--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/pr2/props".
+
+include "pr2/defs.ma".
+
+include "pr0/props.ma".
+
+include "getl/drop.ma".
+
+include "getl/clear.ma".
+
+theorem pr2_thin_dx:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
+(u: T).(\forall (f: F).(pr2 c (THead (Flat f) u t1) (THead (Flat f) u
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
+t2)).(\lambda (u: T).(\lambda (f: F).(pr2_ind (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).(pr2 c0 (THead (Flat f) u t) (THead (Flat f) u t0)))))
+(\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t0
+t3)).(pr2_free c0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u
+(pr0_refl u) t0 t3 H0 (Flat f))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
+Abbr) u0))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (pr0 t0
+t3)).(\lambda (t: T).(\lambda (H2: (subst0 i u0 t3 t)).(pr2_delta c0 d u0 i
+H0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t0
+t3 H1 (Flat f)) (THead (Flat f) u t) (subst0_snd (Flat f) u0 t t3 i H2
+u)))))))))))) c t1 t2 H)))))).
+
+theorem pr2_head_1:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall
+(k: K).(\forall (t: T).(pr2 c (THead k u1 t) (THead k u2 t)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1
+u2)).(\lambda (k: K).(\lambda (t: T).(pr2_ind (\lambda (c0: C).(\lambda (t0:
+T).(\lambda (t1: T).(pr2 c0 (THead k t0 t) (THead k t1 t))))) (\lambda (c0:
+C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(pr2_free c0
+(THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k))))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t2
+t0)).(pr2_delta c0 d u i H0 (THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H1
+t t (pr0_refl t) k) (THead k t0 t) (subst0_fst u t0 t2 i H2 t k)))))))))))) c
+u1 u2 H)))))).
+
+theorem pr2_head_2:
+ \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall
+(k: K).((pr2 (CHead c k u) t1 t2) \to (pr2 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).(K_ind (\lambda (k0: K).((pr2 (CHead c k0 u) t1 t2) \to (pr2 c (THead
+k0 u t1) (THead k0 u t2)))) (\lambda (b: B).(\lambda (H: (pr2 (CHead c (Bind
+b) u) t1 t2)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda
+(t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 (CHead c (Bind
+b) u)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (THead (Bind b) u t1)
+(THead (Bind b) u t2))))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow
+(\lambda (H1: (eq C c0 (CHead c (Bind b) u))).(\lambda (H2: (eq T t0
+t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (_:
+C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (THead (Bind
+b) u t1) (THead (Bind b) u t2)))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1
+(\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (THead (Bind b) u
+t1) (THead (Bind b) u t2))))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2
+(\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b)
+u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead (Bind b) u t1) (THead
+(Bind b) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6 (Bind b)))) t3 (sym_eq T
+t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Bind b) u) H1)
+H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t H2) \Rightarrow (\lambda
+(H3: (eq C c0 (CHead c (Bind b) u))).(\lambda (H4: (eq T t0 t1)).(\lambda
+(H5: (eq T t t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (c1: C).((eq T t0
+t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u0)) \to ((pr0 t0
+t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b)
+u t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T
+t t2) \to ((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0
+t4 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind
+b) u t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4:
+T).((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t3)
+\to ((subst0 i u0 t3 t4) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u
+t2)))))) (\lambda (H8: (getl i (CHead c (Bind b) u) (CHead d (Bind Abbr)
+u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u0 t3 t2)).((match
+i in nat return (\lambda (n: nat).((getl n (CHead c (Bind b) u) (CHead d
+(Bind Abbr) u0)) \to ((subst0 n u0 t3 t2) \to (pr2 c (THead (Bind b) u t1)
+(THead (Bind b) u t2))))) with [O \Rightarrow (\lambda (H11: (getl O (CHead c
+(Bind b) u) (CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 O u0 t3
+t2)).(let H13 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1]))
+(CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d
+(Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)
+H11))) in ((let H14 \def (f_equal C B (\lambda (e: C).(match e in C return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _)
+\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0)
+(CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u
+(getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H11))) in ((let H15
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u0 | (CHead _ _ t4) \Rightarrow t4])) (CHead d
+(Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr)
+u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H11))) in
+(\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H18 \def (eq_ind
+T u0 (\lambda (t4: T).(subst0 O t4 t3 t2)) H12 u H15) in (eq_ind B Abbr
+(\lambda (b0: B).(pr2 c (THead (Bind b0) u t1) (THead (Bind b0) u t2)))
+(pr2_free c (THead (Bind Abbr) u t1) (THead (Bind Abbr) u t2) (pr0_delta u u
+(pr0_refl u) t1 t3 H9 t2 H18)) b H16))))) H14)) H13)))) | (S n) \Rightarrow
+(\lambda (H11: (getl (S n) (CHead c (Bind b) u) (CHead d (Bind Abbr)
+u0))).(\lambda (H12: (subst0 (S n) u0 t3 t2)).(pr2_delta c d u0 (r (Bind b)
+n) (getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H11) (THead (Bind b) u
+t1) (THead (Bind b) u t3) (pr0_comp u u (pr0_refl u) t1 t3 H9 (Bind b))
+(THead (Bind b) u t2) (subst0_snd (Bind b) u0 t2 t3 (r (Bind b) n) H12
+u))))]) H8 H10)))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 (sym_eq
+C c0 (CHead c (Bind b) u) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C
+(CHead c (Bind b) u)) (refl_equal T t1) (refl_equal T t2))))) (\lambda (f:
+F).(\lambda (H: (pr2 (CHead c (Flat f) u) t1 t2)).(let H0 \def (match H in
+pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_:
+(pr2 c0 t t0)).((eq C c0 (CHead c (Flat f) u)) \to ((eq T t t1) \to ((eq T t0
+t2) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))))))) with
+[(pr2_free c0 t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c0 (CHead c (Flat f)
+u))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead
+c (Flat f) u) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0
+t3) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2)))))) (\lambda (H4:
+(eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to
+(pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))) (\lambda (H5: (eq T t3
+t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Flat f) u
+t1) (THead (Flat f) u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead
+(Flat f) u t1) (THead (Flat f) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6
+(Flat f)))) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0
+(CHead c (Flat f) u) H1) H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t
+H2) \Rightarrow (\lambda (H3: (eq C c0 (CHead c (Flat f) u))).(\lambda (H4:
+(eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Flat f) u)
+(\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d
+(Bind Abbr) u0)) \to ((pr0 t0 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead
+(Flat f) u t1) (THead (Flat f) u t2)))))))) (\lambda (H6: (eq T t0
+t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c (Flat
+f) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t4 t3) \to ((subst0 i u0 t3 t) \to
+(pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))))) (\lambda (H7: (eq T
+t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c (Flat f) u) (CHead d
+(Bind Abbr) u0)) \to ((pr0 t1 t3) \to ((subst0 i u0 t3 t4) \to (pr2 c (THead
+(Flat f) u t1) (THead (Flat f) u t2)))))) (\lambda (H8: (getl i (CHead c
+(Flat f) u) (CHead d (Bind Abbr) u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda
+(H10: (subst0 i u0 t3 t2)).((match i in nat return (\lambda (n: nat).((getl n
+(CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to ((subst0 n u0 t3 t2) \to
+(pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))) with [O \Rightarrow
+(\lambda (H11: (getl O (CHead c (Flat f) u) (CHead d (Bind Abbr)
+u0))).(\lambda (H12: (subst0 O u0 t3 t2)).(pr2_delta c d u0 O (getl_intro O c
+(CHead d (Bind Abbr) u0) c (drop_refl c) (clear_gen_flat f c (CHead d (Bind
+Abbr) u0) u (getl_gen_O (CHead c (Flat f) u) (CHead d (Bind Abbr) u0) H11)))
+(THead (Flat f) u t1) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t1 t3
+H9 (Flat f)) (THead (Flat f) u t2) (subst0_snd (Flat f) u0 t2 t3 O H12 u))))
+| (S n) \Rightarrow (\lambda (H11: (getl (S n) (CHead c (Flat f) u) (CHead d
+(Bind Abbr) u0))).(\lambda (H12: (subst0 (S n) u0 t3 t2)).(pr2_delta c d u0
+(r (Flat f) n) (getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H11)
+(THead (Flat f) u t1) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t1 t3
+H9 (Flat f)) (THead (Flat f) u t2) (subst0_snd (Flat f) u0 t2 t3 (r (Flat f)
+n) H12 u))))]) H8 H10)))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0
+(sym_eq C c0 (CHead c (Flat f) u) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal
+C (CHead c (Flat f) u)) (refl_equal T t1) (refl_equal T t2))))) k))))).
+
+theorem clear_pr2_trans:
+ \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to
+(\forall (c1: C).((clear c1 c2) \to (pr2 c1 t1 t2))))))
+\def
+ \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c2 t1
+t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def (match H in
+pr2 return (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2
+c t t0)).((eq C c c2) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c1 t1
+t2)))))))) with [(pr2_free c t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c
+c2)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C c2
+(\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c1
+t1 t2))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3
+t2) \to ((pr0 t t3) \to (pr2 c1 t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind
+T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c1 t1 t2))) (\lambda (H7: (pr0 t1
+t2)).(pr2_free c1 t1 t2 H7)) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1
+H5))) c (sym_eq C c c2 H2) H3 H4 H1)))) | (pr2_delta c d u i H1 t0 t3 H2 t
+H3) \Rightarrow (\lambda (H4: (eq C c c2)).(\lambda (H5: (eq T t0
+t1)).(\lambda (H6: (eq T t t2)).(eq_ind C c2 (\lambda (c0: C).((eq T t0 t1)
+\to ((eq T t t2) \to ((getl i c0 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3)
+\to ((subst0 i u t3 t) \to (pr2 c1 t1 t2))))))) (\lambda (H7: (eq T t0
+t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i c2 (CHead d
+(Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c1 t1
+t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i c2
+(CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c1
+t1 t2))))) (\lambda (H9: (getl i c2 (CHead d (Bind Abbr) u))).(\lambda (H10:
+(pr0 t1 t3)).(\lambda (H11: (subst0 i u t3 t2)).(pr2_delta c1 d u i
+(clear_getl_trans i c2 (CHead d (Bind Abbr) u) H9 c1 H0) t1 t3 H10 t2 H11))))
+t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c (sym_eq C c c2 H4) H5 H6 H1
+H2 H3))))]) in (H1 (refl_equal C c2) (refl_equal T t1) (refl_equal T
+t2)))))))).
+
+theorem pr2_cflat:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
+(f: F).(\forall (v: T).(pr2 (CHead c (Flat f) v) t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (f:
+F).(\forall (v: T).(pr2 (CHead c0 (Flat f) v) t t0)))))) (\lambda (c0:
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (f:
+F).(\lambda (v: T).(pr2_free (CHead c0 (Flat f) v) t3 t4 H0))))))) (\lambda
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl
+i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda
+(H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda
+(f: F).(\lambda (v: T).(pr2_delta (CHead c0 (Flat f) v) d u i (getl_flat c0
+(CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))))) c t1 t2 H)))).
+
+theorem pr2_ctail:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
+(k: K).(\forall (u: T).(pr2 (CTail k u c) t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
+t2)).(\lambda (k: K).(\lambda (u: T).(pr2_ind (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).(pr2 (CTail k u c0) t t0)))) (\lambda (c0: C).(\lambda
+(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free (CTail k u c0)
+t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2:
+(subst0 i u0 t4 t)).(pr2_delta (CTail k u c0) (CTail k u d) u0 i (getl_ctail
+Abbr c0 d u0 i H0 k u) t3 t4 H1 t H2))))))))))) c t1 t2 H)))))).
+
+theorem pr2_lift:
+ \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h
+d c e) \to (\forall (t1: T).(\forall (t2: T).((pr2 e t1 t2) \to (pr2 c (lift
+h d t1) (lift h d t2)))))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 e t1
+t2)).(let H1 \def (match H0 in pr2 return (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 e) \to ((eq T t t1)
+\to ((eq T t0 t2) \to (pr2 c (lift h d t1) (lift h d t2))))))))) with
+[(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0 e)).(\lambda (H3:
+(eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C e (\lambda (_: C).((eq T
+t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (lift h d t1) (lift h d
+t2)))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3
+t2) \to ((pr0 t t3) \to (pr2 c (lift h d t1) (lift h d t2))))) (\lambda (H6:
+(eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (lift h d
+t1) (lift h d t2)))) (\lambda (H7: (pr0 t1 t2)).(pr2_free c (lift h d t1)
+(lift h d t2) (pr0_lift t1 t2 H7 h d))) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T
+t0 t1 H5))) c0 (sym_eq C c0 e H2) H3 H4 H1)))) | (pr2_delta c0 d0 u i H1 t0
+t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 e)).(\lambda (H5: (eq T t0
+t1)).(\lambda (H6: (eq T t t2)).(eq_ind C e (\lambda (c1: C).((eq T t0 t1)
+\to ((eq T t t2) \to ((getl i c1 (CHead d0 (Bind Abbr) u)) \to ((pr0 t0 t3)
+\to ((subst0 i u t3 t) \to (pr2 c (lift h d t1) (lift h d t2)))))))) (\lambda
+(H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i e
+(CHead d0 (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c
+(lift h d t1) (lift h d t2))))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2
+(\lambda (t4: T).((getl i e (CHead d0 (Bind Abbr) u)) \to ((pr0 t1 t3) \to
+((subst0 i u t3 t4) \to (pr2 c (lift h d t1) (lift h d t2)))))) (\lambda (H9:
+(getl i e (CHead d0 (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda
+(H11: (subst0 i u t3 t2)).(lt_le_e i d (pr2 c (lift h d t1) (lift h d t2))
+(\lambda (H12: (lt i d)).(let H13 \def (drop_getl_trans_le i d (le_S_n i d
+(le_S (S i) d H12)) c e h H (CHead d0 (Bind Abbr) u) H9) in (ex3_2_ind C C
+(\lambda (e0: C).(\lambda (_: C).(drop i O c e0))) (\lambda (e0: C).(\lambda
+(e1: C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear
+e1 (CHead d0 (Bind Abbr) u)))) (pr2 c (lift h d t1) (lift h d t2)) (\lambda
+(x0: C).(\lambda (x1: C).(\lambda (H14: (drop i O c x0)).(\lambda (H15: (drop
+h (minus d i) x0 x1)).(\lambda (H16: (clear x1 (CHead d0 (Bind Abbr)
+u))).(let H17 \def (eq_ind nat (minus d i) (\lambda (n: nat).(drop h n x0
+x1)) H15 (S (minus d (S i))) (minus_x_Sy d i H12)) in (let H18 \def
+(drop_clear_S x1 x0 h (minus d (S i)) H17 Abbr d0 u H16) in (ex2_ind C
+(\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h (minus d (S i))
+u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c (lift h d t1)
+(lift h d t2)) (\lambda (x: C).(\lambda (H19: (clear x0 (CHead x (Bind Abbr)
+(lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S i)) x
+d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead x
+(Bind Abbr) (lift h (minus d (S i)) u)) x0 H14 H19) (lift h d t1) (lift h d
+t3) (pr0_lift t1 t3 H10 h d) (lift h d t2) (subst0_lift_lt t3 t2 u i H11 d
+H12 h))))) H18)))))))) H13))) (\lambda (H12: (le d i)).(pr2_delta c d0 u
+(plus i h) (drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H9 H12)
+(lift h d t1) (lift h d t3) (pr0_lift t1 t3 H10 h d) (lift h d t2)
+(subst0_lift_ge t3 t2 u i h H11 d H12))))))) t (sym_eq T t t2 H8))) t0
+(sym_eq T t0 t1 H7))) c0 (sym_eq C c0 e H4) H5 H6 H1 H2 H3))))]) in (H1
+(refl_equal C e) (refl_equal T t1) (refl_equal T t2)))))))))).
+