t0 (THead (Flat Appl) x x0)))) x1 H34)))) (\lambda (H34: 
+(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H6 x1 H34 (pr3_pr2 c t0 x1 H15) 
+(\lambda (u2: T).(\lambda (H35: (pr3 c (THead (Flat Appl) x x0) u2)).(\lambda 
+(H36: (((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: 
+Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H7 u2 H35 H36) (THead 
+(Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2) (THead (Flat Appl) x1 
+u2) (pr2_head_1 c t0 x1 H15 (Flat Appl) u2)))))))) H33))) x3 H28))) x4 
+H27))))) H26))) (\lambda (H25: (((eq T (THead (Flat Appl) x x0) (THead (Flat 
+Appl) x3 x4)) \to (\forall (P: Prop).P)))).(H8 (THead (Flat Appl) x3 x4) H25 
+(pr3_head_12 c x x3 (pr3_pr2 c x x3 H21) (Flat Appl) x0 x4 (pr3_pr2 (CHead c 
+(Flat Appl) x3) x0 x4 (pr2_cflat c x0 x4 H22 Appl x3))) x3 x4 (refl_equal T 
+(THead (Flat Appl) x3 x4)) x1 (sn3_pr3_trans c t0 (sn3_sing c t0 H5) x1 
+(pr3_pr2 c t0 x1 H15)) (\lambda (u2: T).(\lambda (H26: (pr3 c (THead (Flat 
+Appl) x3 x4) u2)).(\lambda (H27: (((iso (THead (Flat Appl) x3 x4) u2) \to 
+(\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H7 u2 
+(pr3_sing c (THead (Flat Appl) x x4) (THead (Flat Appl) x x0) (pr2_thin_dx c 
+x0 x4 H22 x Appl) u2 (pr3_sing c (THead (Flat Appl) x3 x4) (THead (Flat Appl) 
+x x4) (pr2_head_1 c x x3 H21 (Flat Appl) x4) u2 H26)) (\lambda (H28: (iso 
+(THead (Flat Appl) x x0) u2)).(\lambda (P: Prop).(H27 (iso_trans (THead (Flat 
+Appl) x3 x4) (THead (Flat Appl) x x0) (iso_head (Flat Appl) x3 x x4 x0) u2 
+H28) P)))) (THead (Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2) 
+(THead (Flat Appl) x1 u2) (pr2_head_1 c t0 x1 H15 (Flat Appl) u2)))))))) 
+H24))) x2 H20))))))) H19)) (\lambda (H19: (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 
+(CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 
+(CHead c (Bind b) u) z1 t4))))))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda 
+(x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H20: (eq 
+T x0 (THead (Bind Abst) x3 x4))).(\lambda (H21: (eq T x2 (THead (Bind Abbr) 
+x5 x6))).(\lambda (H22: (pr2 c x x5)).(\lambda (H23: ((\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x4 x6))))).(let H24 \def (eq_ind 
+T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) 
+(THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H17 (THead (Bind Abbr) 
+x5 x6) H21) in (eq_ind_r T (THead (Bind Abbr) x5 x6) (\lambda (t: T).(sn3 c 
+(THead (Flat Appl) x1 t))) (let H25 \def (eq_ind T x0 (\lambda (t: T).((eq T 
+(THead (Flat Appl) t0 (THead (Flat Appl) x t)) (THead (Flat Appl) x1 (THead 
+(Bind Abbr) x5 x6))) \to (\forall (P: Prop).P))) H24 (THead (Bind Abst) x3 
+x4) H20) in (let H26 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4: 
+T).((((eq T (THead (Flat Appl) x t) t4) \to (\forall (P: Prop).P))) \to ((pr3 
+c (THead (Flat Appl) x t) t4) \to (sn3 c t4))))) H9 (THead (Bind Abst) x3 x4) 
+H20) in (let H27 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4: T).((((eq T 
+(THead (Flat Appl) x t) t4) \to (\forall (P: Prop).P))) \to ((pr3 c (THead 
+(Flat Appl) x t) t4) \to (\forall (x7: T).(\forall (x8: T).((eq T t4 (THead 
+(Flat Appl) x7 x8)) \to (\forall (v3: T).((sn3 c v3) \to (((\forall (u2: 
+T).((pr3 c (THead (Flat Appl) x7 x8) u2) \to ((((iso (THead (Flat Appl) x7 
+x8) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) 
+\to (sn3 c (THead (Flat Appl) v3 (THead (Flat Appl) x7 x8))))))))))))) H8 
+(THead (Bind Abst) x3 x4) H20) in (let H28 \def (eq_ind T x0 (\lambda (t: 
+T).(\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead 
+(Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat 
+Appl) t0 u2)))))) H7 (THead (Bind Abst) x3 x4) H20) in (let H29 \def (eq_ind 
+T x0 (\lambda (t: T).(\forall (t4: T).((((eq T t0 t4) \to (\forall (P: 
+Prop).P))) \to ((pr3 c t0 t4) \to (((\forall (u2: T).((pr3 c (THead (Flat 
+Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P: 
+Prop).P))) \to (sn3 c (THead (Flat Appl) t4 u2)))))) \to (sn3 c (THead (Flat 
+Appl) t4 (THead (Flat Appl) x t)))))))) H6 (THead (Bind Abst) x3 x4) H20) in 
+(sn3_pr3_trans c (THead (Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (H28 (THead 
+(Bind Abbr) x5 x6) (pr3_sing c (THead (Bind Abbr) x x4) (THead (Flat Appl) x 
+(THead (Bind Abst) x3 x4)) (pr2_free c (THead (Flat Appl) x (THead (Bind 
+Abst) x3 x4)) (THead (Bind Abbr) x x4) (pr0_beta x3 x x (pr0_refl x) x4 x4 
+(pr0_refl x4))) (THead (Bind Abbr) x5 x6) (pr3_head_12 c x x5 (pr3_pr2 c x x5 
+H22) (Bind Abbr) x4 x6 (pr3_pr2 (CHead c (Bind Abbr) x5) x4 x6 (H23 Abbr 
+x5)))) (\lambda (H30: (iso (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) 
+(THead (Bind Abbr) x5 x6))).(\lambda (P: Prop).(let H31 \def (match H30 in 
+iso return (\lambda (t: T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t 
+(THead (Flat Appl) x (THead (Bind Abst) x3 x4))) \to ((eq T t4 (THead (Bind 
+Abbr) x5 x6)) \to P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H31: 
+(eq T (TSort n1) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda 
+(H32: (eq T (TSort n2) (THead (Bind Abbr) x5 x6))).((let H33 \def (eq_ind T 
+(TSort n1) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow False])) I (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H31) 
+in (False_ind ((eq T (TSort n2) (THead (Bind Abbr) x5 x6)) \to P) H33)) 
+H32))) | (iso_lref i1 i2) \Rightarrow (\lambda (H31: (eq T (TLRef i1) (THead 
+(Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H32: (eq T (TLRef i2) 
+(THead (Bind Abbr) x5 x6))).((let H33 \def (eq_ind T (TLRef i1) (\lambda (e: 
+T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
+(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H31) in (False_ind ((eq T 
+(TLRef i2) (THead (Bind Abbr) x5 x6)) \to P) H33)) H32))) | (iso_head k v4 v5 
+t4 t5) \Rightarrow (\lambda (H31: (eq T (THead k v4 t4) (THead (Flat Appl) x 
+(THead (Bind Abst) x3 x4)))).(\lambda (H32: (eq T (THead k v5 t5) (THead 
+(Bind Abbr) x5 x6))).((let H33 \def (f_equal T T (\lambda (e: T).(match e in 
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t4 | (TLRef _) 
+\Rightarrow t4 | (THead _ _ t) \Rightarrow t])) (THead k v4 t4) (THead (Flat 
+Appl) x (THead (Bind Abst) x3 x4)) H31) in ((let H34 \def (f_equal T T 
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow v4 | (TLRef _) \Rightarrow v4 | (THead _ t _) \Rightarrow t])) 
+(THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H31) in ((let 
+H35 \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 v4 t4) (THead (Flat Appl) x (THead (Bind Abst) x3 
+x4)) H31) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x) \to ((eq T 
+t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead k0 v5 t5) (THead (Bind Abbr) 
+x5 x6)) \to P)))) (\lambda (H36: (eq T v4 x)).(eq_ind T x (\lambda (_: 
+T).((eq T t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead (Flat Appl) v5 t5) 
+(THead (Bind Abbr) x5 x6)) \to P))) (\lambda (H37: (eq T t4 (THead (Bind 
+Abst) x3 x4))).(eq_ind T (THead (Bind Abst) x3 x4) (\lambda (_: T).((eq T 
+(THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6)) \to P)) (\lambda (H38: 
+(eq T (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6))).(let H39 \def 
+(eq_ind T (THead (Flat Appl) v5 t5) (\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 Abbr) x5 x6) H38) in (False_ind P H39))) t4 (sym_eq 
+T t4 (THead (Bind Abst) x3 x4) H37))) v4 (sym_eq T v4 x H36))) k (sym_eq K k 
+(Flat Appl) H35))) H34)) H33)) H32)))]) in (H31 (refl_equal T (THead (Flat 
+Appl) x (THead (Bind Abst) x3 x4))) (refl_equal T (THead (Bind Abbr) x5 
+x6))))))) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6)) (pr3_pr2 c (THead 
+(Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (THead (Flat Appl) x1 (THead (Bind 
+Abbr) x5 x6)) (pr2_head_1 c t0 x1 H15 (Flat Appl) (THead (Bind Abbr) x5 
+x6))))))))) x2 H21)))))))))) H19)) (\lambda (H19: (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 x0 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 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 x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind 
+B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+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 x u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) 
+(sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3: B).(\lambda (x4: T).(\lambda 
+(x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H20: 
+(not (eq B x3 Abst))).(\lambda (H21: (eq T x0 (THead (Bind x3) x4 
+x5))).(\lambda (H22: (eq T x2 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S 
+O) O x7) x6)))).(\lambda (H23: (pr2 c x x7)).(\lambda (H24: (pr2 c x4 
+x8)).(\lambda (H25: (pr2 (CHead c (Bind x3) x8) x5 x6)).(let H26 \def (eq_ind 
+T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) 
+(THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H17 (THead (Bind x3) x8 
+(THead (Flat Appl) (lift (S O) O x7) x6)) H22) in (eq_ind_r T (THead (Bind 
+x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (\lambda (t: T).(sn3 c 
+(THead (Flat Appl) x1 t))) (let H27 \def (eq_ind T x0 (\lambda (t: T).((eq T 
+(THead (Flat Appl) t0 (THead (Flat Appl) x t)) (THead (Flat Appl) x1 (THead 
+(Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))) \to (\forall (P: 
+Prop).P))) H26 (THead (Bind x3) x4 x5) H21) in (let H28 \def (eq_ind T x0 
+(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to 
+(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c 
+t4))))) H9 (THead (Bind x3) x4 x5) H21) in (let H29 \def (eq_ind T x0 
+(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to 
+(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall 
+(x9: T).(\forall (x10: T).((eq T t4 (THead (Flat Appl) x9 x10)) \to (\forall 
+(v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x9 x10) 
+u2) \to ((((iso (THead (Flat Appl) x9 x10) u2) \to (\forall (P: Prop).P))) 
+\to (sn3 c (THead (Flat Appl) v3 u2)))))) \to (sn3 c (THead (Flat Appl) v3 
+(THead (Flat Appl) x9 x10))))))))))))) H8 (THead (Bind x3) x4 x5) H21) in 
+(let H30 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c (THead 
+(Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P: 
+Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H7 (THead (Bind x3) x4 
+x5) H21) in (let H31 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4: 
+T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t4) \to 
+(((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead 
+(Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat 
+Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x 
+t)))))))) H6 (THead (Bind x3) x4 x5) H21) in (sn3_pr3_trans c (THead (Flat 
+Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (H30 
+(THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (pr3_sing c 
+(THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) x5)) (THead (Flat 
+Appl) x (THead (Bind x3) x4 x5)) (pr2_free c (THead (Flat Appl) x (THead 
+(Bind x3) x4 x5)) (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) 
+x5)) (pr0_upsilon x3 H20 x x (pr0_refl x) x4 x4 (pr0_refl x4) x5 x5 (pr0_refl 
+x5))) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) 
+(pr3_head_12 c x4 x8 (pr3_pr2 c x4 x8 H24) (Bind x3) (THead (Flat Appl) (lift 
+(S O) O x) x5) (THead (Flat Appl) (lift (S O) O x7) x6) (pr3_head_12 (CHead c 
+(Bind x3) x8) (lift (S O) O x) (lift (S O) O x7) (pr3_lift (CHead c (Bind x3) 
+x8) c (S O) O (drop_drop (Bind x3) O c c (drop_refl c) x8) x x7 (pr3_pr2 c x 
+x7 H23)) (Flat Appl) x5 x6 (pr3_pr2 (CHead (CHead c (Bind x3) x8) (Flat Appl) 
+(lift (S O) O x7)) x5 x6 (pr2_cflat (CHead c (Bind x3) x8) x5 x6 H25 Appl 
+(lift (S O) O x7)))))) (\lambda (H32: (iso (THead (Flat Appl) x (THead (Bind 
+x3) x4 x5)) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) 
+x6)))).(\lambda (P: Prop).(let H33 \def (match H32 in iso return (\lambda (t: 
+T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x 
+(THead (Bind x3) x4 x5))) \to ((eq T t4 (THead (Bind x3) x8 (THead (Flat 
+Appl) (lift (S O) O x7) x6))) \to P))))) with [(iso_sort n1 n2) \Rightarrow 
+(\lambda (H33: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind x3) x4 
+x5)))).(\lambda (H34: (eq T (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) 
+(lift (S O) O x7) x6)))).((let H35 \def (eq_ind T (TSort n1) (\lambda (e: 
+T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I 
+(THead (Flat Appl) x (THead (Bind x3) x4 x5)) H33) in (False_ind ((eq T 
+(TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to 
+P) H35)) H34))) | (iso_lref i1 i2) \Rightarrow (\lambda (H33: (eq T (TLRef 
+i1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H34: (eq T 
+(TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) 
+x6)))).((let H35 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T 
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x 
+(THead (Bind x3) x4 x5)) H33) in (False_ind ((eq T (TLRef i2) (THead (Bind 
+x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H35)) H34))) | 
+(iso_head k v4 v5 t4 t5) \Rightarrow (\lambda (H33: (eq T (THead k v4 t4) 
+(THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H34: (eq T (THead k 
+v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let 
+H35 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
+with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) 
+\Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4 
+x5)) H33) in ((let H36 \def (f_equal T T (\lambda (e: T).(match e in T return 
+(\lambda (_: T).T) with [(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4 
+| (THead _ t _) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead 
+(Bind x3) x4 x5)) H33) in ((let H37 \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 v4 t4) (THead (Flat 
+Appl) x (THead (Bind x3) x4 x5)) H33) in (eq_ind K (Flat Appl) (\lambda (k0: 
+K).((eq T v4 x) \to ((eq T t4 (THead (Bind x3) x4 x5)) \to ((eq T (THead k0 
+v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to 
+P)))) (\lambda (H38: (eq T v4 x)).(eq_ind T x (\lambda (_: T).((eq T t4 
+(THead (Bind x3) x4 x5)) \to ((eq T (THead (Flat Appl) v5 t5) (THead (Bind 
+x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P))) (\lambda (H39: (eq 
+T t4 (THead (Bind x3) x4 x5))).(eq_ind T (THead (Bind x3) x4 x5) (\lambda (_: 
+T).((eq T (THead (Flat Appl) v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) 
+(lift (S O) O x7) x6))) \to P)) (\lambda (H40: (eq T (THead (Flat Appl) v5 
+t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(let H41 
+\def (eq_ind T (THead (Flat Appl) v5 t5) (\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 x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) 
+H40) in (False_ind P H41))) t4 (sym_eq T t4 (THead (Bind x3) x4 x5) H39))) v4 
+(sym_eq T v4 x H38))) k (sym_eq K k (Flat Appl) H37))) H36)) H35)) H34)))]) 
+in (H33 (refl_equal T (THead (Flat Appl) x (THead (Bind x3) x4 x5))) 
+(refl_equal T (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) 
+x6)))))))) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift 
+(S O) O x7) x6))) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x3) x8 (THead 
+(Flat Appl) (lift (S O) O x7) x6))) (THead (Flat Appl) x1 (THead (Bind x3) x8 
+(THead (Flat Appl) (lift (S O) O x7) x6))) (pr2_head_1 c t0 x1 H15 (Flat 
+Appl) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))))))))) 
+x2 H22)))))))))))))) H19)) H18)) t3 H14))))))) H13)) (\lambda (H13: (ex4_4 T 
+T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Flat Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
+Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c t0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+z1 t4))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: 
+T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c t3) (\lambda (x1: 
+T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H14: (eq T 
+(THead (Flat Appl) x x0) (THead (Bind Abst) x1 x2))).(\lambda (H15: (eq T t3 
+(THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c t0 x3)).(\lambda (_: 
+((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let 
+H18 \def (eq_ind T t3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead 
+(Flat Appl) x x0)) t) \to (\forall (P: Prop).P))) H10 (THead (Bind Abbr) x3 
+x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4) (\lambda (t: T).(sn3 c t)) 
+(let H19 \def (eq_ind T (THead (Flat Appl) x x0) (\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) x1 x2) H14) in (False_ind (sn3 c (THead (Bind 
+Abbr) x3 x4)) H19)) t3 H15)))))))))) H13)) (\lambda (H13: (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 (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) 
+y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3) 
+(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda 
+(x5: T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15: 
+(eq T (THead (Flat Appl) x x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T 
+t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda 
+(_: (pr2 c t0 x5)).(\lambda (_: (pr2 c x2 x6)).(\lambda (_: (pr2 (CHead c 
+(Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t: T).((eq T 
+(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: 
+Prop).P))) H10 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) 
+H16) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) 
+x4)) (\lambda (t: T).(sn3 c t)) (let H21 \def (eq_ind T (THead (Flat Appl) x 
+x0) (\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 x1) x2 x3) 
+H15) in (False_ind (sn3 c (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) 
+O x5) x4))) H21)) t3 H16)))))))))))))) H13)) H12)))))))))))) v2 H4))))))))) y 
+H0))))) H))))).
+
+theorem sn3_appl_appls:
+ \forall (v1: T).(\forall (t1: T).(\forall (vs: TList).(let u1 \def (THeads 
+(Flat Appl) (TCons v1 vs) t1) in (\forall (c: C).((sn3 c u1) \to (\forall 
+(v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) 
+\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to 
+(sn3 c (THead (Flat Appl) v2 u1))))))))))
+\def
+ \lambda (v1: T).(\lambda (t1: T).(\lambda (vs: TList).(let u1 \def (THeads 
+(Flat Appl) (TCons v1 vs) t1) in (\lambda (c: C).(\lambda (H: (sn3 c (THead 
+(Flat Appl) v1 (THeads (Flat Appl) vs t1)))).(\lambda (v2: T).(\lambda (H0: 
+(sn3 c v2)).(\lambda (H1: ((\forall (u2: T).((pr3 c (THead (Flat Appl) v1 
+(THeads (Flat Appl) vs t1)) u2) \to ((((iso (THead (Flat Appl) v1 (THeads 
+(Flat Appl) vs t1)) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat 
+Appl) v2 u2))))))).(sn3_appl_appl v1 (THeads (Flat Appl) vs t1) c H v2 H0 
+H1))))))))).
+
+theorem sn3_appls_lref:
+ \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (us: 
+TList).((sns3 c us) \to (sn3 c (THeads (Flat Appl) us (TLRef i)))))))
+\def
+ \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda 
+(us: TList).(TList_ind (\lambda (t: TList).((sns3 c t) \to (sn3 c (THeads 
+(Flat Appl) t (TLRef i))))) (\lambda (_: True).(sn3_nf2 c (TLRef i) H)) 
+(\lambda (t: T).(\lambda (t0: TList).(TList_ind (\lambda (t1: TList).((((sns3 
+c t1) \to (sn3 c (THeads (Flat Appl) t1 (TLRef i))))) \to ((land (sn3 c t) 
+(sns3 c t1)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (TLRef 
+i))))))) (\lambda (_: (((sns3 c TNil) \to (sn3 c (THeads (Flat Appl) TNil 
+(TLRef i)))))).(\lambda (H1: (land (sn3 c t) (sns3 c TNil))).(let H2 \def H1 
+in (and_ind (sn3 c t) True (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) 
+TNil (TLRef i)))) (\lambda (H3: (sn3 c t)).(\lambda (_: True).(sn3_appl_lref 
+c i H t H3))) H2)))) (\lambda (t1: T).(\lambda (t2: TList).(\lambda (_: 
+(((((sns3 c t2) \to (sn3 c (THeads (Flat Appl) t2 (TLRef i))))) \to ((land 
+(sn3 c t) (sns3 c t2)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 
+(TLRef i)))))))).(\lambda (H1: (((sns3 c (TCons t1 t2)) \to (sn3 c (THeads 
+(Flat Appl) (TCons t1 t2) (TLRef i)))))).(\lambda (H2: (land (sn3 c t) (sns3 
+c (TCons t1 t2)))).(let H3 \def H2 in (and_ind (sn3 c t) (land (sn3 c t1) 
+(sns3 c t2)) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) 
+(TLRef i)))) (\lambda (H4: (sn3 c t)).(\lambda (H5: (land (sn3 c t1) (sns3 c 
+t2))).(and_ind (sn3 c t1) (sns3 c t2) (sn3 c (THead (Flat Appl) t (THeads 
+(Flat Appl) (TCons t1 t2) (TLRef i)))) (\lambda (H6: (sn3 c t1)).(\lambda 
+(H7: (sns3 c t2)).(sn3_appl_appls t1 (TLRef i) t2 c (H1 (conj (sn3 c t1) 
+(sns3 c t2) H6 H7)) t H4 (\lambda (u2: T).(\lambda (H8: (pr3 c (THeads (Flat 
+Appl) (TCons t1 t2) (TLRef i)) u2)).(\lambda (H9: (((iso (THeads (Flat Appl) 
+(TCons t1 t2) (TLRef i)) u2) \to (\forall (P: Prop).P)))).(H9 
+(nf2_iso_appls_lref c i H (TCons t1 t2) u2 H8) (sn3 c (THead (Flat Appl) t 
+u2))))))))) H5))) H3))))))) t0))) us)))).
+
+theorem sn3_appls_cast:
+ \forall (c: C).(\forall (vs: TList).(\forall (u: T).((sn3 c (THeads (Flat 
+Appl) vs u)) \to (\forall (t: T).((sn3 c (THeads (Flat Appl) vs t)) \to (sn3 
+c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))
+\def
+ \lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: TList).(\forall 
+(u: T).((sn3 c (THeads (Flat Appl) t u)) \to (\forall (t0: T).((sn3 c (THeads 
+(Flat Appl) t t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Flat Cast) u 
+t0)))))))) (\lambda (u: T).(\lambda (H: (sn3 c u)).(\lambda (t: T).(\lambda 
+(H0: (sn3 c t)).(sn3_cast c u H t H0))))) (\lambda (t: T).(\lambda (t0: 
+TList).(TList_ind (\lambda (t1: TList).(((\forall (u: T).((sn3 c (THeads 
+(Flat Appl) t1 u)) \to (\forall (t2: T).((sn3 c (THeads (Flat Appl) t1 t2)) 
+\to (sn3 c (THeads (Flat Appl) t1 (THead (Flat Cast) u t2)))))))) \to 
+(\forall (u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 u))) \to 
+(\forall (t2: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 t2))) 
+\to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (THead (Flat Cast) u 
+t2)))))))))) (\lambda (_: ((\forall (u: T).((sn3 c (THeads (Flat Appl) TNil 
+u)) \to (\forall (t1: T).((sn3 c (THeads (Flat Appl) TNil t1)) \to (sn3 c 
+(THeads (Flat Appl) TNil (THead (Flat Cast) u t1))))))))).(\lambda (u: 
+T).(\lambda (H0: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) TNil 
+u)))).(\lambda (t1: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads 
+(Flat Appl) TNil t1)))).(sn3_appl_cast c t u H0 t1 H1)))))) (\lambda (t1: 
+T).(\lambda (t2: TList).(\lambda (_: ((((\forall (u: T).((sn3 c (THeads (Flat 
+Appl) t2 u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) t2 t3)) \to 
+(sn3 c (THeads (Flat Appl) t2 (THead (Flat Cast) u t3)))))))) \to (\forall 
+(u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 u))) \to (\forall 
+(t3: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 t3))) \to (sn3 c 
+(THead (Flat Appl) t (THeads (Flat Appl) t2 (THead (Flat Cast) u 
+t3))))))))))).(\lambda (H0: ((\forall (u: T).((sn3 c (THeads (Flat Appl) 
+(TCons t1 t2) u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) (TCons t1 
+t2) t3)) \to (sn3 c (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u 
+t3))))))))).(\lambda (u: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads 
+(Flat Appl) (TCons t1 t2) u)))).(\lambda (t3: T).(\lambda (H2: (sn3 c (THead 
+(Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) t3)))).(let H3 \def 
+(sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) t3) H2) in (and_ind 
+(sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads (Flat Appl) t2 t3))) (sn3 c 
+(THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u 
+t3)))) (\lambda (_: (sn3 c t)).(\lambda (H5: (sn3 c (THead (Flat Appl) t1 
+(THeads (Flat Appl) t2 t3)))).(let H6 \def H5 in (let H7 \def (sn3_gen_flat 
+Appl c t (THeads (Flat Appl) (TCons t1 t2) u) H1) in (and_ind (sn3 c t) (sn3 
+c (THead (Flat Appl) t1 (THeads (Flat Appl) t2 u))) (sn3 c (THead (Flat Appl) 
+t (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u t3)))) (\lambda (H8: 
+(sn3 c t)).(\lambda (H9: (sn3 c (THead (Flat Appl) t1 (THeads (Flat Appl) t2 
+u)))).(let H10 \def H9 in (sn3_appl_appls t1 (THead (Flat Cast) u t3) t2 c 
+(H0 u H10 t3 H6) t H8 (\lambda (u2: T).(\lambda (H11: (pr3 c (THeads (Flat 
+Appl) (TCons t1 t2) (THead (Flat Cast) u t3)) u2)).(\lambda (H12: (((iso 
+(THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u t3)) u2) \to (\forall 
+(P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t (THeads (Flat Appl) 
+(TCons t1 t2) t3)) H2 (THead (Flat Appl) t u2) (pr3_thin_dx c (THeads (Flat 
+Appl) (TCons t1 t2) t3) u2 (pr3_iso_appls_cast c u t3 (TCons t1 t2) u2 H11 
+H12) t Appl))))))))) H7))))) H3)))))))))) t0))) vs)).
+
+theorem sn3_lift:
+ \forall (d: C).(\forall (t: T).((sn3 d t) \to (\forall (c: C).(\forall (h: 
+nat).(\forall (i: nat).((drop h i c d) \to (sn3 c (lift h i t))))))))
+\def
+ \lambda (d: C).(\lambda (t: T).(\lambda (H: (sn3 d t)).(sn3_ind d (\lambda 
+(t0: T).(\forall (c: C).(\forall (h: nat).(\forall (i: nat).((drop h i c d) 
+\to (sn3 c (lift h i t0))))))) (\lambda (t1: T).(\lambda (_: ((\forall (t2: 
+T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 d t1 t2) \to (sn3 d 
+t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
+Prop).P))) \to ((pr3 d t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall 
+(i: nat).((drop h i c d) \to (sn3 c (lift h i t2))))))))))).(\lambda (c: 
+C).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H2: (drop h i c 
+d)).(sn3_pr2_intro c (lift h i t1) (\lambda (t2: T).(\lambda (H3: (((eq T 
+(lift h i t1) t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr2 c (lift h i 
+t1) t2)).(let H5 \def (pr2_gen_lift c t1 t2 h i H4 d H2) in (ex2_ind T 
+(\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t1 t3)) 
+(sn3 c t2) (\lambda (x: T).(\lambda (H6: (eq T t2 (lift h i x))).(\lambda 
+(H7: (pr2 d t1 x)).(let H8 \def (eq_ind T t2 (\lambda (t0: T).((eq T (lift h 
+i t1) t0) \to (\forall (P: Prop).P))) H3 (lift h i x) H6) in (eq_ind_r T 
+(lift h i x) (\lambda (t0: T).(sn3 c t0)) (H1 x (\lambda (H9: (eq T t1 
+x)).(\lambda (P: Prop).(let H10 \def (eq_ind_r T x (\lambda (t0: T).((eq T 
+(lift h i t1) (lift h i t0)) \to (\forall (P0: Prop).P0))) H8 t1 H9) in (let 
+H11 \def (eq_ind_r T x (\lambda (t0: T).(pr2 d t1 t0)) H7 t1 H9) in (H10 
+(refl_equal T (lift h i t1)) P))))) (pr3_pr2 d t1 x H7) c h i H2) t2 H6))))) 
+H5))))))))))))) t H))).
+
+theorem sn3_abbr:
+ \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abbr) v)) \to ((sn3 d v) \to (sn3 c (TLRef i)))))))
+\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 d 
+v)).(sn3_pr2_intro c (TLRef i) (\lambda (t2: T).(\lambda (H1: (((eq T (TLRef 
+i) t2) \to (\forall (P: Prop).P)))).(\lambda (H2: (pr2 c (TLRef i) t2)).(let 
+H3 \def (pr2_gen_lref c t2 i H2) in (or_ind (eq T t2 (TLRef i)) (ex2_2 C T 
+(\lambda (d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) 
+(\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S i) O u))))) (sn3 c t2) 
+(\lambda (H4: (eq T t2 (TLRef i))).(let H5 \def (eq_ind T t2 (\lambda (t: 
+T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H4) in 
+(eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c t)) (H5 (refl_equal T (TLRef i)) 
+(sn3 c (TLRef i))) t2 H4))) (\lambda (H4: (ex2_2 C T (\lambda (d0: 
+C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: 
+C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda 
+(d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: 
+C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))) (sn3 c t2) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (H5: (getl i c (CHead x0 (Bind Abbr) 
+x1))).(\lambda (H6: (eq T t2 (lift (S i) O x1))).(let H7 \def (eq_ind T t2 
+(\lambda (t: T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (lift (S 
+i) O x1) H6) in (eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(sn3 c t)) (let 
+H8 \def (eq_ind C (CHead d (Bind Abbr) v) (\lambda (c0: C).(getl i c c0)) H 
+(CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 
+(Bind Abbr) x1) H5)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in 
+C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) 
+\Rightarrow c0])) (CHead d (Bind Abbr) v) (CHead x0 (Bind Abbr) x1) 
+(getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 (Bind Abbr) x1) H5)) in 
+((let H10 \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 d 
+(Bind Abbr) v) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) 
+i H (CHead x0 (Bind Abbr) x1) H5)) in (\lambda (H11: (eq C d x0)).(let H12 
+\def (eq_ind_r T x1 (\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 v 
+H10) in (eq_ind T v (\lambda (t: T).(sn3 c (lift (S i) O t))) (let H13 \def 
+(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) v))) H12 d 
+H11) in (sn3_lift d v H0 c (S i) O (getl_drop Abbr c d v i H13))) x1 H10)))) 
+H9))) t2 H6)))))) H4)) H3))))))))))).
+
+theorem sns3_lifts:
+ \forall (c: C).(\forall (d: C).(\forall (h: nat).(\forall (i: nat).((drop h 
+i c d) \to (\forall (ts: TList).((sns3 d ts) \to (sns3 c (lifts h i ts))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (h: nat).(\lambda (i: nat).(\lambda 
+(H: (drop h i c d)).(\lambda (ts: TList).(TList_ind (\lambda (t: 
+TList).((sns3 d t) \to (sns3 c (lifts h i t)))) (\lambda (H0: True).H0) 
+(\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: (((sns3 d t0) \to (sns3 c 
+(lifts h i t0))))).(\lambda (H1: (land (sn3 d t) (sns3 d t0))).(let H2 \def 
+H1 in (and_ind (sn3 d t) (sns3 d t0) (land (sn3 c (lift h i t)) (sns3 c 
+(lifts h i t0))) (\lambda (H3: (sn3 d t)).(\lambda (H4: (sns3 d t0)).(conj 
+(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)))))).
+
diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma
index f479a8baa..7f7761df1 100644
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma
@@ -266,3 +266,61 @@ include "pr3/subst1.ma".
 
 include "pr3/iso.ma".
 
+include "csuba/defs.ma".
+
+include "csuba/fwd.ma".
+
+include "csuba/props.ma".
+
+include "csuba/clear.ma".
+
+include "csuba/drop.ma".
+
+include "csuba/getl.ma".
+
+include "csuba/arity.ma".
+
+include "arity/pr3.ma".
+
+include "nf2/defs.ma".
+
+include "nf2/fwd.ma".
+
+include "nf2/props.ma".
+
+include "nf2/pr3.ma".
+
+include "nf2/lift1.ma".
+
+include "nf2/iso.ma".
+
+include "nf2/dec.ma".
+
+include "sn3/defs.ma".
+
+include "sn3/fwd.ma".
+
+include "sn3/nf2.ma".
+
+include "sn3/props.ma".
+
+include "sn3/lift1.ma".
+
+include "sc3/defs.ma".
+
+include "sc3/props.ma".
+
+include "ceqc/defs.ma".
+
+include "ceqc/props.ma".
+
+include "sc3/arity.ma".
+
+include "pc1/defs.ma".
+
+include "pc1/props.ma".
+
+include "pc3/defs.ma".
+
+include "pc3/props.ma".
+