new makefiles

[helm.git] / helm / software / matita / contribs / LAMBDA-TYPES / Level-1 / LambdaDelta.ma

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta.ma
index df8a3d5341e4aa0ce93c135fce78a7233efccca6..8642561d21806c4ccddeb42fb9efba6e1503589b 100644 (file)
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta.ma
@@ -18,2467 +18,3 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta".
 
 include "LambdaDelta/theory.ma".
 
-definition TApp:
- TList \to (T \to TList)
-\def
- let rec TApp (ts: TList) on ts: (T \to TList) \def (\lambda (v: T).(match ts 
-with [TNil \Rightarrow (TCons v TNil) | (TCons t ts0) \Rightarrow (TCons t 
-(TApp ts0 v))])) in TApp.
-
-definition tslen:
- TList \to nat
-\def
- let rec tslen (ts: TList) on ts: nat \def (match ts with [TNil \Rightarrow O 
-| (TCons _ ts0) \Rightarrow (S (tslen ts0))]) in tslen.
-
-definition tslt:
- TList \to (TList \to Prop)
-\def
- \lambda (ts1: TList).(\lambda (ts2: TList).(lt (tslen ts1) (tslen ts2))).
-
-theorem tslt_wf__q_ind:
- \forall (P: ((TList \to Prop))).(((\forall (n: nat).((\lambda (P0: ((TList 
-\to Prop))).(\lambda (n0: nat).(\forall (ts: TList).((eq nat (tslen ts) n0) 
-\to (P0 ts))))) P n))) \to (\forall (ts: TList).(P ts)))
-\def
- let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts: 
-TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to 
-Prop))).(\lambda (H: ((\forall (n: nat).(\forall (ts: TList).((eq nat (tslen 
-ts) n) \to (P ts)))))).(\lambda (ts: TList).(H (tslen ts) ts (refl_equal nat 
-(tslen ts)))))).
-
-theorem tslt_wf_ind:
- \forall (P: ((TList \to Prop))).(((\forall (ts2: TList).(((\forall (ts1: 
-TList).((tslt ts1 ts2) \to (P ts1)))) \to (P ts2)))) \to (\forall (ts: 
-TList).(P ts)))
-\def
- let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts: 
-TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to 
-Prop))).(\lambda (H: ((\forall (ts2: TList).(((\forall (ts1: TList).((lt 
-(tslen ts1) (tslen ts2)) \to (P ts1)))) \to (P ts2))))).(\lambda (ts: 
-TList).(tslt_wf__q_ind (\lambda (t: TList).(P t)) (\lambda (n: 
-nat).(lt_wf_ind n (Q (\lambda (t: TList).(P t))) (\lambda (n0: nat).(\lambda 
-(H0: ((\forall (m: nat).((lt m n0) \to (Q (\lambda (t: TList).(P t)) 
-m))))).(\lambda (ts0: TList).(\lambda (H1: (eq nat (tslen ts0) n0)).(let H2 
-\def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall (m: nat).((lt m n1) \to 
-(\forall (ts1: TList).((eq nat (tslen ts1) m) \to (P ts1)))))) H0 (tslen ts0) 
-H1) in (H ts0 (\lambda (ts1: TList).(\lambda (H3: (lt (tslen ts1) (tslen 
-ts0))).(H2 (tslen ts1) H3 ts1 (refl_equal nat (tslen ts1))))))))))))) ts)))).
-
-theorem theads_tapp:
- \forall (k: K).(\forall (vs: TList).(\forall (v: T).(\forall (t: T).(eq T 
-(THeads k (TApp vs v) t) (THeads k vs (THead k v t))))))
-\def
- \lambda (k: K).(\lambda (vs: TList).(TList_ind (\lambda (t: TList).(\forall 
-(v: T).(\forall (t0: T).(eq T (THeads k (TApp t v) t0) (THeads k t (THead k v 
-t0)))))) (\lambda (v: T).(\lambda (t: T).(refl_equal T (THead k v t)))) 
-(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (v: T).(\forall 
-(t1: T).(eq T (THeads k (TApp t0 v) t1) (THeads k t0 (THead k v 
-t1))))))).(\lambda (v: T).(\lambda (t1: T).(eq_ind_r T (THeads k t0 (THead k 
-v t1)) (\lambda (t2: T).(eq T (THead k t t2) (THead k t (THeads k t0 (THead k 
-v t1))))) (refl_equal T (THead k t (THeads k t0 (THead k v t1)))) (THeads k 
-(TApp t0 v) t1) (H v t1))))))) vs)).
-
-theorem tcons_tapp_ex:
- \forall (ts1: TList).(\forall (t1: T).(ex2_2 TList T (\lambda (ts2: 
-TList).(\lambda (t2: T).(eq TList (TCons t1 ts1) (TApp ts2 t2)))) (\lambda 
-(ts2: TList).(\lambda (_: T).(eq nat (tslen ts1) (tslen ts2))))))
-\def
- \lambda (ts1: TList).(TList_ind (\lambda (t: TList).(\forall (t1: T).(ex2_2 
-TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t) (TApp 
-ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t) (tslen 
-ts2))))))) (\lambda (t1: T).(ex2_2_intro TList T (\lambda (ts2: 
-TList).(\lambda (t2: T).(eq TList (TCons t1 TNil) (TApp ts2 t2)))) (\lambda 
-(ts2: TList).(\lambda (_: T).(eq nat O (tslen ts2)))) TNil t1 (refl_equal 
-TList (TApp TNil t1)) (refl_equal nat (tslen TNil)))) (\lambda (t: 
-T).(\lambda (t0: TList).(\lambda (H: ((\forall (t1: T).(ex2_2 TList T 
-(\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t0) (TApp ts2 
-t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) (tslen 
-ts2)))))))).(\lambda (t1: T).(let H_x \def (H t) in (let H0 \def H_x in 
-(ex2_2_ind TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t 
-t0) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) 
-(tslen ts2)))) (ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq 
-TList (TCons t1 (TCons t t0)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda 
-(_: T).(eq nat (S (tslen t0)) (tslen ts2))))) (\lambda (x0: TList).(\lambda 
-(x1: T).(\lambda (H1: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H2: (eq 
-nat (tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t2: 
-TList).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t3: T).(eq TList (TCons 
-t1 t2) (TApp ts2 t3)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (S 
-(tslen t0)) (tslen ts2)))))) (eq_ind_r nat (tslen x0) (\lambda (n: 
-nat).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons 
-t1 (TApp x0 x1)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq 
-nat (S n) (tslen ts2)))))) (ex2_2_intro TList T (\lambda (ts2: 
-TList).(\lambda (t2: T).(eq TList (TCons t1 (TApp x0 x1)) (TApp ts2 t2)))) 
-(\lambda (ts2: TList).(\lambda (_: T).(eq nat (S (tslen x0)) (tslen ts2)))) 
-(TCons t1 x0) x1 (refl_equal TList (TApp (TCons t1 x0) x1)) (refl_equal nat 
-(tslen (TCons t1 x0)))) (tslen t0) H2) (TCons t t0) H1))))) H0))))))) ts1).
-
-theorem tlist_ind_rew:
- \forall (P: ((TList \to Prop))).((P TNil) \to (((\forall (ts: 
-TList).(\forall (t: T).((P ts) \to (P (TApp ts t)))))) \to (\forall (ts: 
-TList).(P ts))))
-\def
- \lambda (P: ((TList \to Prop))).(\lambda (H: (P TNil)).(\lambda (H0: 
-((\forall (ts: TList).(\forall (t: T).((P ts) \to (P (TApp ts 
-t))))))).(\lambda (ts: TList).(tslt_wf_ind (\lambda (t: TList).(P t)) 
-(\lambda (ts2: TList).(match ts2 in TList return (\lambda (t: 
-TList).(((\forall (ts1: TList).((tslt ts1 t) \to (P ts1)))) \to (P t))) with 
-[TNil \Rightarrow (\lambda (_: ((\forall (ts1: TList).((tslt ts1 TNil) \to (P 
-ts1))))).H) | (TCons t t0) \Rightarrow (\lambda (H1: ((\forall (ts1: 
-TList).((tslt ts1 (TCons t t0)) \to (P ts1))))).(let H_x \def (tcons_tapp_ex 
-t0 t) in (let H2 \def H_x in (ex2_2_ind TList T (\lambda (ts3: 
-TList).(\lambda (t2: T).(eq TList (TCons t t0) (TApp ts3 t2)))) (\lambda 
-(ts3: TList).(\lambda (_: T).(eq nat (tslen t0) (tslen ts3)))) (P (TCons t 
-t0)) (\lambda (x0: TList).(\lambda (x1: T).(\lambda (H3: (eq TList (TCons t 
-t0) (TApp x0 x1))).(\lambda (H4: (eq nat (tslen t0) (tslen x0))).(eq_ind_r 
-TList (TApp x0 x1) (\lambda (t1: TList).(P t1)) (H0 x0 x1 (H1 x0 (eq_ind nat 
-(tslen t0) (\lambda (n: nat).(lt n (tslen (TCons t t0)))) (le_n (tslen (TCons 
-t t0))) (tslen x0) H4))) (TCons t t0) H3))))) H2))))])) ts)))).
-
-theorem iso_gen_sort:
- \forall (u2: T).(\forall (n1: nat).((iso (TSort n1) u2) \to (ex nat (\lambda 
-(n2: nat).(eq T u2 (TSort n2))))))
-\def
- \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TSort n1) u2)).(let H0 
-\def (match H in iso return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: 
-(iso t t0)).((eq T t (TSort n1)) \to ((eq T t0 u2) \to (ex nat (\lambda (n2: 
-nat).(eq T u2 (TSort n2))))))))) with [(iso_sort n0 n2) \Rightarrow (\lambda 
-(H0: (eq T (TSort n0) (TSort n1))).(\lambda (H1: (eq T (TSort n2) u2)).((let 
-H2 \def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: 
-T).nat) with [(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _ 
-_) \Rightarrow n0])) (TSort n0) (TSort n1) H0) in (eq_ind nat n1 (\lambda (_: 
-nat).((eq T (TSort n2) u2) \to (ex nat (\lambda (n3: nat).(eq T u2 (TSort 
-n3)))))) (\lambda (H3: (eq T (TSort n2) u2)).(eq_ind T (TSort n2) (\lambda 
-(t: T).(ex nat (\lambda (n3: nat).(eq T t (TSort n3))))) (ex_intro nat 
-(\lambda (n3: nat).(eq T (TSort n2) (TSort n3))) n2 (refl_equal T (TSort 
-n2))) u2 H3)) n0 (sym_eq nat n0 n1 H2))) H1))) | (iso_lref i1 i2) \Rightarrow 
-(\lambda (H0: (eq T (TLRef i1) (TSort n1))).(\lambda (H1: (eq T (TLRef i2) 
-u2)).((let H2 \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 (TSort n1) H0) in 
-(False_ind ((eq T (TLRef i2) u2) \to (ex nat (\lambda (n2: nat).(eq T u2 
-(TSort n2))))) H2)) H1))) | (iso_head v1 v2 t1 t2 k) \Rightarrow (\lambda 
-(H0: (eq T (THead k v1 t1) (TSort n1))).(\lambda (H1: (eq T (THead k v2 t2) 
-u2)).((let H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: T).(match e in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n1) H0) in 
-(False_ind ((eq T (THead k v2 t2) u2) \to (ex nat (\lambda (n2: nat).(eq T u2 
-(TSort n2))))) H2)) H1)))]) in (H0 (refl_equal T (TSort n1)) (refl_equal T 
-u2))))).
-
-theorem iso_gen_lref:
- \forall (u2: T).(\forall (n1: nat).((iso (TLRef n1) u2) \to (ex nat (\lambda 
-(n2: nat).(eq T u2 (TLRef n2))))))
-\def
- \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TLRef n1) u2)).(let H0 
-\def (match H in iso return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: 
-(iso t t0)).((eq T t (TLRef n1)) \to ((eq T t0 u2) \to (ex nat (\lambda (n2: 
-nat).(eq T u2 (TLRef n2))))))))) with [(iso_sort n0 n2) \Rightarrow (\lambda 
-(H0: (eq T (TSort n0) (TLRef n1))).(\lambda (H1: (eq T (TSort n2) u2)).((let 
-H2 \def (eq_ind T (TSort n0) (\lambda (e: T).(match e in T return (\lambda 
-(_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow False])) I (TLRef n1) H0) in (False_ind ((eq T 
-(TSort n2) u2) \to (ex nat (\lambda (n3: nat).(eq T u2 (TLRef n3))))) H2)) 
-H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (TLRef 
-n1))).(\lambda (H1: (eq T (TLRef i2) u2)).((let H2 \def (f_equal T nat 
-(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) 
-\Rightarrow i1 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i1])) 
-(TLRef i1) (TLRef n1) H0) in (eq_ind nat n1 (\lambda (_: nat).((eq T (TLRef 
-i2) u2) \to (ex nat (\lambda (n2: nat).(eq T u2 (TLRef n2)))))) (\lambda (H3: 
-(eq T (TLRef i2) u2)).(eq_ind T (TLRef i2) (\lambda (t: T).(ex nat (\lambda 
-(n2: nat).(eq T t (TLRef n2))))) (ex_intro nat (\lambda (n2: nat).(eq T 
-(TLRef i2) (TLRef n2))) i2 (refl_equal T (TLRef i2))) u2 H3)) i1 (sym_eq nat 
-i1 n1 H2))) H1))) | (iso_head v1 v2 t1 t2 k) \Rightarrow (\lambda (H0: (eq T 
-(THead k v1 t1) (TLRef n1))).(\lambda (H1: (eq T (THead k v2 t2) u2)).((let 
-H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: T).(match e in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n1) H0) in 
-(False_ind ((eq T (THead k v2 t2) u2) \to (ex nat (\lambda (n2: nat).(eq T u2 
-(TLRef n2))))) H2)) H1)))]) in (H0 (refl_equal T (TLRef n1)) (refl_equal T 
-u2))))).
-
-theorem iso_gen_head:
- \forall (k: K).(\forall (v1: T).(\forall (t1: T).(\forall (u2: T).((iso 
-(THead k v1 t1) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2 
-(THead k v2 t2)))))))))
-\def
- \lambda (k: K).(\lambda (v1: T).(\lambda (t1: T).(\lambda (u2: T).(\lambda 
-(H: (iso (THead k v1 t1) u2)).(let H0 \def (match H in iso return (\lambda 
-(t: T).(\lambda (t0: T).(\lambda (_: (iso t t0)).((eq T t (THead k v1 t1)) 
-\to ((eq T t0 u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2 
-(THead k v2 t2)))))))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq 
-T (TSort n1) (THead k v1 t1))).(\lambda (H1: (eq T (TSort n2) u2)).((let H2 
-\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 k v1 t1) H0) in (False_ind ((eq T 
-(TSort n2) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2 
-(THead k v2 t2)))))) H2)) H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: 
-(eq T (TLRef i1) (THead k v1 t1))).(\lambda (H1: (eq T (TLRef i2) u2)).((let 
-H2 \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 k v1 t1) H0) in (False_ind ((eq T 
-(TLRef i2) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2 
-(THead k v2 t2)))))) H2)) H1))) | (iso_head v0 v2 t0 t2 k0) \Rightarrow 
-(\lambda (H0: (eq T (THead k0 v0 t0) (THead k v1 t1))).(\lambda (H1: (eq T 
-(THead k0 v2 t2) u2)).((let H2 \def (f_equal T T (\lambda (e: T).(match e in 
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) 
-\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k0 v0 t0) (THead k v1 
-t1) H0) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 
-| (THead _ t _) \Rightarrow t])) (THead k0 v0 t0) (THead k v1 t1) H0) in 
-((let H4 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: 
-T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ 
-_) \Rightarrow k1])) (THead k0 v0 t0) (THead k v1 t1) H0) in (eq_ind K k 
-(\lambda (k1: K).((eq T v0 v1) \to ((eq T t0 t1) \to ((eq T (THead k1 v2 t2) 
-u2) \to (ex_2 T T (\lambda (v3: T).(\lambda (t3: T).(eq T u2 (THead k v3 
-t3))))))))) (\lambda (H5: (eq T v0 v1)).(eq_ind T v1 (\lambda (_: T).((eq T 
-t0 t1) \to ((eq T (THead k v2 t2) u2) \to (ex_2 T T (\lambda (v3: T).(\lambda 
-(t3: T).(eq T u2 (THead k v3 t3)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T 
-t1 (\lambda (_: T).((eq T (THead k v2 t2) u2) \to (ex_2 T T (\lambda (v3: 
-T).(\lambda (t3: T).(eq T u2 (THead k v3 t3))))))) (\lambda (H7: (eq T (THead 
-k v2 t2) u2)).(eq_ind T (THead k v2 t2) (\lambda (t: T).(ex_2 T T (\lambda 
-(v3: T).(\lambda (t3: T).(eq T t (THead k v3 t3)))))) (ex_2_intro T T 
-(\lambda (v3: T).(\lambda (t3: T).(eq T (THead k v2 t2) (THead k v3 t3)))) v2 
-t2 (refl_equal T (THead k v2 t2))) u2 H7)) t0 (sym_eq T t0 t1 H6))) v0 
-(sym_eq T v0 v1 H5))) k0 (sym_eq K k0 k H4))) H3)) H2)) H1)))]) in (H0 
-(refl_equal T (THead k v1 t1)) (refl_equal T u2))))))).
-
-theorem iso_refl:
- \forall (t: T).(iso t t)
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(iso t0 t0)) (\lambda (n: 
-nat).(iso_sort n n)) (\lambda (n: nat).(iso_lref n n)) (\lambda (k: 
-K).(\lambda (t0: T).(\lambda (_: (iso t0 t0)).(\lambda (t1: T).(\lambda (_: 
-(iso t1 t1)).(iso_head t0 t0 t1 t1 k)))))) t).
-
-theorem lifts_tapp:
- \forall (h: nat).(\forall (d: nat).(\forall (v: T).(\forall (vs: TList).(eq 
-TList (lifts h d (TApp vs v)) (TApp (lifts h d vs) (lift h d v))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (v: T).(\lambda (vs: 
-TList).(TList_ind (\lambda (t: TList).(eq TList (lifts h d (TApp t v)) (TApp 
-(lifts h d t) (lift h d v)))) (refl_equal TList (TCons (lift h d v) TNil)) 
-(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq TList (lifts h d (TApp 
-t0 v)) (TApp (lifts h d t0) (lift h d v)))).(eq_ind_r TList (TApp (lifts h d 
-t0) (lift h d v)) (\lambda (t1: TList).(eq TList (TCons (lift h d t) t1) 
-(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v))))) (refl_equal TList 
-(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v)))) (lifts h d (TApp t0 
-v)) H)))) vs)))).
-
-theorem pr3_flat:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
-(t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (f: F).(pr3 c (THead 
-(Flat f) u1 t1) (THead (Flat f) u2 t2)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 
-u2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda 
-(f: F).(pr3_head_12 c u1 u2 H (Flat f) t1 t2 (pr3_cflat c t1 t2 H0 f 
-u2))))))))).
-
-theorem pr3_gen_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u1: 
-T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind b) u1 t1) x) \to (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind b) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 (CHead c (Bind b) u1) t1 t2)))) (pr3 (CHead c (Bind 
-b) u1) t1 (lift (S O) O x)))))))))
-\def
- \lambda (b: B).(match b in B return (\lambda (b0: B).((not (eq B b0 Abst)) 
-\to (\forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
-(THead (Bind b0) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind b0) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c 
-u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b0) u1) t1 
-t2)))) (pr3 (CHead c (Bind b0) u1) t1 (lift (S O) O x)))))))))) with [Abbr 
-\Rightarrow (\lambda (_: (not (eq B Abbr Abst))).(\lambda (c: C).(\lambda 
-(u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H0: (pr3 c (THead (Bind 
-Abbr) u1 t1) x)).(let H1 \def (pr3_gen_abbr c u1 t1 x H0) in (or_ind (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) 
-t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
-(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 
-(CHead c (Bind Abbr) u1) t1 (lift (S O) O x))) (\lambda (H2: (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind Abbr) u1) t1 t2))) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) 
-u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x))) (\lambda 
-(x0: T).(\lambda (x1: T).(\lambda (H3: (eq T x (THead (Bind Abbr) x0 
-x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: (pr3 (CHead c (Bind Abbr) 
-u1) t1 x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
-(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 
-(CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (ex3_2_intro T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind Abbr) u1) t1 t2))) x0 x1 H3 H4 H5))))))) H2)) (\lambda (H2: 
-(pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x))).(or_intror (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) 
-t1 (lift (S O) O x)) H2)) H1)))))))) | Abst \Rightarrow (\lambda (H: (not (eq 
-B Abst Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: 
-T).(\lambda (_: (pr3 c (THead (Bind Abst) u1 t1) x)).(let H1 \def (match (H 
-(refl_equal B Abst)) in False return (\lambda (_: False).(or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 (CHead c (Bind Abst) u1) t1 t2)))) (pr3 (CHead c (Bind Abst) u1) 
-t1 (lift (S O) O x)))) with []) in H1))))))) | Void \Rightarrow (\lambda (_: 
-(not (eq B Void Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: 
-T).(\lambda (x: T).(\lambda (H0: (pr3 c (THead (Bind Void) u1 t1) x)).(let H1 
-\def (pr3_gen_void c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
-(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t2)))))) (pr3 (CHead c 
-(Bind Void) u1) t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) 
-u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda 
-(H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) 
-u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) 
-u) t1 t2))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
-(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead 
-c (Bind b0) u) t1 t2))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 
-t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H3: (eq T x (THead (Bind Void) x0 
-x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: ((\forall (b0: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(or_introl (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind Void) u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S 
-O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
-(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 t2))) x0 x1 
-H3 H4 (H5 Void u1)))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Void) u1) t1 
-(lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 
-t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x)) H2)) H1))))))))]).
-
-theorem pr3_iso_appls_abbr:
- \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
-(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).(let u1 \def (THeads (Flat 
-Appl) vs (TLRef i)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
-(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) vs (lift (S i) O w)) 
-u2))))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda 
-(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind 
-(\lambda (t: TList).(let u1 \def (THeads (Flat Appl) t (TLRef i)) in (\forall 
-(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to 
-(pr3 c (THeads (Flat Appl) t (lift (S i) O w)) u2)))))) (\lambda (u2: 
-T).(\lambda (H0: (pr3 c (TLRef i) u2)).(\lambda (H1: (((iso (TLRef i) u2) \to 
-(\forall (P: Prop).P)))).(let H2 \def (pr3_gen_lref c u2 i H0) in (or_ind (eq 
-T u2 (TLRef i)) (ex3_3 C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_: 
-T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u: 
-T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(v: T).(eq T u2 (lift (S i) O v)))))) (pr3 c (lift (S i) O w) u2) (\lambda 
-(H3: (eq T u2 (TLRef i))).(let H4 \def (eq_ind T u2 (\lambda (t: T).((iso 
-(TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H3) in (eq_ind_r T 
-(TLRef i) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (H4 (iso_refl (TLRef 
-i)) (pr3 c (lift (S i) O w) (TLRef i))) u2 H3))) (\lambda (H3: (ex3_3 C T T 
-(\lambda (d0: C).(\lambda (u: T).(\lambda (_: T).(getl i c (CHead d0 (Bind 
-Abbr) u))))) (\lambda (d0: C).(\lambda (u: T).(\lambda (v: T).(pr3 d0 u v)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T u2 (lift (S i) O 
-v))))))).(ex3_3_ind C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_: 
-T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u: 
-T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(v: T).(eq T u2 (lift (S i) O v))))) (pr3 c (lift (S i) O w) u2) (\lambda 
-(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H4: (getl i c (CHead x0 
-(Bind Abbr) x1))).(\lambda (H5: (pr3 x0 x1 x2)).(\lambda (H6: (eq T u2 (lift 
-(S i) O x2))).(let H7 \def (eq_ind T u2 (\lambda (t: T).((iso (TLRef i) t) 
-\to (\forall (P: Prop).P))) H1 (lift (S i) O x2) H6) in (eq_ind_r T (lift (S 
-i) O x2) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (let H8 \def (eq_ind C 
-(CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H (CHead x0 (Bind 
-Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) 
-H4)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return 
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow 
-c0])) (CHead d (Bind Abbr) w) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d 
-(Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) H4)) in ((let H10 \def (f_equal 
-C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
-\Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) w) (CHead 
-x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind 
-Abbr) x1) H4)) in (\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 
-(\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 w H10) in (let H13 
-\def (eq_ind_r T x1 (\lambda (t: T).(pr3 x0 t x2)) H5 w H10) in (let H14 \def 
-(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) w))) H12 d 
-H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c0: C).(pr3 c0 w x2)) H13 d 
-H11) in (pr3_lift c d (S i) O (getl_drop Abbr c d w i H14) w x2 H15))))))) 
-H9))) u2 H6)))))))) H3)) H2))))) (\lambda (t: T).(\lambda (t0: 
-TList).(\lambda (H0: ((\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (TLRef 
-i)) u2) \to ((((iso (THeads (Flat Appl) t0 (TLRef i)) u2) \to (\forall (P: 
-Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (lift (S i) O w)) 
-u2)))))).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads 
-(Flat Appl) t0 (TLRef i))) u2)).(\lambda (H2: (((iso (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (TLRef i))) u2) \to (\forall (P: Prop).P)))).(let H3 
-\def (pr3_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) u2 H1) in (or3_ind 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 
-t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2)))) (ex4_4 T 
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 
-c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (H4: (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: 
-T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))))).(ex3_2_ind T T (\lambda 
-(u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
-(THeads (Flat Appl) t0 (TLRef i)) t2))) (pr3 c (THead (Flat Appl) t (THeads 
-(Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H5: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c t 
-x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(let H8 \def 
-(eq_ind T u2 (\lambda (t1: T).((iso (THead (Flat Appl) t (THeads (Flat Appl) 
-t0 (TLRef i))) t1) \to (\forall (P: Prop).P))) H2 (THead (Flat Appl) x0 x1) 
-H5) in (eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(pr3 c (THead 
-(Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) t1)) (H8 (iso_head t 
-x0 (THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) (pr3 c (THead (Flat 
-Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) (THead (Flat Appl) x0 x1))) 
-u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t 
-u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T 
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 
-c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O 
-w))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H6: (pr3 c t 
-x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
-Abst) x0 x1))).(\lambda (H8: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c 
-(Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t 
-(THead (Bind Abst) x0 x1)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift 
-(S i) O w))) c (pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead 
-(Bind Abst) x0 x1) (H0 (THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso 
-(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (P: 
-Prop).(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H9 P)))) t Appl) (THead 
-(Bind Abbr) t x1) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) 
-(THead (Bind Abbr) t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) 
-x0 x1)) (THead (Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 
-(pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t 
-x1) c (pr3_head_12 c t x2 H6 (Bind Abbr) x1 x3 (H8 Abbr x2)) u2 H5)))))))))) 
-H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind 
-B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
-(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift 
-(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
-(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat 
-Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda 
-(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not 
-(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
-(THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c (THead (Bind x0) x5 (THead 
-(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H8: (pr3 c t x4)).(\lambda 
-(H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t 
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat 
-Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Bind x0) 
-x1 (THead (Flat Appl) (lift (S O) O t) x2)) (THead (Flat Appl) t (THeads 
-(Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t (THead (Bind 
-x0) x1 x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c 
-(pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead (Bind x0) x1 
-x2) (H0 (THead (Bind x0) x1 x2) H6 (\lambda (H11: (iso (THeads (Flat Appl) t0 
-(TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (P: 
-Prop).(iso_flats_lref_bind_false Appl x0 i x1 x2 t0 H11 P)))) t Appl) (THead 
-(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr3_pr2 c (THead (Flat 
-Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift 
-(S O) O t) x2)) (pr2_free c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) 
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr0_upsilon x0 
-H5 t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2))))) (THead (Bind 
-x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr3_head_12 c x1 x1 
-(pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift (S O) O t) x2) (THead 
-(Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead c (Bind x0) x1) (lift 
-(S O) O t) (lift (S O) O x4) (pr3_lift (CHead c (Bind x0) x1) c (S O) O 
-(drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H8) (Flat Appl) x2 x2 
-(pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) 
-u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) 
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c (pr3_head_12 
-c x1 x5 H9 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat 
-Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 
-(lift (S O) O x4) Appl)) u2 H7)))))))))))))) H4)) H3)))))))) vs)))))).
-
-theorem pr3_iso_appl_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: 
-T).(\forall (t: T).(let u1 \def (THead (Flat Appl) v1 (THead (Bind b) v2 t)) 
-in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
-(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) v2 (THead (Flat Appl) 
-(lift (S O) O v1) t)) u2))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v1: T).(\lambda 
-(v2: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H0: (pr3 c 
-(THead (Flat Appl) v1 (THead (Bind b) v2 t)) u2)).(\lambda (H1: (((iso (THead 
-(Flat Appl) v1 (THead (Bind b) v2 t)) u2) \to (\forall (P: Prop).P)))).(let 
-H2 \def (pr3_gen_appl c v1 (THead (Bind b) v2 t) u2 H0) in (or3_ind (ex3_2 T 
-T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 c (THead (Bind b) v2 t) t2)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1 
-t2)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) 
-(\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 
-z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
-T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat 
-Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b0: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c (THead (Bind b) v2 
-(THead (Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (H3: (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 c (THead (Bind b) v2 t) t2))))).(ex3_2_ind T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
-(THead (Bind b) v2 t) t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) 
-(lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq 
-T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v1 x0)).(\lambda (_: 
-(pr3 c (THead (Bind b) v2 t) x1)).(let H7 \def (eq_ind T u2 (\lambda (t0: 
-T).((iso (THead (Flat Appl) v1 (THead (Bind b) v2 t)) t0) \to (\forall (P: 
-Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl) 
-x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S 
-O) O v1) t)) t0)) (H7 (iso_head v1 x0 (THead (Bind b) v2 t) x1 (Flat Appl)) 
-(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead 
-(Flat Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1 
-t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))) 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
-(THead (Bind b) v2 t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: 
-T).(pr3 (CHead c (Bind b0) u) z1 t2))))))) (pr3 c (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c (THead (Bind Abbr) 
-x2 x3) u2)).(\lambda (H5: (pr3 c v1 x2)).(\lambda (H6: (pr3 c (THead (Bind b) 
-v2 t) (THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b0: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b0) u) x1 x3))))).(pr3_t (THead (Bind Abbr) x2 x3) 
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) c (let H_x \def 
-(pr3_gen_bind b H c v2 t (THead (Bind Abst) x0 x1) H6) in (let H8 \def H_x in 
-(or_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst) 
-x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) 
-(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind Abst) x0 x1))) (pr3 c 
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind 
-Abbr) x2 x3)) (\lambda (H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
-T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: 
-T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) 
-(lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3)) (\lambda (x4: T).(\lambda 
-(x5: T).(\lambda (H10: (eq T (THead (Bind Abst) x0 x1) (THead (Bind b) x4 
-x5))).(\lambda (H11: (pr3 c v2 x4)).(\lambda (H12: (pr3 (CHead c (Bind b) v2) 
-t x5)).(let H13 \def (f_equal T B (\lambda (e: T).(match e in T return 
-(\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow 
-Abst | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with 
-[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (THead (Bind Abst) 
-x0 x1) (THead (Bind b) x4 x5) H10) in ((let H14 \def (f_equal T T (\lambda 
-(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 
-| (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind 
-Abst) x0 x1) (THead (Bind b) x4 x5) H10) in ((let H15 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) \Rightarrow t0])) 
-(THead (Bind Abst) x0 x1) (THead (Bind b) x4 x5) H10) in (\lambda (H16: (eq T 
-x0 x4)).(\lambda (H17: (eq B Abst b)).(let H18 \def (eq_ind_r T x5 (\lambda 
-(t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H12 x1 H15) in (let H19 \def 
-(eq_ind_r T x4 (\lambda (t0: T).(pr3 c v2 t0)) H11 x0 H16) in (let H20 \def 
-(eq_ind_r B b (\lambda (b0: B).(pr3 (CHead c (Bind b0) v2) t x1)) H18 Abst 
-H17) in (let H21 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H 
-Abst H17) in (eq_ind B Abst (\lambda (b0: B).(pr3 c (THead (Bind b0) v2 
-(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3))) (let H22 
-\def (match (H21 (refl_equal B Abst)) in False return (\lambda (_: 
-False).(pr3 c (THead (Bind Abst) v2 (THead (Flat Appl) (lift (S O) O v1) t)) 
-(THead (Bind Abbr) x2 x3))) with []) in H22) b H17)))))))) H14)) H13))))))) 
-H9)) (\lambda (H9: (pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind 
-Abst) x0 x1)))).(pr3_t (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O 
-x2) (lift (S O) O (THead (Bind Abst) x0 x1)))) (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O v1) t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift 
-(S O) O v1) t) (THead (Flat Appl) (lift (S O) O x2) (lift (S O) O (THead 
-(Bind Abst) x0 x1))) (Bind b) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O 
-v1) (lift (S O) O x2) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop 
-(Bind b) O c c (drop_refl c) v2) v1 x2 H5) t (lift (S O) O (THead (Bind Abst) 
-x0 x1)) H9 Appl)) (THead (Bind Abbr) x2 x3) (eq_ind T (lift (S O) O (THead 
-(Flat Appl) x2 (THead (Bind Abst) x0 x1))) (\lambda (t0: T).(pr3 c (THead 
-(Bind b) v2 t0) (THead (Bind Abbr) x2 x3))) (pr3_sing c (THead (Bind Abbr) x2 
-x1) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst) 
-x0 x1)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 
-(THead (Bind Abst) x0 x1)))) (THead (Bind Abbr) x2 x1) (pr0_zeta b H (THead 
-(Flat Appl) x2 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x1) (pr0_beta 
-x0 x2 x2 (pr0_refl x2) x1 x1 (pr0_refl x1)) v2)) (THead (Bind Abbr) x2 x3) 
-(pr3_head_12 c x2 x2 (pr3_refl c x2) (Bind Abbr) x1 x3 (H7 Abbr x2))) (THead 
-(Flat Appl) (lift (S O) O x2) (lift (S O) O (THead (Bind Abst) x0 x1))) 
-(lift_flat Appl x2 (THead (Bind Abst) x0 x1) (S O) O)))) H8))) u2 H4))))))))) 
-H3)) (\lambda (H3: (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead 
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) 
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c v1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind 
-B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 z1)))))))) (\lambda 
-(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: 
-T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) 
-O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) 
-y2) z1 z2))))))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O 
-v1) t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0 
-Abst))).(\lambda (H5: (pr3 c (THead (Bind b) v2 t) (THead (Bind x0) x1 
-x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) 
-O x4) x3)) u2)).(\lambda (H7: (pr3 c v1 x4)).(\lambda (H8: (pr3 c x1 
-x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t (THead (Bind 
-x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O v1) t)) c (let H_x \def (pr3_gen_bind b H c v2 t 
-(THead (Bind x0) x1 x2) H5) in (let H10 \def H_x in (or_ind (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) (pr3 (CHead c (Bind 
-b) v2) t (lift (S O) O (THead (Bind x0) x1 x2))) (pr3 c (THead (Bind b) v2 
-(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat 
-Appl) (lift (S O) O x4) x3))) (\lambda (H11: (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 (CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 (CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat Appl) 
-(lift (S O) O x4) x3))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H12: (eq 
-T (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7))).(\lambda (H13: (pr3 c v2 
-x6)).(\lambda (H14: (pr3 (CHead c (Bind b) v2) t x7)).(let H15 \def (f_equal 
-T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) 
-\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) \Rightarrow (match 
-k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
-\Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in 
-((let H16 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ t0 
-_) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in 
-((let H17 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x2 | (TLRef _) \Rightarrow x2 | (THead _ _ 
-t0) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in 
-(\lambda (H18: (eq T x1 x6)).(\lambda (H19: (eq B x0 b)).(let H20 \def 
-(eq_ind_r T x7 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H14 x2 H17) 
-in (let H21 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c v2 t0)) H13 x1 H18) 
-in (let H22 \def (eq_ind B x0 (\lambda (b0: B).(pr3 (CHead c (Bind b0) x5) x2 
-x3)) H9 b H19) in (let H23 \def (eq_ind B x0 (\lambda (b0: B).(not (eq B b0 
-Abst))) H4 b H19) in (eq_ind_r B b (\lambda (b0: B).(pr3 c (THead (Bind b) v2 
-(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind b0) x5 (THead (Flat 
-Appl) (lift (S O) O x4) x3)))) (pr3_head_21 c v2 x5 (pr3_t x1 v2 c H21 x5 H8) 
-(Bind b) (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat Appl) (lift (S 
-O) O x4) x3) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O 
-x4) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c 
-(drop_refl c) v2) v1 x4 H7) t x3 (pr3_t x2 t (CHead c (Bind b) v2) H20 x3 
-(pr3_pr3_pr3_t c v2 x1 H21 x2 x3 (Bind b) (pr3_pr3_pr3_t c x1 x5 H8 x2 x3 
-(Bind b) H22))) Appl)) x0 H19)))))))) H16)) H15))))))) H11)) (\lambda (H11: 
-(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind x0) x1 x2)))).(pr3_t 
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead 
-(Bind x0) x1 x2)))) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) 
-t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat 
-Appl) (lift (S O) O x4) (lift (S O) O (THead (Bind x0) x1 x2))) (Bind b) 
-(pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O x4) (pr3_lift 
-(CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) v2) 
-v1 x4 H7) t (lift (S O) O (THead (Bind x0) x1 x2)) H11 Appl)) (THead (Bind 
-x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (eq_ind T (lift (S O) O 
-(THead (Flat Appl) x4 (THead (Bind x0) x1 x2))) (\lambda (t0: T).(pr3 c 
-(THead (Bind b) v2 t0) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O 
-x4) x3)))) (pr3_sing c (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O 
-x4) x2)) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x4 (THead (Bind 
-x0) x1 x2)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) 
-x4 (THead (Bind x0) x1 x2)))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S 
-O) O x4) x2)) (pr0_zeta b H (THead (Flat Appl) x4 (THead (Bind x0) x1 x2)) 
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr0_upsilon x0 
-H4 x4 x4 (pr0_refl x4) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2)) v2)) (THead 
-(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (pr3_head_12 c x1 x5 
-H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat Appl) 
-(lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H9 (lift (S 
-O) O x4) Appl))) (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead 
-(Bind x0) x1 x2))) (lift_flat Appl x4 (THead (Bind x0) x1 x2) (S O) O)))) 
-H10))) u2 H6))))))))))))) H3)) H2)))))))))).
-
-theorem pr3_iso_appls_appl_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (v: T).(\forall (u: 
-T).(\forall (t: T).(\forall (vs: TList).(let u1 \def (THeads (Flat Appl) vs 
-(THead (Flat Appl) v (THead (Bind b) u t))) in (\forall (c: C).(\forall (u2: 
-T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c 
-(THeads (Flat Appl) vs (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) 
-t))) u2)))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v: T).(\lambda 
-(u: T).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: 
-TList).(let u1 \def (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind 
-b) u t))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 
-u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead 
-(Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))) (\lambda (c: 
-C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) v (THead (Bind b) 
-u t)) u2)).(\lambda (H1: (((iso (THead (Flat Appl) v (THead (Bind b) u t)) 
-u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b H v u t c u2 H0 H1))))) 
-(\lambda (t0: T).(\lambda (t1: TList).(\lambda (H0: ((\forall (c: C).(\forall 
-(u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u 
-t))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind 
-b) u t))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1 
-(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))).(\lambda 
-(c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t0 (THeads 
-(Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t)))) u2)).(\lambda 
-(H2: (((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v 
-(THead (Bind b) u t)))) u2) \to (\forall (P: Prop).P)))).(let H3 \def 
-(pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind 
-b) u t))) u2 H1) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
-T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead 
-(Flat Appl) v (THead (Bind b) u t))) t2)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
-Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: 
-B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2)))))))) (ex6_6 B T T T 
-T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
-c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead 
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) 
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c 
-(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat 
-Appl) (lift (S O) O v) t)))) u2) (\lambda (H4: (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
-(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) 
-t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
-(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
-Appl) v (THead (Bind b) u t))) t2))) (pr3 c (THead (Flat Appl) t0 (THeads 
-(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) 
-u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T u2 (THead (Flat 
-Appl) x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat 
-Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) x1)).(let H8 \def 
-(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl) 
-t1 (THead (Flat Appl) v (THead (Bind b) u t)))) t2) \to (\forall (P: 
-Prop).P))) H2 (THead (Flat Appl) x0 x1) H5) in (eq_ind_r T (THead (Flat Appl) 
-x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 
-(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) t2)) (H8 
-(iso_head t0 x0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u 
-t))) x1 (Flat Appl)) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 
-(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) (THead (Flat 
-Appl) x0 x1))) u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
-Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: 
-B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))))).(ex4_4_ind T T 
-T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c 
-(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
-Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: 
-B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))) (pr3 c (THead 
-(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) 
-(lift (S O) O v) t)))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: 
-T).(\lambda (x3: T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) 
-u2)).(\lambda (H6: (pr3 c t0 x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t1 
-(THead (Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 
-x1))).(\lambda (H8: ((\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind 
-b0) u0) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0 
-(THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) 
-t)))) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Flat 
-Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S 
-O) O v) t)))) c (pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u 
-(THead (Flat Appl) (lift (S O) O v) t))) (THead (Bind Abst) x0 x1) (H0 c 
-(THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso (THeads (Flat Appl) t1 (THead 
-(Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 x1))).(\lambda (P: 
-Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 (THead (Bind b) u t) 
-t1 H9 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c (THead (Flat Appl) 
-t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr2_free c (THead 
-(Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr0_beta 
-x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) 
-x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c t0 x2 H6 (Bind Abbr) x1 x3 
-(H8 Abbr x2)) u2 H5)))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda 
-(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
-(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead 
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) 
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind 
-B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u 
-t))) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b0) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))) (pr3 c 
-(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat 
-Appl) (lift (S O) O v) t)))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda 
-(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not 
-(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t1 (THead (Flat 
-Appl) v (THead (Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c 
-(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda 
-(H8: (pr3 c t0 x4)).(\lambda (H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c 
-(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S 
-O) O x4) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u 
-(THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t (THead (Bind x0) x1 (THead 
-(Flat Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) 
-t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t 
-(THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0 (THeads 
-(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c 
-(pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) 
-(lift (S O) O v) t))) (THead (Bind x0) x1 x2) (H0 c (THead (Bind x0) x1 x2) 
-H6 (\lambda (H11: (iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead 
-(Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (P: 
-Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind b) u t) t1 
-H11 P)))) t0 Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) 
-x2)) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind 
-x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat 
-Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) 
-(lift (S O) O t0) x2)) (pr0_upsilon x0 H5 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl 
-x1) x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S 
-O) O x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat 
-Appl) (lift (S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2) 
-(pr3_head_12 (CHead c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4) 
-(pr3_lift (CHead c (Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c 
-(drop_refl c) x1) t0 x4 H8) (Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind 
-x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 
-(THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat 
-Appl) (lift (S O) O x4) x2)) c (pr3_head_12 c x1 x5 H9 (Bind x0) (THead (Flat 
-Appl) (lift (S O) O x4) x2) (THead (Flat Appl) (lift (S O) O x4) x3) 
-(pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 (lift (S O) O x4) Appl)) u2 
-H7)))))))))))))) H4)) H3))))))))) vs)))))).
-
-theorem pr3_iso_appls_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (vs: TList).(\forall (u: 
-T).(\forall (t: T).(let u1 \def (THeads (Flat Appl) vs (THead (Bind b) u t)) 
-in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
-(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) 
-(lifts (S O) O vs) t)) u2))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (vs: 
-TList).(tlist_ind_rew (\lambda (t: TList).(\forall (u: T).(\forall (t0: 
-T).(let u1 \def (THeads (Flat Appl) t (THead (Bind b) u t0)) in (\forall (c: 
-C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: 
-Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t) 
-t0)) u2))))))))) (\lambda (u: T).(\lambda (t: T).(\lambda (c: C).(\lambda 
-(u2: T).(\lambda (H0: (pr3 c (THead (Bind b) u t) u2)).(\lambda (_: (((iso 
-(THead (Bind b) u t) u2) \to (\forall (P: Prop).P)))).H0)))))) (\lambda (ts: 
-TList).(\lambda (t: T).(\lambda (H0: ((\forall (u: T).(\forall (t0: 
-T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) ts (THead 
-(Bind b) u t0)) u2) \to ((((iso (THeads (Flat Appl) ts (THead (Bind b) u t0)) 
-u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat 
-Appl) (lifts (S O) O ts) t0)) u2))))))))).(\lambda (u: T).(\lambda (t0: 
-T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THeads (Flat Appl) 
-(TApp ts t) (THead (Bind b) u t0)) u2)).(\lambda (H2: (((iso (THeads (Flat 
-Appl) (TApp ts t) (THead (Bind b) u t0)) u2) \to (\forall (P: 
-Prop).P)))).(eq_ind_r TList (TApp (lifts (S O) O ts) (lift (S O) O t)) 
-(\lambda (t1: TList).(pr3 c (THead (Bind b) u (THeads (Flat Appl) t1 t0)) 
-u2)) (eq_ind_r T (THeads (Flat Appl) (lifts (S O) O ts) (THead (Flat Appl) 
-(lift (S O) O t) t0)) (\lambda (t1: T).(pr3 c (THead (Bind b) u t1) u2)) (let 
-H3 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind b) u t0)) 
-(\lambda (t1: T).(pr3 c t1 u2)) H1 (THeads (Flat Appl) ts (THead (Flat Appl) 
-t (THead (Bind b) u t0))) (theads_tapp (Flat Appl) ts t (THead (Bind b) u 
-t0))) in (let H4 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind 
-b) u t0)) (\lambda (t1: T).((iso t1 u2) \to (\forall (P: Prop).P))) H2 
-(THeads (Flat Appl) ts (THead (Flat Appl) t (THead (Bind b) u t0))) 
-(theads_tapp (Flat Appl) ts t (THead (Bind b) u t0))) in ((match ts in TList 
-return (\lambda (t1: TList).(((\forall (u0: T).(\forall (t2: T).(\forall (c0: 
-C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) 
-u3) \to ((((iso (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to 
-(\forall (P: Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) 
-(lifts (S O) O t1) t2)) u3)))))))) \to ((pr3 c (THeads (Flat Appl) t1 (THead 
-(Flat Appl) t (THead (Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) t1 
-(THead (Flat Appl) t (THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) 
-\to (pr3 c (THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t1) (THead 
-(Flat Appl) (lift (S O) O t) t0))) u2))))) with [TNil \Rightarrow (\lambda 
-(_: ((\forall (u0: T).(\forall (t1: T).(\forall (c0: C).(\forall (u3: 
-T).((pr3 c0 (THeads (Flat Appl) TNil (THead (Bind b) u0 t1)) u3) \to ((((iso 
-(THeads (Flat Appl) TNil (THead (Bind b) u0 t1)) u3) \to (\forall (P: 
-Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O 
-TNil) t1)) u3))))))))).(\lambda (H6: (pr3 c (THeads (Flat Appl) TNil (THead 
-(Flat Appl) t (THead (Bind b) u t0))) u2)).(\lambda (H7: (((iso (THeads (Flat 
-Appl) TNil (THead (Flat Appl) t (THead (Bind b) u t0))) u2) \to (\forall (P: 
-Prop).P)))).(pr3_iso_appl_bind b H t u t0 c u2 H6 H7)))) | (TCons t1 t2) 
-\Rightarrow (\lambda (H5: ((\forall (u0: T).(\forall (t3: T).(\forall (c0: 
-C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) (TCons t1 t2) (THead (Bind 
-b) u0 t3)) u3) \to ((((iso (THeads (Flat Appl) (TCons t1 t2) (THead (Bind b) 
-u0 t3)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 (THead (Bind b) u0 
-(THeads (Flat Appl) (lifts (S O) O (TCons t1 t2)) t3)) u3))))))))).(\lambda 
-(H6: (pr3 c (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Appl) t (THead 
-(Bind b) u t0))) u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t1 t2) 
-(THead (Flat Appl) t (THead (Bind b) u t0))) u2) \to (\forall (P: 
-Prop).P)))).(H5 u (THead (Flat Appl) (lift (S O) O t) t0) c u2 
-(pr3_iso_appls_appl_bind b H t u t0 (TCons t1 t2) c u2 H6 H7) (\lambda (H8: 
-(iso (THeads (Flat Appl) (TCons t1 t2) (THead (Bind b) u (THead (Flat Appl) 
-(lift (S O) O t) t0))) u2)).(\lambda (P: Prop).(H7 (iso_trans (THeads (Flat 
-Appl) (TCons t1 t2) (THead (Flat Appl) t (THead (Bind b) u t0))) (THeads 
-(Flat Appl) (TCons t1 t2) (THead (Bind b) u (THead (Flat Appl) (lift (S O) O 
-t) t0))) (iso_head t1 t1 (THeads (Flat Appl) t2 (THead (Flat Appl) t (THead 
-(Bind b) u t0))) (THeads (Flat Appl) t2 (THead (Bind b) u (THead (Flat Appl) 
-(lift (S O) O t) t0))) (Flat Appl)) u2 H8) P)))))))]) H0 H3 H4))) (THeads 
-(Flat Appl) (TApp (lifts (S O) O ts) (lift (S O) O t)) t0) (theads_tapp (Flat 
-Appl) (lifts (S O) O ts) (lift (S O) O t) t0)) (lifts (S O) O (TApp ts t)) 
-(lifts_tapp (S O) O t ts))))))))))) vs))).
-
-theorem pr3_iso_beta:
- \forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 \def (THead (Flat 
-Appl) v (THead (Bind Abst) w t)) in (\forall (c: C).(\forall (u2: T).((pr3 c 
-u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind 
-Abbr) v t) u2))))))))
-\def
- \lambda (v: T).(\lambda (w: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: 
-T).(\lambda (H: (pr3 c (THead (Flat Appl) v (THead (Bind Abst) w t)) 
-u2)).(\lambda (H0: (((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) u2) 
-\to (\forall (P: Prop).P)))).(let H1 \def (pr3_gen_appl c v (THead (Bind 
-Abst) w t) u2 H) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
-T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v 
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) w t) t2)))) 
-(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: 
-T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) 
-w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind 
-b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c v u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c 
-(THead (Bind Abbr) v t) u2) (\lambda (H2: (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
-(THead (Bind Abst) w t) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: 
-T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) 
-w t) t2))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H3: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v 
-x0)).(\lambda (_: (pr3 c (THead (Bind Abst) w t) x1)).(let H6 \def (eq_ind T 
-u2 (\lambda (t0: T).((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) t0) 
-\to (\forall (P: Prop).P))) H0 (THead (Flat Appl) x0 x1) H3) in (eq_ind_r T 
-(THead (Flat Appl) x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind Abbr) v t) 
-t0)) (H6 (iso_head v x0 (THead (Bind Abst) w t) x1 (Flat Appl)) (pr3 c (THead 
-(Bind Abbr) v t) (THead (Flat Appl) x0 x1))) u2 H3))))))) H2)) (\lambda (H2: 
-(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: 
-T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) 
-w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind 
-b) u) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v 
-u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Bind Abbr) v t) 
-u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H3: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H4: (pr3 c v 
-x2)).(\lambda (H5: (pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) x0 
-x1))).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) 
-u) x1 x3))))).(let H7 \def (pr3_gen_abst c w t (THead (Bind Abst) x0 x1) H5) 
-in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst) 
-x0 x1) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w 
-u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda 
-(x4: T).(\lambda (x5: T).(\lambda (H8: (eq T (THead (Bind Abst) x0 x1) (THead 
-(Bind Abst) x4 x5))).(\lambda (H9: (pr3 c w x4)).(\lambda (H10: ((\forall (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x5))))).(let H11 \def (f_equal 
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) 
-(THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in ((let H12 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) 
-\Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in 
-(\lambda (H13: (eq T x0 x4)).(let H14 \def (eq_ind_r T x5 (\lambda (t0: 
-T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) H10 x1 
-H12) in (let H15 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c w t0)) H9 x0 
-H13) in (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) v t) c 
-(pr3_head_12 c v x2 H4 (Bind Abbr) t x3 (pr3_t x1 t (CHead c (Bind Abbr) x2) 
-(H14 Abbr x2) x3 (H6 Abbr x2))) u2 H3))))) H11))))))) H7)))))))))) H2)) 
-(\lambda (H2: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead (Bind b) y1 
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
-T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat 
-Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
-(THead (Bind Abst) w t) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda 
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) 
-u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2))))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: B).(\lambda 
-(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: 
-T).(\lambda (H3: (not (eq B x0 Abst))).(\lambda (H4: (pr3 c (THead (Bind 
-Abst) w t) (THead (Bind x0) x1 x2))).(\lambda (H5: (pr3 c (THead (Bind x0) x5 
-(THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (_: (pr3 c v 
-x4)).(\lambda (_: (pr3 c x1 x5)).(\lambda (H8: (pr3 (CHead c (Bind x0) x5) x2 
-x3)).(let H9 \def (pr3_gen_abst c w t (THead (Bind x0) x1 x2) H4) in 
-(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 
-x2) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w 
-u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda 
-(x6: T).(\lambda (x7: T).(\lambda (H10: (eq T (THead (Bind x0) x1 x2) (THead 
-(Bind Abst) x6 x7))).(\lambda (H11: (pr3 c w x6)).(\lambda (H12: ((\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x7))))).(let H13 \def 
-(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with 
-[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) 
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) 
-\Rightarrow b | (Flat _) \Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead 
-(Bind Abst) x6 x7) H10) in ((let H14 \def (f_equal T T (\lambda (e: T).(match 
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) 
-\Rightarrow x1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind x0) x1 x2) 
-(THead (Bind Abst) x6 x7) H10) in ((let H15 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 | 
-(TLRef _) \Rightarrow x2 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind x0) 
-x1 x2) (THead (Bind Abst) x6 x7) H10) in (\lambda (H16: (eq T x1 
-x6)).(\lambda (H17: (eq B x0 Abst)).(let H18 \def (eq_ind_r T x7 (\lambda 
-(t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) 
-H12 x2 H15) in (let H19 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c w t0)) 
-H11 x1 H16) in (let H20 \def (eq_ind B x0 (\lambda (b: B).(pr3 (CHead c (Bind 
-b) x5) x2 x3)) H8 Abst H17) in (let H21 \def (eq_ind B x0 (\lambda (b: 
-B).(pr3 c (THead (Bind b) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)) 
-H5 Abst H17) in (let H22 \def (eq_ind B x0 (\lambda (b: B).(not (eq B b 
-Abst))) H3 Abst H17) in (let H23 \def (match (H22 (refl_equal B Abst)) in 
-False return (\lambda (_: False).(pr3 c (THead (Bind Abbr) v t) u2)) with []) 
-in H23))))))))) H14)) H13))))))) H9)))))))))))))) H2)) H1)))))))).
-
-theorem pr3_iso_appls_beta:
- \forall (us: TList).(\forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 
-\def (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) w t))) in 
-(\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
-(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) us (THead (Bind Abbr) 
-v t)) u2)))))))))
-\def
- \lambda (us: TList).(TList_ind (\lambda (t: TList).(\forall (v: T).(\forall 
-(w: T).(\forall (t0: T).(let u1 \def (THeads (Flat Appl) t (THead (Flat Appl) 
-v (THead (Bind Abst) w t0))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 
-u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat 
-Appl) t (THead (Bind Abbr) v t0)) u2)))))))))) (\lambda (v: T).(\lambda (w: 
-T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H: (pr3 c 
-(THead (Flat Appl) v (THead (Bind Abst) w t)) u2)).(\lambda (H0: (((iso 
-(THead (Flat Appl) v (THead (Bind Abst) w t)) u2) \to (\forall (P: 
-Prop).P)))).(pr3_iso_beta v w t c u2 H H0)))))))) (\lambda (t: T).(\lambda 
-(t0: TList).(\lambda (H: ((\forall (v: T).(\forall (w: T).(\forall (t1: 
-T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (THead 
-(Flat Appl) v (THead (Bind Abst) w t1))) u2) \to ((((iso (THeads (Flat Appl) 
-t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) u2) \to (\forall (P: 
-Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1)) 
-u2)))))))))).(\lambda (v: T).(\lambda (w: T).(\lambda (t1: T).(\lambda (c: 
-C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) t (THeads (Flat 
-Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) u2)).(\lambda (H1: 
-(((iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Flat Appl) v 
-(THead (Bind Abst) w t1)))) u2) \to (\forall (P: Prop).P)))).(let H2 \def 
-(pr3_gen_appl c t (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind 
-Abst) w t1))) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
-T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) 
-t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2)))) (ex4_4 T T T T 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c 
-(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
-Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
-(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c 
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) 
-(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
-(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
-Appl) v (THead (Bind Abst) w t1))) t2))))).(ex3_2_ind T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
-(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2))) 
-(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) 
-u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat 
-Appl) x0 x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat 
-Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) x1)).(let H7 \def 
-(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t (THeads (Flat Appl) 
-t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) t2) \to (\forall (P: 
-Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl) 
-x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 
-(THead (Bind Abbr) v t1))) t2)) (H7 (iso_head t x0 (THeads (Flat Appl) t0 
-(THead (Flat Appl) v (THead (Bind Abst) w t1))) x1 (Flat Appl)) (pr3 c (THead 
-(Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) (THead (Flat 
-Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
-Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T T T 
-T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c 
-(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
-Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Flat 
-Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c 
-(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t x2)).(\lambda (H6: (pr3 
-c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) 
-(THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b: B).(\forall (u: 
-T).(pr3 (CHead c (Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) 
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c 
-(pr3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads 
-(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind Abst) x0 x1) (H v w t1 
-c (THead (Bind Abst) x0 x1) H6 (\lambda (H8: (iso (THeads (Flat Appl) t0 
-(THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) x0 
-x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 
-(THead (Bind Abst) w t1) t0 H8 P)))) t Appl) (THead (Bind Abbr) t x1) 
-(pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) 
-t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead 
-(Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 (pr0_refl x1))))) u2 
-(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t x1) c (pr3_head_12 c t 
-x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3: 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) 
-w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c 
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 
-z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
-Appl) v (THead (Bind Abst) w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda 
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) 
-u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead 
-(Bind Abbr) v t1))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: 
-T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq 
-B x0 Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v 
-(THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c 
-(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda 
-(H7: (pr3 c t x4)).(\lambda (H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c 
-(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S 
-O) O x4) x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) 
-v t1))) c (pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) 
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c 
-(pr3_t (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads 
-(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind x0) x1 x2) (H v w t1 c 
-(THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads (Flat Appl) t0 (THead 
-(Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda 
-(P: Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind Abst) 
-w t1) t0 H10 P)))) t Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) 
-O t) x2)) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead 
-(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr2_free c (THead 
-(Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) 
-(lift (S O) O t) x2)) (pr0_upsilon x0 H4 t t (pr0_refl t) x1 x1 (pr0_refl x1) 
-x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O 
-x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl) 
-(lift (S O) O t) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 
-(CHead c (Bind x0) x1) (lift (S O) O t) (lift (S O) O x4) (pr3_lift (CHead c 
-(Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H7) 
-(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift 
-(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S 
-O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c 
-(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) 
-(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) 
-x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))))))) us).
-
-theorem sn3_cpr3_trans:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
-(k: K).(\forall (t: T).((sn3 (CHead c k u1) t) \to (sn3 (CHead c k u2) 
-t)))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 
-u2)).(\lambda (k: K).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c k u1) 
-t)).(sn3_ind (CHead c k u1) (\lambda (t0: T).(sn3 (CHead c k u2) t0)) 
-(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
-(P: Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u1) 
-t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u2) 
-t2)))))).(sn3_sing (CHead c k u2) t1 (\lambda (t2: T).(\lambda (H3: (((eq T 
-t1 t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 (CHead c k u2) t1 
-t2)).(H2 t2 H3 (pr3_pr3_pr3_t c u1 u2 H t1 t2 k H4))))))))) t H0))))))).
-
-theorem sn3_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: 
-T).((sn3 c u) \to (\forall (t: T).((sn3 (CHead c (Bind b) u) t) \to (sn3 c 
-(THead (Bind b) u t))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda 
-(u: T).(\lambda (H0: (sn3 c u)).(sn3_ind c (\lambda (t: T).(\forall (t0: 
-T).((sn3 (CHead c (Bind b) t) t0) \to (sn3 c (THead (Bind b) t t0))))) 
-(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
-(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall 
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
-(\forall (t: T).((sn3 (CHead c (Bind b) t2) t) \to (sn3 c (THead (Bind b) t2 
-t))))))))).(\lambda (t: T).(\lambda (H3: (sn3 (CHead c (Bind b) t1) 
-t)).(sn3_ind (CHead c (Bind b) t1) (\lambda (t0: T).(sn3 c (THead (Bind b) t1 
-t0))) (\lambda (t2: T).(\lambda (H4: ((\forall (t3: T).((((eq T t2 t3) \to 
-(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 
-(CHead c (Bind b) t1) t3)))))).(\lambda (H5: ((\forall (t3: T).((((eq T t2 
-t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to 
-(sn3 c (THead (Bind b) t1 t3))))))).(sn3_sing c (THead (Bind b) t1 t2) 
-(\lambda (t3: T).(\lambda (H6: (((eq T (THead (Bind b) t1 t2) t3) \to 
-(\forall (P: Prop).P)))).(\lambda (H7: (pr3 c (THead (Bind b) t1 t2) 
-t3)).(let H_x \def (pr3_gen_bind b H c t1 t2 t3 H7) in (let H8 \def H_x in 
-(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) 
-u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) t2 t4)))) (pr3 (CHead c (Bind 
-b) t1) t2 (lift (S O) O t3)) (sn3 c t3) (\lambda (H9: (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 
-(CHead c (Bind b) t1) t2 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda 
-(t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 (CHead c (Bind b) 
-t1) t2 t4))) (sn3 c t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq 
-T t3 (THead (Bind b) x0 x1))).(\lambda (H11: (pr3 c t1 x0)).(\lambda (H12: 
-(pr3 (CHead c (Bind b) t1) t2 x1)).(let H13 \def (eq_ind T t3 (\lambda (t0: 
-T).((eq T (THead (Bind b) t1 t2) t0) \to (\forall (P: Prop).P))) H6 (THead 
-(Bind b) x0 x1) H10) in (eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: 
-T).(sn3 c t0)) (let H_x0 \def (term_dec t1 x0) in (let H14 \def H_x0 in 
-(or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: Prop).P)) (sn3 c (THead 
-(Bind b) x0 x1)) (\lambda (H15: (eq T t1 x0)).(let H16 \def (eq_ind_r T x0 
-(\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t0 x1)) \to 
-(\forall (P: Prop).P))) H13 t1 H15) in (let H17 \def (eq_ind_r T x0 (\lambda 
-(t0: T).(pr3 c t1 t0)) H11 t1 H15) in (eq_ind T t1 (\lambda (t0: T).(sn3 c 
-(THead (Bind b) t0 x1))) (let H_x1 \def (term_dec t2 x1) in (let H18 \def 
-H_x1 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: Prop).P)) (sn3 c 
-(THead (Bind b) t1 x1)) (\lambda (H19: (eq T t2 x1)).(let H20 \def (eq_ind_r 
-T x1 (\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t1 t0)) 
-\to (\forall (P: Prop).P))) H16 t2 H19) in (let H21 \def (eq_ind_r T x1 
-(\lambda (t0: T).(pr3 (CHead c (Bind b) t1) t2 t0)) H12 t2 H19) in (eq_ind T 
-t2 (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (H20 (refl_equal T (THead 
-(Bind b) t1 t2)) (sn3 c (THead (Bind b) t1 t2))) x1 H19)))) (\lambda (H19: 
-(((eq T t2 x1) \to (\forall (P: Prop).P)))).(H5 x1 H19 H12)) H18))) x0 
-H15)))) (\lambda (H15: (((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x1 
-\def (term_dec t2 x1) in (let H16 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 
-x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind b) x0 x1)) (\lambda (H17: 
-(eq T t2 x1)).(let H18 \def (eq_ind_r T x1 (\lambda (t0: T).(pr3 (CHead c 
-(Bind b) t1) t2 t0)) H12 t2 H17) in (eq_ind T t2 (\lambda (t0: T).(sn3 c 
-(THead (Bind b) x0 t0))) (H2 x0 H15 H11 t2 (sn3_cpr3_trans c t1 x0 H11 (Bind 
-b) t2 (sn3_sing (CHead c (Bind b) t1) t2 H4))) x1 H17))) (\lambda (H17: (((eq 
-T t2 x1) \to (\forall (P: Prop).P)))).(H2 x0 H15 H11 x1 (sn3_cpr3_trans c t1 
-x0 H11 (Bind b) x1 (H4 x1 H17 H12)))) H16)))) H14))) t3 H10))))))) H9)) 
-(\lambda (H9: (pr3 (CHead c (Bind b) t1) t2 (lift (S O) O t3))).(sn3_gen_lift 
-(CHead c (Bind b) t1) t3 (S O) O (sn3_pr3_trans (CHead c (Bind b) t1) t2 
-(sn3_pr2_intro (CHead c (Bind b) t1) t2 (\lambda (t0: T).(\lambda (H10: (((eq 
-T t2 t0) \to (\forall (P: Prop).P)))).(\lambda (H11: (pr2 (CHead c (Bind b) 
-t1) t2 t0)).(H4 t0 H10 (pr3_pr2 (CHead c (Bind b) t1) t2 t0 H11)))))) (lift 
-(S O) O t3) H9) c (drop_drop (Bind b) O c c (drop_refl c) t1))) H8)))))))))) 
-t H3)))))) u H0))))).
-
-theorem sn3_beta:
- \forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c (THead (Bind Abbr) v 
-t)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead 
-(Bind Abst) w t))))))))
-\def
- \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead 
-(Bind Abbr) v t))).(insert_eq T (THead (Bind Abbr) v t) (\lambda (t0: T).(sn3 
-c t0)) (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead 
-(Bind Abst) w t))))) (\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t 
-(\lambda (t0: T).((eq T y (THead (Bind Abbr) v t0)) \to (\forall (w: T).((sn3 
-c w) \to (sn3 c (THead (Flat Appl) v (THead (Bind Abst) w t0))))))) (unintro 
-T v (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind Abbr) t0 x)) \to 
-(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) t0 (THead (Bind 
-Abst) w x)))))))) (sn3_ind c (\lambda (t0: T).(\forall (x: T).(\forall (x0: 
-T).((eq T t0 (THead (Bind Abbr) x x0)) \to (\forall (w: T).((sn3 c w) \to 
-(sn3 c (THead (Flat Appl) x (THead (Bind Abst) w x0))))))))) (\lambda (t1: 
-T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall 
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
-(\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Bind Abbr) x x0)) \to 
-(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) 
-w x0))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1 
-(THead (Bind Abbr) x x0))).(\lambda (w: T).(\lambda (H4: (sn3 c w)).(let H5 
-\def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to 
-(\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (\forall (x1: T).(\forall (x2: 
-T).((eq T t2 (THead (Bind Abbr) x1 x2)) \to (\forall (w0: T).((sn3 c w0) \to 
-(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) w0 x2)))))))))))) H2 (THead 
-(Bind Abbr) x x0) H3) in (let H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall 
-(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to 
-(sn3 c t2))))) H1 (THead (Bind Abbr) x x0) H3) in (sn3_ind c (\lambda (t0: 
-T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t0 x0)))) (\lambda (t2: 
-T).(\lambda (H7: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: 
-Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H8: ((\forall 
-(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to 
-(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t3 x0)))))))).(sn3_pr2_intro c 
-(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (\lambda (t3: T).(\lambda 
-(H9: (((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t3) \to (\forall 
-(P: Prop).P)))).(\lambda (H10: (pr2 c (THead (Flat Appl) x (THead (Bind Abst) 
-t2 x0)) t3)).(let H11 \def (pr2_gen_appl c x (THead (Bind Abst) t2 x0) t3 
-H10) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) t2 x0) t4)))) 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (THead (Bind Abst) t2 x0) (THead (Bind Abst) y1 z1)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
-(Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind 
-b) u) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c t3) 
-(\lambda (H12: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) t2 x0) 
-t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) t2 x0) t4))) (sn3 
-c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H13: (eq T t3 (THead (Flat 
-Appl) x1 x2))).(\lambda (H14: (pr2 c x x1)).(\lambda (H15: (pr2 c (THead 
-(Bind Abst) t2 x0) x2)).(let H16 \def (eq_ind T t3 (\lambda (t0: T).((eq T 
-(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P: 
-Prop).P))) H9 (THead (Flat Appl) x1 x2) H13) in (eq_ind_r T (THead (Flat 
-Appl) x1 x2) (\lambda (t0: T).(sn3 c t0)) (let H17 \def (pr2_gen_abst c t2 x0 
-x2 H15) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead 
-(Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t2 u2))) 
-(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead 
-c (Bind b) u) x0 t4))))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3: 
-T).(\lambda (x4: T).(\lambda (H18: (eq T x2 (THead (Bind Abst) x3 
-x4))).(\lambda (H19: (pr2 c t2 x3)).(\lambda (H20: ((\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) x0 x4))))).(let H21 \def (eq_ind T x2 
-(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) 
-(THead (Flat Appl) x1 t0)) \to (\forall (P: Prop).P))) H16 (THead (Bind Abst) 
-x3 x4) H18) in (eq_ind_r T (THead (Bind Abst) x3 x4) (\lambda (t0: T).(sn3 c 
-(THead (Flat Appl) x1 t0))) (let H_x \def (term_dec t2 x3) in (let H22 \def 
-H_x in (or_ind (eq T t2 x3) ((eq T t2 x3) \to (\forall (P: Prop).P)) (sn3 c 
-(THead (Flat Appl) x1 (THead (Bind Abst) x3 x4))) (\lambda (H23: (eq T t2 
-x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: T).((eq T (THead (Flat Appl) 
-x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) x1 (THead (Bind Abst) t0 
-x4))) \to (\forall (P: Prop).P))) H21 t2 H23) in (let H25 \def (eq_ind_r T x3 
-(\lambda (t0: T).(pr2 c t2 t0)) H19 t2 H23) in (eq_ind T t2 (\lambda (t0: 
-T).(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t0 x4)))) (let H_x0 \def 
-(term_dec x x1) in (let H26 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to 
-(\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t2 
-x4))) (\lambda (H27: (eq T x x1)).(let H28 \def (eq_ind_r T x1 (\lambda (t0: 
-T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) 
-t0 (THead (Bind Abst) t2 x4))) \to (\forall (P: Prop).P))) H24 x H27) in (let 
-H29 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H27) in (eq_ind 
-T x (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) t2 
-x4)))) (let H_x1 \def (term_dec x0 x4) in (let H30 \def H_x1 in (or_ind (eq T 
-x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x 
-(THead (Bind Abst) t2 x4))) (\lambda (H31: (eq T x0 x4)).(let H32 \def 
-(eq_ind_r T x4 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind 
-Abst) t2 x0)) (THead (Flat Appl) x (THead (Bind Abst) t2 t0))) \to (\forall 
-(P: Prop).P))) H28 x0 H31) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: 
-T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 
-H31) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead 
-(Bind Abst) t2 t0)))) (H32 (refl_equal T (THead (Flat Appl) x (THead (Bind 
-Abst) t2 x0))) (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)))) x4 
-H31)))) (\lambda (H31: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead 
-(Bind Abbr) x x4) (\lambda (H32: (eq T (THead (Bind Abbr) x x0) (THead (Bind 
-Abbr) x x4))).(\lambda (P: Prop).(let H33 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind 
-Abbr) x x0) (THead (Bind Abbr) x x4) H32) in (let H34 \def (eq_ind_r T x4 
-(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H31 x0 H33) in 
-(let H35 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H33) in (H34 (refl_equal T x0) 
-P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) 
-(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead 
-(Bind Abbr) x x4)) t2 (sn3_sing c t2 H7))) H30))) x1 H27)))) (\lambda (H27: 
-(((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind Abbr) x1 x4) 
-(\lambda (H28: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x1 
-x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) 
-\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
-(THead (Bind Abbr) x1 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind 
-Abbr) x x0) (THead (Bind Abbr) x1 x4) H28) in (\lambda (H31: (eq T x 
-x1)).(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (let H33 \def 
-(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) 
-H27 x H31) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 
-x H31) in (H33 (refl_equal T x) P)))))) H29)))) (pr3_head_12 c x x1 (pr3_pr2 
-c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20 
-Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) t2 (sn3_sing c t2 
-H7))) H26))) x3 H23)))) (\lambda (H23: (((eq T t2 x3) \to (\forall (P: 
-Prop).P)))).(let H_x0 \def (term_dec x x1) in (let H24 \def H_x0 in (or_ind 
-(eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) 
-x1 (THead (Bind Abst) x3 x4))) (\lambda (H25: (eq T x x1)).(let H26 \def 
-(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H25) in (eq_ind T x 
-(\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4)))) 
-(let H_x1 \def (term_dec x0 x4) in (let H27 \def H_x1 in (or_ind (eq T x0 x4) 
-((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x (THead 
-(Bind Abst) x3 x4))) (\lambda (H28: (eq T x0 x4)).(let H29 \def (eq_ind_r T 
-x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-x0 t0)))) H20 x0 H28) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat 
-Appl) x (THead (Bind Abst) x3 t0)))) (H8 x3 H23 (pr3_pr2 c t2 x3 H19)) x4 
-H28))) (\lambda (H28: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead 
-(Bind Abbr) x x4) (\lambda (H29: (eq T (THead (Bind Abbr) x x0) (THead (Bind 
-Abbr) x x4))).(\lambda (P: Prop).(let H30 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind 
-Abbr) x x0) (THead (Bind Abbr) x x4) H29) in (let H31 \def (eq_ind_r T x4 
-(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H28 x0 H30) in 
-(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (H31 (refl_equal T x0) 
-P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) 
-(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead 
-(Bind Abbr) x x4)) x3 (H7 x3 H23 (pr3_pr2 c t2 x3 H19)))) H27))) x1 H25))) 
-(\lambda (H25: (((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind 
-Abbr) x1 x4) (\lambda (H26: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
-x1 x4))).(\lambda (P: Prop).(let H27 \def (f_equal T T (\lambda (e: T).(match 
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) 
-\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
-(THead (Bind Abbr) x1 x4) H26) in ((let H28 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind 
-Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in (\lambda (H29: (eq T x 
-x1)).(let H30 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (let H31 \def 
-(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) 
-H25 x H29) in (let H32 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 
-x H29) in (H31 (refl_equal T x) P)))))) H27)))) (pr3_head_12 c x x1 (pr3_pr2 
-c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20 
-Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) x3 (H7 x3 H23 
-(pr3_pr2 c t2 x3 H19)))) H24)))) H22))) x2 H18))))))) H17)) t3 H13))))))) 
-H12)) (\lambda (H12: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T 
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind Abst) t2 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
-Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-z1 t4))))))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (x4: T).(\lambda (H13: (eq T (THead (Bind Abst) t2 x0) (THead 
-(Bind Abst) x1 x2))).(\lambda (H14: (eq T t3 (THead (Bind Abbr) x3 
-x4))).(\lambda (H15: (pr2 c x x3)).(\lambda (H16: ((\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H17 \def (eq_ind T t3 
-(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) 
-\to (\forall (P: Prop).P))) H9 (THead (Bind Abbr) x3 x4) H14) in (eq_ind_r T 
-(THead (Bind Abbr) x3 x4) (\lambda (t0: T).(sn3 c t0)) (let H18 \def (f_equal 
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) \Rightarrow t0])) 
-(THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in ((let H19 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) 
-\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in 
-(\lambda (_: (eq T t2 x1)).(let H21 \def (eq_ind_r T x2 (\lambda (t0: 
-T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t0 x4)))) H16 x0 
-H19) in (let H_x \def (term_dec x x3) in (let H22 \def H_x in (or_ind (eq T x 
-x3) ((eq T x x3) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abbr) x3 x4)) 
-(\lambda (H23: (eq T x x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: 
-T).(pr2 c x t0)) H15 x H23) in (eq_ind T x (\lambda (t0: T).(sn3 c (THead 
-(Bind Abbr) t0 x4))) (let H_x0 \def (term_dec x0 x4) in (let H25 \def H_x0 in 
-(or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead 
-(Bind Abbr) x x4)) (\lambda (H26: (eq T x0 x4)).(let H27 \def (eq_ind_r T x4 
-(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 
-t0)))) H21 x0 H26) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Bind Abbr) 
-x t0))) (sn3_sing c (THead (Bind Abbr) x x0) H6) x4 H26))) (\lambda (H26: 
-(((eq T x0 x4) \to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x x4) 
-(\lambda (H27: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x 
-x4))).(\lambda (P: Prop).(let H28 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) 
-\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
-(THead (Bind Abbr) x x4) H27) in (let H29 \def (eq_ind_r T x4 (\lambda (t0: 
-T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H26 x0 H28) in (let H30 \def 
-(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c 
-(Bind b) u) x0 t0)))) H21 x0 H28) in (H29 (refl_equal T x0) P)))))) (pr3_pr2 
-c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4 
-(Bind Abbr) (H21 Abbr x))))) H25))) x3 H23))) (\lambda (H23: (((eq T x x3) 
-\to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x3 x4) (\lambda (H24: (eq 
-T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x3 x4))).(\lambda (P: 
-Prop).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | 
-(THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
-x3 x4) H24) in ((let H26 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) 
-\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
-(THead (Bind Abbr) x3 x4) H24) in (\lambda (H27: (eq T x x3)).(let H28 \def 
-(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c 
-(Bind b) u) x0 t0)))) H21 x0 H26) in (let H29 \def (eq_ind_r T x3 (\lambda 
-(t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H23 x H27) in (let H30 
-\def (eq_ind_r T x3 (\lambda (t0: T).(pr2 c x t0)) H15 x H27) in (H29 
-(refl_equal T x) P)))))) H25)))) (pr3_head_12 c x x3 (pr3_pr2 c x x3 H15) 
-(Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x3) x0 x4 (H21 Abbr x3))))) 
-H22)))))) H18)) t3 H14)))))))))) H12)) (\lambda (H12: (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) 
-y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3) 
-(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda 
-(x5: T).(\lambda (x6: T).(\lambda (H13: (not (eq B x1 Abst))).(\lambda (H14: 
-(eq T (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3))).(\lambda (H15: (eq 
-T t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda 
-(_: (pr2 c x x5)).(\lambda (H17: (pr2 c x2 x6)).(\lambda (H18: (pr2 (CHead c 
-(Bind x1) x6) x3 x4)).(let H19 \def (eq_ind T t3 (\lambda (t0: T).((eq T 
-(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P: 
-Prop).P))) H9 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) 
-H15) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) 
-x4)) (\lambda (t0: T).(sn3 c t0)) (let H20 \def (f_equal T B (\lambda (e: 
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | 
-(TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return 
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
-Abst])])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in ((let H21 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in 
-((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ 
-t0) \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) 
-in (\lambda (H23: (eq T t2 x2)).(\lambda (H24: (eq B Abst x1)).(let H25 \def 
-(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H18 x0 
-H22) in (let H26 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H17 t2 
-H23) in (let H27 \def (eq_ind_r B x1 (\lambda (b: B).(pr2 (CHead c (Bind b) 
-x6) x0 x4)) H25 Abst H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b: 
-B).(not (eq B b Abst))) H13 Abst H24) in (eq_ind B Abst (\lambda (b: B).(sn3 
-c (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (let H29 
-\def (match (H28 (refl_equal B Abst)) in False return (\lambda (_: 
-False).(sn3 c (THead (Bind Abst) x6 (THead (Flat Appl) (lift (S O) O x5) 
-x4)))) with []) in H29) x1 H24)))))))) H21)) H20)) t3 H15)))))))))))))) H12)) 
-H11))))))))) w H4))))))))))) y H0))))) H)))).
-
-theorem nf3_appl_abbr:
- \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
-(CHead d (Bind Abbr) w)) \to (\forall (v: T).((sn3 c (THead (Flat Appl) v 
-(lift (S i) O w))) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda 
-(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (v: T).(\lambda (H0: (sn3 c 
-(THead (Flat Appl) v (lift (S i) O w)))).(insert_eq T (THead (Flat Appl) v 
-(lift (S i) O w)) (\lambda (t: T).(sn3 c t)) (sn3 c (THead (Flat Appl) v 
-(TLRef i))) (\lambda (y: T).(\lambda (H1: (sn3 c y)).(unintro T v (\lambda 
-(t: T).((eq T y (THead (Flat Appl) t (lift (S i) O w))) \to (sn3 c (THead 
-(Flat Appl) t (TLRef i))))) (sn3_ind c (\lambda (t: T).(\forall (x: T).((eq T 
-t (THead (Flat Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x 
-(TLRef i)))))) (\lambda (t1: T).(\lambda (H2: ((\forall (t2: T).((((eq T t1 
-t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c 
-t2)))))).(\lambda (H3: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).((eq T t2 (THead (Flat 
-Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x (TLRef 
-i)))))))))).(\lambda (x: T).(\lambda (H4: (eq T t1 (THead (Flat Appl) x (lift 
-(S i) O w)))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: 
-T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall 
-(x0: T).((eq T t2 (THead (Flat Appl) x0 (lift (S i) O w))) \to (sn3 c (THead 
-(Flat Appl) x0 (TLRef i))))))))) H3 (THead (Flat Appl) x (lift (S i) O w)) 
-H4) in (let H6 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: T).((((eq T t 
-t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (sn3 c t2))))) H2 
-(THead (Flat Appl) x (lift (S i) O w)) H4) in (sn3_pr2_intro c (THead (Flat 
-Appl) x (TLRef i)) (\lambda (t2: T).(\lambda (H7: (((eq T (THead (Flat Appl) 
-x (TLRef i)) t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr2 c (THead 
-(Flat Appl) x (TLRef i)) t2)).(let H9 \def (pr2_gen_appl c x (TLRef i) t2 H8) 
-in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x 
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq 
-T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) 
-(sn3 c t2) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T 
-T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H12: (pr2 c 
-x x0)).(\lambda (H13: (pr2 c (TLRef i) x1)).(let H14 \def (eq_ind T t2 
-(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: 
-Prop).P))) H7 (THead (Flat Appl) x0 x1) H11) in (eq_ind_r T (THead (Flat 
-Appl) x0 x1) (\lambda (t: T).(sn3 c t)) (let H15 \def (pr2_gen_lref c x1 i 
-H13) in (or_ind (eq T x1 (TLRef i)) (ex2_2 C T (\lambda (d0: C).(\lambda (u: 
-T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq 
-T x1 (lift (S i) O u))))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda (H16: 
-(eq T x1 (TLRef i))).(let H17 \def (eq_ind T x1 (\lambda (t: T).((eq T (THead 
-(Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P: 
-Prop).P))) H14 (TLRef i) H16) in (eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c 
-(THead (Flat Appl) x0 t))) (let H_x \def (term_dec x x0) in (let H18 \def H_x 
-in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c (THead 
-(Flat Appl) x0 (TLRef i))) (\lambda (H19: (eq T x x0)).(let H20 \def 
-(eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead 
-(Flat Appl) t (TLRef i))) \to (\forall (P: Prop).P))) H17 x H19) in (let H21 
-\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H19) in (eq_ind T x 
-(\lambda (t: T).(sn3 c (THead (Flat Appl) t (TLRef i)))) (H20 (refl_equal T 
-(THead (Flat Appl) x (TLRef i))) (sn3 c (THead (Flat Appl) x (TLRef i)))) x0 
-H19)))) (\lambda (H19: (((eq T x x0) \to (\forall (P: Prop).P)))).(H5 (THead 
-(Flat Appl) x0 (lift (S i) O w)) (\lambda (H20: (eq T (THead (Flat Appl) x 
-(lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)))).(\lambda (P: 
-Prop).(let H21 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | 
-(THead _ t _) \Rightarrow t])) (THead (Flat Appl) x (lift (S i) O w)) (THead 
-(Flat Appl) x0 (lift (S i) O w)) H20) in (let H22 \def (eq_ind_r T x0 
-(\lambda (t: T).((eq T x t) \to (\forall (P0: Prop).P0))) H19 x H21) in (let 
-H23 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H21) in (H22 
-(refl_equal T x) P)))))) (pr3_pr2 c (THead (Flat Appl) x (lift (S i) O w)) 
-(THead (Flat Appl) x0 (lift (S i) O w)) (pr2_head_1 c x x0 H12 (Flat Appl) 
-(lift (S i) O w))) x0 (refl_equal T (THead (Flat Appl) x0 (lift (S i) O 
-w))))) H18))) x1 H16))) (\lambda (H16: (ex2_2 C T (\lambda (d0: C).(\lambda 
-(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
-T).(eq T x1 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda 
-(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
-T).(eq T x1 (lift (S i) O u)))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda 
-(x2: C).(\lambda (x3: T).(\lambda (H17: (getl i c (CHead x2 (Bind Abbr) 
-x3))).(\lambda (H18: (eq T x1 (lift (S i) O x3))).(let H19 \def (eq_ind T x1 
-(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 
-t)) \to (\forall (P: Prop).P))) H14 (lift (S i) O x3) H18) in (eq_ind_r T 
-(lift (S i) O x3) (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H20 
-\def (eq_ind C (CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H 
-(CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 
-(Bind Abbr) x3) H17)) in (let H21 \def (f_equal C C (\lambda (e: C).(match e 
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) 
-\Rightarrow c0])) (CHead d (Bind Abbr) w) (CHead x2 (Bind Abbr) x3) 
-(getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 (Bind Abbr) x3) H17)) in 
-((let H22 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: 
-C).T) with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d 
-(Bind Abbr) w) (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) 
-i H (CHead x2 (Bind Abbr) x3) H17)) in (\lambda (H23: (eq C d x2)).(let H24 
-\def (eq_ind_r T x3 (\lambda (t: T).(getl i c (CHead x2 (Bind Abbr) t))) H20 
-w H22) in (eq_ind T w (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 (lift (S 
-i) O t)))) (let H25 \def (eq_ind_r C x2 (\lambda (c0: C).(getl i c (CHead c0 
-(Bind Abbr) w))) H24 d H23) in (let H_x \def (term_dec x x0) in (let H26 \def 
-H_x in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c 
-(THead (Flat Appl) x0 (lift (S i) O w))) (\lambda (H27: (eq T x x0)).(let H28 
-\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H27) in (eq_ind T x 
-(\lambda (t: T).(sn3 c (THead (Flat Appl) t (lift (S i) O w)))) (sn3_sing c 
-(THead (Flat Appl) x (lift (S i) O w)) H6) x0 H27))) (\lambda (H27: (((eq T x 
-x0) \to (\forall (P: Prop).P)))).(H6 (THead (Flat Appl) x0 (lift (S i) O w)) 
-(\lambda (H28: (eq T (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat 
-Appl) x0 (lift (S i) O w)))).(\lambda (P: Prop).(let H29 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t])) 
-(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O 
-w)) H28) in (let H30 \def (eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to 
-(\forall (P0: Prop).P0))) H27 x H29) in (let H31 \def (eq_ind_r T x0 (\lambda 
-(t: T).(pr2 c x t)) H12 x H29) in (H30 (refl_equal T x) P)))))) (pr3_pr2 c 
-(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O 
-w)) (pr2_head_1 c x x0 H12 (Flat Appl) (lift (S i) O w))))) H26)))) x3 
-H22)))) H21))) x1 H18)))))) H16)) H15)) t2 H11))))))) H10)) (\lambda (H10: 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 
-t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind 
-b) u) z1 t3))))))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: 
-T).(\lambda (x3: T).(\lambda (H11: (eq T (TLRef i) (THead (Bind Abst) x0 
-x1))).(\lambda (H12: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c 
-x x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) 
-u) x1 x3))))).(let H15 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat 
-Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 (THead (Bind Abbr) x2 
-x3) H12) in (eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) 
-(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 
-x1) H11) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H16)) t2 
-H12)))))))))) H10)) (\lambda (H10: (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind 
-B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq 
-T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) 
-(sn3 c t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 
-Abst))).(\lambda (H12: (eq T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda 
-(H13: (eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) 
-x3)))).(\lambda (_: (pr2 c x x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: 
-(pr2 (CHead c (Bind x0) x5) x2 x3)).(let H17 \def (eq_ind T t2 (\lambda (t: 
-T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 
-(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) H13) in 
-(eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) 
-(\lambda (t: T).(sn3 c t)) (let H18 \def (eq_ind T (TLRef i) (\lambda (ee: 
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
-(THead (Bind x0) x1 x2) H12) in (False_ind (sn3 c (THead (Bind x0) x5 (THead 
-(Flat Appl) (lift (S O) O x4) x3))) H18)) t2 H13)))))))))))))) H10)) 
-H9))))))))))))) y H1)))) H0))))))).
-
-theorem sn3_appl_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: 
-T).((sn3 c u) \to (\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) u) 
-(THead (Flat Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v 
-(THead (Bind b) u t))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda 
-(u: T).(\lambda (H0: (sn3 c u)).(sn3_ind c (\lambda (t: T).(\forall (t0: 
-T).(\forall (v: T).((sn3 (CHead c (Bind b) t) (THead (Flat Appl) (lift (S O) 
-O v) t0)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t t0))))))) 
-(\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
-(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall 
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
-(\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) t2) (THead (Flat 
-Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t2 
-t))))))))))).(\lambda (t: T).(\lambda (v: T).(\lambda (H3: (sn3 (CHead c 
-(Bind b) t1) (THead (Flat Appl) (lift (S O) O v) t))).(insert_eq T (THead 
-(Flat Appl) (lift (S O) O v) t) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) 
-t0)) (sn3 c (THead (Flat Appl) v (THead (Bind b) t1 t))) (\lambda (y: 
-T).(\lambda (H4: (sn3 (CHead c (Bind b) t1) y)).(unintro T t (\lambda (t0: 
-T).((eq T y (THead (Flat Appl) (lift (S O) O v) t0)) \to (sn3 c (THead (Flat 
-Appl) v (THead (Bind b) t1 t0))))) (unintro T v (\lambda (t0: T).(\forall (x: 
-T).((eq T y (THead (Flat Appl) (lift (S O) O t0) x)) \to (sn3 c (THead (Flat 
-Appl) t0 (THead (Bind b) t1 x)))))) (sn3_ind (CHead c (Bind b) t1) (\lambda 
-(t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat Appl) (lift 
-(S O) O x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) t1 x0))))))) 
-(\lambda (t2: T).(\lambda (H5: ((\forall (t3: T).((((eq T t2 t3) \to (\forall 
-(P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind 
-b) t1) t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t2 t3) \to (\forall 
-(P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (\forall (x: 
-T).(\forall (x0: T).((eq T t3 (THead (Flat Appl) (lift (S O) O x) x0)) \to 
-(sn3 c (THead (Flat Appl) x (THead (Bind b) t1 x0))))))))))).(\lambda (x: 
-T).(\lambda (x0: T).(\lambda (H7: (eq T t2 (THead (Flat Appl) (lift (S O) O 
-x) x0))).(let H8 \def (eq_ind T t2 (\lambda (t0: T).(\forall (t3: T).((((eq T 
-t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t0 t3) \to 
-(\forall (x1: T).(\forall (x2: T).((eq T t3 (THead (Flat Appl) (lift (S O) O 
-x1) x2)) \to (sn3 c (THead (Flat Appl) x1 (THead (Bind b) t1 x2)))))))))) H6 
-(THead (Flat Appl) (lift (S O) O x) x0) H7) in (let H9 \def (eq_ind T t2 
-(\lambda (t0: T).(\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) 
-\to ((pr3 (CHead c (Bind b) t1) t0 t3) \to (sn3 (CHead c (Bind b) t1) t3))))) 
-H5 (THead (Flat Appl) (lift (S O) O x) x0) H7) in (sn3_pr2_intro c (THead 
-(Flat Appl) x (THead (Bind b) t1 x0)) (\lambda (t3: T).(\lambda (H10: (((eq T 
-(THead (Flat Appl) x (THead (Bind b) t1 x0)) t3) \to (\forall (P: 
-Prop).P)))).(\lambda (H11: (pr2 c (THead (Flat Appl) x (THead (Bind b) t1 
-x0)) t3)).(let H12 \def (pr2_gen_appl c x (THead (Bind b) t1 x0) t3 H11) in 
-(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat 
-Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4)))) (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
-Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) 
-u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead 
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
-b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2)))))))) (sn3 c t3) 
-(\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) 
-t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))) (sn3 c 
-t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H14: (eq T t3 (THead (Flat 
-Appl) x1 x2))).(\lambda (H15: (pr2 c x x1)).(\lambda (H16: (pr2 c (THead 
-(Bind b) t1 x0) x2)).(let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T 
-(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) 
-H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat Appl) x1 x2) 
-(\lambda (t0: T).(sn3 c t0)) (let H_x \def (pr3_gen_bind b H c t1 x0 x2) in 
-(let H18 \def (H_x (pr3_pr2 c (THead (Bind b) t1 x0) x2 H16)) in (or_ind 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4)))) (pr3 (CHead c (Bind 
-b) t1) x0 (lift (S O) O x2)) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (H19: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 t4)))) (\lambda 
-(u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 
-(CHead c (Bind b) t1) x0 t4))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda 
-(x3: T).(\lambda (x4: T).(\lambda (H20: (eq T x2 (THead (Bind b) x3 
-x4))).(\lambda (H21: (pr3 c t1 x3)).(\lambda (H22: (pr3 (CHead c (Bind b) t1) 
-x0 x4)).(let H23 \def (eq_ind T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) 
-x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 t0)) \to (\forall (P: 
-Prop).P))) H17 (THead (Bind b) x3 x4) H20) in (eq_ind_r T (THead (Bind b) x3 
-x4) (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 t0))) (let H_x0 \def 
-(term_dec t1 x3) in (let H24 \def H_x0 in (or_ind (eq T t1 x3) ((eq T t1 x3) 
-\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind b) x3 
-x4))) (\lambda (H25: (eq T t1 x3)).(let H26 \def (eq_ind_r T x3 (\lambda (t0: 
-T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 
-(THead (Bind b) t0 x4))) \to (\forall (P: Prop).P))) H23 t1 H25) in (let H27 
-\def (eq_ind_r T x3 (\lambda (t0: T).(pr3 c t1 t0)) H21 t1 H25) in (eq_ind T 
-t1 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 (THead (Bind b) t0 x4)))) 
-(let H_x1 \def (term_dec x0 x4) in (let H28 \def H_x1 in (or_ind (eq T x0 x4) 
-((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead 
-(Bind b) t1 x4))) (\lambda (H29: (eq T x0 x4)).(let H30 \def (eq_ind_r T x4 
-(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead 
-(Flat Appl) x1 (THead (Bind b) t1 t0))) \to (\forall (P: Prop).P))) H26 x0 
-H29) in (let H31 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) 
-t1) x0 t0)) H22 x0 H29) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat 
-Appl) x1 (THead (Bind b) t1 t0)))) (let H_x2 \def (term_dec x x1) in (let H32 
-\def H_x2 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 
-c (THead (Flat Appl) x1 (THead (Bind b) t1 x0))) (\lambda (H33: (eq T x 
-x1)).(let H34 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) 
-x (THead (Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to 
-(\forall (P: Prop).P))) H30 x H33) in (let H35 \def (eq_ind_r T x1 (\lambda 
-(t0: T).(pr2 c x t0)) H15 x H33) in (eq_ind T x (\lambda (t0: T).(sn3 c 
-(THead (Flat Appl) t0 (THead (Bind b) t1 x0)))) (H34 (refl_equal T (THead 
-(Flat Appl) x (THead (Bind b) t1 x0))) (sn3 c (THead (Flat Appl) x (THead 
-(Bind b) t1 x0)))) x1 H33)))) (\lambda (H33: (((eq T x x1) \to (\forall (P: 
-Prop).P)))).(H8 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H34: (eq T 
-(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
-x0))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map 
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
-lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
-(lift (S O) O x1) x0) H34) in (let H36 \def (eq_ind_r T x1 (\lambda (t0: 
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H33 x (lift_inj x x1 (S O) O 
-H35)) in (let H37 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x 
-(lift_inj x x1 (S O) O H35)) in (H36 (refl_equal T x) P)))))) (pr3_flat 
-(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c 
-(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 
-(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl) x1 x0 
-(refl_equal T (THead (Flat Appl) (lift (S O) O x1) x0)))) H32))) x4 H29)))) 
-(\lambda (H29: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H8 (THead (Flat 
-Appl) (lift (S O) O x1) x4) (\lambda (H30: (eq T (THead (Flat Appl) (lift (S 
-O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P: 
-Prop).(let H31 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat 
-\to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
-lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
-(lift (S O) O x1) x4) H30) in ((let H32 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat 
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H30) in 
-(\lambda (H33: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H34 \def 
-(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) 
-H29 x0 H32) in (let H35 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T (THead 
-(Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 (THead (Bind b) 
-t1 t0))) \to (\forall (P0: Prop).P0))) H26 x0 H32) in (let H36 \def (eq_ind_r 
-T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) H22 x0 H32) in (let 
-H37 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead 
-(Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to (\forall 
-(P0: Prop).P0))) H35 x (lift_inj x x1 (S O) O H33)) in (let H38 \def 
-(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O 
-H33)) in (H34 (refl_equal T x0) P)))))))) H31)))) (pr3_flat (CHead c (Bind b) 
-t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S 
-O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) 
-x0 x4 H22 Appl) x1 x4 (refl_equal T (THead (Flat Appl) (lift (S O) O x1) 
-x4)))) H28))) x3 H25)))) (\lambda (H25: (((eq T t1 x3) \to (\forall (P: 
-Prop).P)))).(H2 x3 H25 H21 x4 x1 (sn3_cpr3_trans c t1 x3 H21 (Bind b) (THead 
-(Flat Appl) (lift (S O) O x1) x4) (let H_x1 \def (term_dec x0 x4) in (let H26 
-\def H_x1 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) 
-(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x4)) (\lambda 
-(H27: (eq T x0 x4)).(let H28 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead 
-c (Bind b) t1) x0 t0)) H22 x0 H27) in (eq_ind T x0 (\lambda (t0: T).(sn3 
-(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) t0))) (let H_x2 
-\def (term_dec x x1) in (let H29 \def H_x2 in (or_ind (eq T x x1) ((eq T x 
-x1) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) 
-(lift (S O) O x1) x0)) (\lambda (H30: (eq T x x1)).(let H31 \def (eq_ind_r T 
-x1 (\lambda (t0: T).(pr2 c x t0)) H15 x H30) in (eq_ind T x (\lambda (t0: 
-T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0))) 
-(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9) 
-x1 H30))) (\lambda (H30: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9 
-(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H31: (eq T (THead (Flat 
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
-x0))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map 
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
-lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
-(lift (S O) O x1) x0) H31) in (let H33 \def (eq_ind_r T x1 (\lambda (t0: 
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H30 x (lift_inj x x1 (S O) O 
-H32)) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x 
-(lift_inj x x1 (S O) O H32)) in (H33 (refl_equal T x) P)))))) (pr3_flat 
-(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c 
-(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 
-(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl))) 
-H29))) x4 H27))) (\lambda (H27: (((eq T x0 x4) \to (\forall (P: 
-Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x1) x4) (\lambda (H28: (eq T 
-(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
-x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map 
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
-lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
-(lift (S O) O x1) x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat 
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H28) in 
-(\lambda (H31: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H32 \def 
-(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) 
-H27 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c 
-(Bind b) t1) x0 t0)) H22 x0 H30) in (let H34 \def (eq_ind_r T x1 (\lambda 
-(t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H31)) in (H32 (refl_equal 
-T x0) P)))))) H29)))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift 
-(S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c 
-c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x4 H22 Appl))) H26)))))) 
-H24))) x2 H20))))))) H19)) (\lambda (H19: (pr3 (CHead c (Bind b) t1) x0 (lift 
-(S O) O x2))).(sn3_gen_lift (CHead c (Bind b) t1) (THead (Flat Appl) x1 x2) 
-(S O) O (eq_ind_r T (THead (Flat Appl) (lift (S O) O x1) (lift (S O) (s (Flat 
-Appl) O) x2)) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) t0)) (sn3_pr3_trans 
-(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x0) (let H_x0 \def 
-(term_dec x x1) in (let H20 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to 
-(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S 
-O) O x1) x0)) (\lambda (H21: (eq T x x1)).(let H22 \def (eq_ind_r T x1 
-(\lambda (t0: T).(pr2 c x t0)) H15 x H21) in (eq_ind T x (\lambda (t0: 
-T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0))) 
-(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9) 
-x1 H21))) (\lambda (H21: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9 
-(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H22: (eq T (THead (Flat 
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
-x0))).(\lambda (P: Prop).(let H23 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map 
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
-lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t4))]) in lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
-(lift (S O) O x1) x0) H22) in (let H24 \def (eq_ind_r T x1 (\lambda (t0: 
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H21 x (lift_inj x x1 (S O) O 
-H23)) in (let H25 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x 
-(lift_inj x x1 (S O) O H23)) in (H24 (refl_equal T x) P)))))) (pr3_flat 
-(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c 
-(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 
-(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl))) 
-H20))) (THead (Flat Appl) (lift (S O) O x1) (lift (S O) O x2)) (pr3_thin_dx 
-(CHead c (Bind b) t1) x0 (lift (S O) O x2) H19 (lift (S O) O x1) Appl)) (lift 
-(S O) O (THead (Flat Appl) x1 x2)) (lift_head (Flat Appl) x1 x2 (S O) O)) c 
-(drop_drop (Bind b) O c c (drop_refl c) t1))) H18))) t3 H14))))))) H13)) 
-(\lambda (H13: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: 
-T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: 
-T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))))).(ex4_4_ind T T T T (\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
-b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x 
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: 
-T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))) 
-(sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (H14: (eq T (THead (Bind b) t1 x0) (THead (Bind Abst) x1 
-x2))).(\lambda (H15: (eq T t3 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c 
-x x3)).(\lambda (H17: ((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind 
-b0) u0) x2 x4))))).(let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T (THead 
-(Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) H10 
-(THead (Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4) 
-(\lambda (t0: T).(sn3 c t0)) (let H19 \def (f_equal T B (\lambda (e: 
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | 
-(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return 
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
-b])])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in ((let H20 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in 
-((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ 
-t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) 
-in (\lambda (_: (eq T t1 x1)).(\lambda (H23: (eq B b Abst)).(let H24 \def 
-(eq_ind_r T x2 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead 
-c (Bind b0) u0) t0 x4)))) H17 x0 H21) in (let H25 \def (eq_ind B b (\lambda 
-(b0: B).((eq T (THead (Flat Appl) x (THead (Bind b0) t1 x0)) (THead (Bind 
-Abbr) x3 x4)) \to (\forall (P: Prop).P))) H18 Abst H23) in (let H26 \def 
-(eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T (THead (Flat Appl) 
-(lift (S O) O x) x0) t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind 
-b0) t1) (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (sn3 (CHead c (Bind 
-b0) t1) t4))))) H9 Abst H23) in (let H27 \def (eq_ind B b (\lambda (b0: 
-B).(\forall (t4: T).((((eq T (THead (Flat Appl) (lift (S O) O x) x0) t4) \to 
-(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) (THead (Flat Appl) 
-(lift (S O) O x) x0) t4) \to (\forall (x5: T).(\forall (x6: T).((eq T t4 
-(THead (Flat Appl) (lift (S O) O x5) x6)) \to (sn3 c (THead (Flat Appl) x5 
-(THead (Bind b0) t1 x6)))))))))) H8 Abst H23) in (let H28 \def (eq_ind B b 
-(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P))) 
-\to ((pr3 c t1 t4) \to (\forall (t0: T).(\forall (v0: T).((sn3 (CHead c (Bind 
-b0) t4) (THead (Flat Appl) (lift (S O) O v0) t0)) \to (sn3 c (THead (Flat 
-Appl) v0 (THead (Bind b0) t4 t0)))))))))) H2 Abst H23) in (let H29 \def 
-(eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H Abst H23) in (let H30 
-\def (match (H29 (refl_equal B Abst)) in False return (\lambda (_: 
-False).(sn3 c (THead (Bind Abbr) x3 x4))) with []) in H30)))))))))) H20)) 
-H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 B T T T T T (\lambda (b0: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) 
-t1 x0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-t3 (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
-y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 
-z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 
-Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind 
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b0) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b0: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2))))))) (sn3 c t3) (\lambda (x1: 
-B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: 
-T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15: (eq T 
-(THead (Bind b) t1 x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T t3 
-(THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda 
-(H17: (pr2 c x x5)).(\lambda (H18: (pr2 c x2 x6)).(\lambda (H19: (pr2 (CHead 
-c (Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t0: T).((eq T 
-(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) 
-H10 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) H16) in 
-(eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) 
-(\lambda (t0: T).(sn3 c t0)) (let H21 \def (f_equal T B (\lambda (e: 
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | 
-(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return 
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
-b])])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in ((let H22 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in 
-((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ 
-t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in 
-(\lambda (H24: (eq T t1 x2)).(\lambda (H25: (eq B b x1)).(let H26 \def 
-(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H19 x0 
-H23) in (let H27 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H18 t1 
-H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b0: B).(pr2 (CHead c (Bind b0) 
-x6) x0 x4)) H26 b H25) in (eq_ind B b (\lambda (b0: B).(sn3 c (THead (Bind 
-b0) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (sn3_pr3_trans c (THead 
-(Bind b) t1 (THead (Flat Appl) (lift (S O) O x5) x4)) (sn3_bind b H c t1 
-(sn3_sing c t1 H1) (THead (Flat Appl) (lift (S O) O x5) x4) (let H_x \def 
-(term_dec x x5) in (let H29 \def H_x in (or_ind (eq T x x5) ((eq T x x5) \to 
-(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S 
-O) O x5) x4)) (\lambda (H30: (eq T x x5)).(let H31 \def (eq_ind_r T x5 
-(\lambda (t0: T).(pr2 c x t0)) H17 x H30) in (eq_ind T x (\lambda (t0: 
-T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x4))) (let 
-H_x0 \def (term_dec x0 x4) in (let H32 \def H_x0 in (or_ind (eq T x0 x4) ((eq 
-T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat 
-Appl) (lift (S O) O x) x4)) (\lambda (H33: (eq T x0 x4)).(let H34 \def 
-(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0 
-H33) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat 
-Appl) (lift (S O) O x) t0))) (sn3_sing (CHead c (Bind b) t1) (THead (Flat 
-Appl) (lift (S O) O x) x0) H9) x4 H33))) (\lambda (H33: (((eq T x0 x4) \to 
-(\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x) x4) (\lambda 
-(H34: (eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift 
-(S O) O x) x4))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat 
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x) x4) H34) in 
-(let H36 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: 
-Prop).P0))) H33 x0 H35) in (let H37 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2 
-(CHead c (Bind b) x6) x0 t0)) H28 x0 H35) in (H36 (refl_equal T x0) P)))))) 
-(pr3_pr3_pr3_t c t1 x6 (pr3_pr2 c t1 x6 H27) (THead (Flat Appl) (lift (S O) O 
-x) x0) (THead (Flat Appl) (lift (S O) O x) x4) (Bind b) (pr3_pr2 (CHead c 
-(Bind b) x6) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift 
-(S O) O x) x4) (pr2_thin_dx (CHead c (Bind b) x6) x0 x4 H28 (lift (S O) O x) 
-Appl))))) H32))) x5 H30))) (\lambda (H30: (((eq T x x5) \to (\forall (P: 
-Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x5) x4) (\lambda (H31: (eq T 
-(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) 
-x4))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map 
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) 
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with 
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) 
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in 
-lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (TLRef _) \Rightarrow 
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match 
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef 
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | 
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
-t4))]) in lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) 
-(lift (S O) O x5) x4) H31) in ((let H33 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat 
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) x4) H31) in 
-(\lambda (H34: (eq T (lift (S O) O x) (lift (S O) O x5))).(let H35 \def 
-(eq_ind_r T x5 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) 
-H30 x (lift_inj x x5 (S O) O H34)) in (let H36 \def (eq_ind_r T x5 (\lambda 
-(t0: T).(pr2 c x t0)) H17 x (lift_inj x x5 (S O) O H34)) in (let H37 \def 
-(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0 
-H33) in (H35 (refl_equal T x) P)))))) H32)))) (pr3_pr3_pr3_t c t1 x6 (pr3_pr2 
-c t1 x6 H27) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift 
-(S O) O x5) x4) (Bind b) (pr3_flat (CHead c (Bind b) x6) (lift (S O) O x) 
-(lift (S O) O x5) (pr3_lift (CHead c (Bind b) x6) c (S O) O (drop_drop (Bind 
-b) O c c (drop_refl c) x6) x x5 (pr3_pr2 c x x5 H17)) x0 x4 (pr3_pr2 (CHead c 
-(Bind b) x6) x0 x4 H28) Appl)))) H29)))) (THead (Bind b) x6 (THead (Flat 
-Appl) (lift (S O) O x5) x4)) (pr3_pr2 c (THead (Bind b) t1 (THead (Flat Appl) 
-(lift (S O) O x5) x4)) (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O 
-x5) x4)) (pr2_head_1 c t1 x6 H27 (Bind b) (THead (Flat Appl) (lift (S O) O 
-x5) x4)))) x1 H25))))))) H22)) H21)) t3 H16)))))))))))))) H13)) 
-H12)))))))))))))) y H4))))) H3))))))) u H0))))).
-
-theorem sn3_appl_beta:
- \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((sn3 c 
-(THead (Flat Appl) u (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) 
-\to (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind Abst) w 
-t))))))))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H: 
-(sn3 c (THead (Flat Appl) u (THead (Bind Abbr) v t)))).(\lambda (w: 
-T).(\lambda (H0: (sn3 c w)).(let H1 \def (sn3_gen_flat Appl c u (THead (Bind 
-Abbr) v t) H) in (and_ind (sn3 c u) (sn3 c (THead (Bind Abbr) v t)) (sn3 c 
-(THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind Abst) w t)))) (\lambda 
-(H2: (sn3 c u)).(\lambda (H3: (sn3 c (THead (Bind Abbr) v t))).(sn3_appl_appl 
-v (THead (Bind Abst) w t) c (sn3_beta c v t H3 w H0) u H2 (\lambda (u2: 
-T).(\lambda (H4: (pr3 c (THead (Flat Appl) v (THead (Bind Abst) w t)) 
-u2)).(\lambda (H5: (((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) u2) 
-\to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) u (THead 
-(Bind Abbr) v t)) H (THead (Flat Appl) u u2) (pr3_thin_dx c (THead (Bind 
-Abbr) v t) u2 (pr3_iso_beta v w t c u2 H4 H5) u Appl)))))))) H1)))))))).
-
-theorem sn3_appls_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: 
-T).((sn3 c u) \to (\forall (vs: TList).(\forall (t: T).((sn3 (CHead c (Bind 
-b) u) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to (sn3 c (THeads (Flat 
-Appl) vs (THead (Bind b) u t))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda 
-(u: T).(\lambda (H0: (sn3 c u)).(\lambda (vs: TList).(TList_ind (\lambda (t: 
-TList).(\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) (lifts 
-(S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u t0)))))) 
-(\lambda (t: T).(\lambda (H1: (sn3 (CHead c (Bind b) u) t)).(sn3_bind b H c u 
-H0 t H1))) (\lambda (v: T).(\lambda (vs0: TList).(TList_ind (\lambda (t: 
-TList).(((\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) 
-(lifts (S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u 
-t0)))))) \to (\forall (t0: T).((sn3 (CHead c (Bind b) u) (THead (Flat Appl) 
-(lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t) t0))) \to (sn3 c 
-(THead (Flat Appl) v (THeads (Flat Appl) t (THead (Bind b) u t0)))))))) 
-(\lambda (_: ((\forall (t: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) 
-(lifts (S O) O TNil) t)) \to (sn3 c (THeads (Flat Appl) TNil (THead (Bind b) 
-u t))))))).(\lambda (t: T).(\lambda (H2: (sn3 (CHead c (Bind b) u) (THead 
-(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O TNil) 
-t)))).(sn3_appl_bind b H c u H0 t v H2)))) (\lambda (t: T).(\lambda (t0: 
-TList).(\lambda (_: ((((\forall (t1: T).((sn3 (CHead c (Bind b) u) (THeads 
-(Flat Appl) (lifts (S O) O t0) t1)) \to (sn3 c (THeads (Flat Appl) t0 (THead 
-(Bind b) u t1)))))) \to (\forall (t1: T).((sn3 (CHead c (Bind b) u) (THead 
-(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t0) t1))) \to 
-(sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (THead (Bind b) u 
-t1))))))))).(\lambda (H2: ((\forall (t1: T).((sn3 (CHead c (Bind b) u) 
-(THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) \to (sn3 c (THeads 
-(Flat Appl) (TCons t t0) (THead (Bind b) u t1))))))).(\lambda (t1: 
-T).(\lambda (H3: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O 
-v) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))).(let H4 \def 
-(sn3_gen_flat Appl (CHead c (Bind b) u) (lift (S O) O v) (THeads (Flat Appl) 
-(lifts (S O) O (TCons t t0)) t1) H3) in (and_ind (sn3 (CHead c (Bind b) u) 
-(lift (S O) O v)) (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O 
-t) (THeads (Flat Appl) (lifts (S O) O t0) t1))) (sn3 c (THead (Flat Appl) v 
-(THeads (Flat Appl) (TCons t t0) (THead (Bind b) u t1)))) (\lambda (H5: (sn3 
-(CHead c (Bind b) u) (lift (S O) O v))).(\lambda (H6: (sn3 (CHead c (Bind b) 
-u) (THead (Flat Appl) (lift (S O) O t) (THeads (Flat Appl) (lifts (S O) O t0) 
-t1)))).(let H_y \def (sn3_gen_lift (CHead c (Bind b) u) v (S O) O H5 c) in 
-(sn3_appl_appls t (THead (Bind b) u t1) t0 c (H2 t1 H6) v (H_y (drop_drop 
-(Bind b) O c c (drop_refl c) u)) (\lambda (u2: T).(\lambda (H7: (pr3 c 
-(THeads (Flat Appl) (TCons t t0) (THead (Bind b) u t1)) u2)).(\lambda (H8: 
-(((iso (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u t1)) u2) \to 
-(\forall (P: Prop).P)))).(let H9 \def (pr3_iso_appls_bind b H (TCons t t0) u 
-t1 c u2 H7 H8) in (sn3_pr3_trans c (THead (Flat Appl) v (THead (Bind b) u 
-(THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1))) (sn3_appl_bind b H c u 
-H0 (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1) v H3) (THead (Flat 
-Appl) v u2) (pr3_flat c v v (pr3_refl c v) (THead (Bind b) u (THeads (Flat 
-Appl) (lifts (S O) O (TCons t t0)) t1)) u2 H9 Appl)))))))))) H4)))))))) 
-vs0))) vs)))))).
-
-theorem sn3_appls_beta:
- \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (us: TList).((sn3 c 
-(THeads (Flat Appl) us (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c 
-w) \to (sn3 c (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) 
-w t))))))))))
-\def
- \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (us: 
-TList).(TList_ind (\lambda (t0: TList).((sn3 c (THeads (Flat Appl) t0 (THead 
-(Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat 
-Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) (\lambda (H: 
-(sn3 c (THead (Bind Abbr) v t))).(\lambda (w: T).(\lambda (H0: (sn3 c 
-w)).(sn3_beta c v t H w H0)))) (\lambda (u: T).(\lambda (us0: 
-TList).(TList_ind (\lambda (t0: TList).((((sn3 c (THeads (Flat Appl) t0 
-(THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads 
-(Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 
-c (THead (Flat Appl) u (THeads (Flat Appl) t0 (THead (Bind Abbr) v t)))) \to 
-(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat 
-Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))) (\lambda (_: 
-(((sn3 c (THeads (Flat Appl) TNil (THead (Bind Abbr) v t))) \to (\forall (w: 
-T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) TNil (THead (Flat Appl) v (THead 
-(Bind Abst) w t))))))))).(\lambda (H0: (sn3 c (THead (Flat Appl) u (THeads 
-(Flat Appl) TNil (THead (Bind Abbr) v t))))).(\lambda (w: T).(\lambda (H1: 
-(sn3 c w)).(sn3_appl_beta c u v t H0 w H1))))) (\lambda (t0: T).(\lambda (t1: 
-TList).(\lambda (_: (((((sn3 c (THeads (Flat Appl) t1 (THead (Bind Abbr) v 
-t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) t1 (THead 
-(Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 c (THead (Flat Appl) u 
-(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)))) \to (\forall (w: T).((sn3 c 
-w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) t1 (THead (Flat Appl) 
-v (THead (Bind Abst) w t))))))))))).(\lambda (H0: (((sn3 c (THeads (Flat 
-Appl) (TCons t0 t1) (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) 
-\to (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead 
-(Bind Abst) w t))))))))).(\lambda (H1: (sn3 c (THead (Flat Appl) u (THeads 
-(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t))))).(\lambda (w: 
-T).(\lambda (H2: (sn3 c w)).(let H3 \def (sn3_gen_flat Appl c u (THeads (Flat 
-Appl) (TCons t0 t1) (THead (Bind Abbr) v t)) H1) in (and_ind (sn3 c u) (sn3 c 
-(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)))) (sn3 
-c (THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v 
-(THead (Bind Abst) w t))))) (\lambda (H4: (sn3 c u)).(\lambda (H5: (sn3 c 
-(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v 
-t))))).(sn3_appl_appls t0 (THead (Flat Appl) v (THead (Bind Abst) w t)) t1 c 
-(H0 H5 w H2) u H4 (\lambda (u2: T).(\lambda (H6: (pr3 c (THeads (Flat Appl) 
-(TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w t))) u2)).(\lambda 
-(H7: (((iso (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead 
-(Bind Abst) w t))) u2) \to (\forall (P: Prop).P)))).(let H8 \def 
-(pr3_iso_appls_beta (TCons t0 t1) v w t c u2 H6 H7) in (sn3_pr3_trans c 
-(THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v 
-t))) H1 (THead (Flat Appl) u u2) (pr3_thin_dx c (THeads (Flat Appl) (TCons t0 
-t1) (THead (Bind Abbr) v t)) u2 H8 u Appl))))))))) H3))))))))) us0))) us)))).
-
-theorem sn3_appls_abbr:
- \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
-(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).((sn3 c (THeads (Flat Appl) 
-vs (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda 
-(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind 
-(\lambda (t: TList).((sn3 c (THeads (Flat Appl) t (lift (S i) O w))) \to (sn3 
-c (THeads (Flat Appl) t (TLRef i))))) (\lambda (H0: (sn3 c (lift (S i) O 
-w))).(let H_y \def (sn3_gen_lift c w (S i) O H0 d (getl_drop Abbr c d w i H)) 
-in (sn3_abbr c d w i H H_y))) (\lambda (v: T).(\lambda (vs0: 
-TList).(TList_ind (\lambda (t: TList).((((sn3 c (THeads (Flat Appl) t (lift 
-(S i) O w))) \to (sn3 c (THeads (Flat Appl) t (TLRef i))))) \to ((sn3 c 
-(THead (Flat Appl) v (THeads (Flat Appl) t (lift (S i) O w)))) \to (sn3 c 
-(THead (Flat Appl) v (THeads (Flat Appl) t (TLRef i))))))) (\lambda (_: 
-(((sn3 c (THeads (Flat Appl) TNil (lift (S i) O w))) \to (sn3 c (THeads (Flat 
-Appl) TNil (TLRef i)))))).(\lambda (H1: (sn3 c (THead (Flat Appl) v (THeads 
-(Flat Appl) TNil (lift (S i) O w))))).(nf3_appl_abbr c d w i H v H1))) 
-(\lambda (t: T).(\lambda (t0: TList).(\lambda (_: (((((sn3 c (THeads (Flat 
-Appl) t0 (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) t0 (TLRef i))))) 
-\to ((sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (lift (S i) O w)))) 
-\to (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (TLRef 
-i)))))))).(\lambda (H1: (((sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i) 
-O w))) \to (sn3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)))))).(\lambda 
-(H2: (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (lift (S i) 
-O w))))).(let H3 \def (sn3_gen_flat Appl c v (THeads (Flat Appl) (TCons t t0) 
-(lift (S i) O w)) H2) in (and_ind (sn3 c v) (sn3 c (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (lift (S i) O w)))) (sn3 c (THead (Flat Appl) v 
-(THeads (Flat Appl) (TCons t t0) (TLRef i)))) (\lambda (H4: (sn3 c 
-v)).(\lambda (H5: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S 
-i) O w))))).(sn3_appl_appls t (TLRef i) t0 c (H1 H5) v H4 (\lambda (u2: 
-T).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)) 
-u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2) \to 
-(\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) v (THeads (Flat 
-Appl) (TCons t t0) (lift (S i) O w))) H2 (THead (Flat Appl) v u2) 
-(pr3_thin_dx c (THeads (Flat Appl) (TCons t t0) (lift (S i) O w)) u2 
-(pr3_iso_appls_abbr c d w i H (TCons t t0) u2 H6 H7) v Appl)))))))) H3))))))) 
-vs0))) vs)))))).
-