include "LambdaDelta/theory.ma".
-theorem bind_dec_not:
- \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) (not (eq B b1 b2))))
-\def
- \lambda (b1: B).(\lambda (b2: B).(let H_x \def (terms_props__bind_dec b1 b2)
-in (let H \def H_x in (or_ind (eq B b1 b2) ((eq B b1 b2) \to (\forall (P:
-Prop).P)) (or (eq B b1 b2) ((eq B b1 b2) \to False)) (\lambda (H0: (eq B b1
-b2)).(or_introl (eq B b1 b2) ((eq B b1 b2) \to False) H0)) (\lambda (H0:
-(((eq B b1 b2) \to (\forall (P: Prop).P)))).(or_intror (eq B b1 b2) ((eq B b1
-b2) \to False) (\lambda (H1: (eq B b1 b2)).(H0 H1 False)))) H)))).
-
-definition TApp:
- TList \to (T \to TList)
-\def
- let rec TApp (ts: TList) on ts: (T \to TList) \def (\lambda (v: T).(match ts
-with [TNil \Rightarrow (TCons v TNil) | (TCons t ts0) \Rightarrow (TCons t
-(TApp ts0 v))])) in TApp.
-
-definition tslen:
- TList \to nat
-\def
- let rec tslen (ts: TList) on ts: nat \def (match ts with [TNil \Rightarrow O
-| (TCons _ ts0) \Rightarrow (S (tslen ts0))]) in tslen.
-
-definition tslt:
- TList \to (TList \to Prop)
-\def
- \lambda (ts1: TList).(\lambda (ts2: TList).(lt (tslen ts1) (tslen ts2))).
-
-theorem tslt_wf__q_ind:
- \forall (P: ((TList \to Prop))).(((\forall (n: nat).((\lambda (P0: ((TList
-\to Prop))).(\lambda (n0: nat).(\forall (ts: TList).((eq nat (tslen ts) n0)
-\to (P0 ts))))) P n))) \to (\forall (ts: TList).(P ts)))
-\def
- let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts:
-TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to
-Prop))).(\lambda (H: ((\forall (n: nat).(\forall (ts: TList).((eq nat (tslen
-ts) n) \to (P ts)))))).(\lambda (ts: TList).(H (tslen ts) ts (refl_equal nat
-(tslen ts)))))).
-
-theorem tslt_wf_ind:
- \forall (P: ((TList \to Prop))).(((\forall (ts2: TList).(((\forall (ts1:
-TList).((tslt ts1 ts2) \to (P ts1)))) \to (P ts2)))) \to (\forall (ts:
-TList).(P ts)))
-\def
- let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts:
-TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to
-Prop))).(\lambda (H: ((\forall (ts2: TList).(((\forall (ts1: TList).((lt
-(tslen ts1) (tslen ts2)) \to (P ts1)))) \to (P ts2))))).(\lambda (ts:
-TList).(tslt_wf__q_ind (\lambda (t: TList).(P t)) (\lambda (n:
-nat).(lt_wf_ind n (Q (\lambda (t: TList).(P t))) (\lambda (n0: nat).(\lambda
-(H0: ((\forall (m: nat).((lt m n0) \to (Q (\lambda (t: TList).(P t))
-m))))).(\lambda (ts0: TList).(\lambda (H1: (eq nat (tslen ts0) n0)).(let H2
-\def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall (m: nat).((lt m n1) \to
-(\forall (ts1: TList).((eq nat (tslen ts1) m) \to (P ts1)))))) H0 (tslen ts0)
-H1) in (H ts0 (\lambda (ts1: TList).(\lambda (H3: (lt (tslen ts1) (tslen
-ts0))).(H2 (tslen ts1) H3 ts1 (refl_equal nat (tslen ts1))))))))))))) ts)))).
-
-theorem theads_tapp:
- \forall (k: K).(\forall (vs: TList).(\forall (v: T).(\forall (t: T).(eq T
-(THeads k (TApp vs v) t) (THeads k vs (THead k v t))))))
-\def
- \lambda (k: K).(\lambda (vs: TList).(TList_ind (\lambda (t: TList).(\forall
-(v: T).(\forall (t0: T).(eq T (THeads k (TApp t v) t0) (THeads k t (THead k v
-t0)))))) (\lambda (v: T).(\lambda (t: T).(refl_equal T (THead k v t))))
-(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (v: T).(\forall
-(t1: T).(eq T (THeads k (TApp t0 v) t1) (THeads k t0 (THead k v
-t1))))))).(\lambda (v: T).(\lambda (t1: T).(eq_ind_r T (THeads k t0 (THead k
-v t1)) (\lambda (t2: T).(eq T (THead k t t2) (THead k t (THeads k t0 (THead k
-v t1))))) (refl_equal T (THead k t (THeads k t0 (THead k v t1)))) (THeads k
-(TApp t0 v) t1) (H v t1))))))) vs)).
-
-theorem tcons_tapp_ex:
- \forall (ts1: TList).(\forall (t1: T).(ex2_2 TList T (\lambda (ts2:
-TList).(\lambda (t2: T).(eq TList (TCons t1 ts1) (TApp ts2 t2)))) (\lambda
-(ts2: TList).(\lambda (_: T).(eq nat (tslen ts1) (tslen ts2))))))
-\def
- \lambda (ts1: TList).(TList_ind (\lambda (t: TList).(\forall (t1: T).(ex2_2
-TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t) (TApp
-ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t) (tslen
-ts2))))))) (\lambda (t1: T).(ex2_2_intro TList T (\lambda (ts2:
-TList).(\lambda (t2: T).(eq TList (TCons t1 TNil) (TApp ts2 t2)))) (\lambda
-(ts2: TList).(\lambda (_: T).(eq nat O (tslen ts2)))) TNil t1 (refl_equal
-TList (TApp TNil t1)) (refl_equal nat (tslen TNil)))) (\lambda (t:
-T).(\lambda (t0: TList).(\lambda (H: ((\forall (t1: T).(ex2_2 TList T
-(\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t0) (TApp ts2
-t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) (tslen
-ts2)))))))).(\lambda (t1: T).(let H_x \def (H t) in (let H0 \def H_x in
-(ex2_2_ind TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t
-t0) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0)
-(tslen ts2)))) (ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq
-TList (TCons t1 (TCons t t0)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda
-(_: T).(eq nat (S (tslen t0)) (tslen ts2))))) (\lambda (x0: TList).(\lambda
-(x1: T).(\lambda (H1: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H2: (eq
-nat (tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t2:
-TList).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t3: T).(eq TList (TCons
-t1 t2) (TApp ts2 t3)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (S
-(tslen t0)) (tslen ts2)))))) (eq_ind_r nat (tslen x0) (\lambda (n:
-nat).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons
-t1 (TApp x0 x1)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq
-nat (S n) (tslen ts2)))))) (ex2_2_intro TList T (\lambda (ts2:
-TList).(\lambda (t2: T).(eq TList (TCons t1 (TApp x0 x1)) (TApp ts2 t2))))
-(\lambda (ts2: TList).(\lambda (_: T).(eq nat (S (tslen x0)) (tslen ts2))))
-(TCons t1 x0) x1 (refl_equal TList (TApp (TCons t1 x0) x1)) (refl_equal nat
-(tslen (TCons t1 x0)))) (tslen t0) H2) (TCons t t0) H1))))) H0))))))) ts1).
-
-theorem tlist_ind_rew:
- \forall (P: ((TList \to Prop))).((P TNil) \to (((\forall (ts:
-TList).(\forall (t: T).((P ts) \to (P (TApp ts t)))))) \to (\forall (ts:
-TList).(P ts))))
-\def
- \lambda (P: ((TList \to Prop))).(\lambda (H: (P TNil)).(\lambda (H0:
-((\forall (ts: TList).(\forall (t: T).((P ts) \to (P (TApp ts
-t))))))).(\lambda (ts: TList).(tslt_wf_ind (\lambda (t: TList).(P t))
-(\lambda (ts2: TList).(TList_ind (\lambda (t: TList).(((\forall (ts1:
-TList).((tslt ts1 t) \to (P ts1)))) \to (P t))) (\lambda (_: ((\forall (ts1:
-TList).((tslt ts1 TNil) \to (P ts1))))).H) (\lambda (t: T).(\lambda (t0:
-TList).(\lambda (_: ((((\forall (ts1: TList).((tslt ts1 t0) \to (P ts1))))
-\to (P t0)))).(\lambda (H2: ((\forall (ts1: TList).((tslt ts1 (TCons t t0))
-\to (P ts1))))).(let H_x \def (tcons_tapp_ex t0 t) in (let H3 \def H_x in
-(ex2_2_ind TList T (\lambda (ts3: TList).(\lambda (t2: T).(eq TList (TCons t
-t0) (TApp ts3 t2)))) (\lambda (ts3: TList).(\lambda (_: T).(eq nat (tslen t0)
-(tslen ts3)))) (P (TCons t t0)) (\lambda (x0: TList).(\lambda (x1:
-T).(\lambda (H4: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H5: (eq nat
-(tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t1: TList).(P
-t1)) (H0 x0 x1 (H2 x0 (eq_ind nat (tslen t0) (\lambda (n: nat).(lt n (tslen
-(TCons t t0)))) (le_n (tslen (TCons t t0))) (tslen x0) H5))) (TCons t t0)
-H4))))) H3))))))) ts2)) ts)))).
-
-theorem 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 dnf_dec2:
- \forall (t: T).(\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v:
-T).(subst0 d w t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S
-O) d v))))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(or (\forall (w:
-T).(ex T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))))) (ex T (\lambda
-(v: T).(eq T t0 (lift (S O) d v))))))) (\lambda (n: nat).(\lambda (d:
-nat).(or_intror (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (TSort n)
-(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (TSort n) (lift (S O) d
-v)))) (ex_intro T (\lambda (v: T).(eq T (TSort n) (lift (S O) d v))) (TSort
-n) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0)) (refl_equal T
-(TSort n)) (lift (S O) d (TSort n)) (lift_sort n (S O) d)))))) (\lambda (n:
-nat).(\lambda (d: nat).(lt_eq_gt_e n d (or (\forall (w: T).(ex T (\lambda (v:
-T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T
-(TLRef n) (lift (S O) d v))))) (\lambda (H: (lt n d)).(or_intror (\forall (w:
-T).(ex T (\lambda (v: T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T
-(\lambda (v: T).(eq T (TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v:
-T).(eq T (TLRef n) (lift (S O) d v))) (TLRef n) (eq_ind_r T (TLRef n)
-(\lambda (t0: T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d
-(TLRef n)) (lift_lref_lt n (S O) d H))))) (\lambda (H: (eq nat n d)).(eq_ind
-nat n (\lambda (n0: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 n0
-w (TLRef n) (lift (S O) n0 v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift
-(S O) n0 v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v: T).(subst0 n w
-(TLRef n) (lift (S O) n v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift (S
-O) n v)))) (\lambda (w: T).(ex_intro T (\lambda (v: T).(subst0 n w (TLRef n)
-(lift (S O) n v))) (lift n O w) (eq_ind_r T (lift (plus (S O) n) O w)
-(\lambda (t0: T).(subst0 n w (TLRef n) t0)) (subst0_lref w n) (lift (S O) n
-(lift n O w)) (lift_free w n (S O) O n (le_n (plus O n)) (le_O_n n)))))) d
-H)) (\lambda (H: (lt d n)).(or_intror (\forall (w: T).(ex T (\lambda (v:
-T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T
-(TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v: T).(eq T (TLRef n)
-(lift (S O) d v))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t0:
-T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d (TLRef (pred
-n))) (lift_lref_gt d n H)))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda
-(H: ((\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w
-t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d
-v)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(or (\forall (w:
-T).(ex T (\lambda (v: T).(subst0 d w t1 (lift (S O) d v))))) (ex T (\lambda
-(v: T).(eq T t1 (lift (S O) d v)))))))).(\lambda (d: nat).(let H_x \def (H d)
-in (let H1 \def H_x in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0
-d w t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d
-v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1)
-(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O)
-d v))))) (\lambda (H2: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 d w t0
-(lift (S O) d v))))))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in
-(or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w t1 (lift (S
-O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))))
-(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift
-(S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d
-v))))) (\lambda (H4: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w
-t1 (lift (S O) (s k d) v))))))).(or_introl (\forall (w: T).(ex T (\lambda (v:
-T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq
-T (THead k t0 t1) (lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H4 w)
-in (let H5 \def H_x1 in (ex_ind T (\lambda (v: T).(subst0 (s k d) w t1 (lift
-(S O) (s k d) v))) (ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S
-O) d v)))) (\lambda (x: T).(\lambda (H6: (subst0 (s k d) w t1 (lift (S O) (s
-k d) x))).(let H_x2 \def (H2 w) in (let H7 \def H_x2 in (ex_ind T (\lambda
-(v: T).(subst0 d w t0 (lift (S O) d v))) (ex T (\lambda (v: T).(subst0 d w
-(THead k t0 t1) (lift (S O) d v)))) (\lambda (x0: T).(\lambda (H8: (subst0 d
-w t0 (lift (S O) d x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0
-t1) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0)
-(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 t1) t2))
-(subst0_both w t0 (lift (S O) d x0) d H8 k t1 (lift (S O) (s k d) x) H6)
-(lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H7))))) H5))))))
-(\lambda (H4: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d)
-v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or
-(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O)
-d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v)))))
-(\lambda (x: T).(\lambda (H5: (eq T t1 (lift (S O) (s k d) x))).(eq_ind_r T
-(lift (S O) (s k d) x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda
-(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v:
-T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex
-T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O)
-d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 (lift (S O) (s k d) x))
-(lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H2 w) in (let H6 \def
-H_x1 in (ex_ind T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))) (ex T
-(\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O) d
-v)))) (\lambda (x0: T).(\lambda (H7: (subst0 d w t0 (lift (S O) d
-x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d)
-x)) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0)
-(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 (lift (S O)
-(s k d) x)) t2)) (subst0_fst w (lift (S O) d x0) t0 d H7 (lift (S O) (s k d)
-x) k) (lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H6))))) t1
-H5))) H4)) H3)))) (\lambda (H2: (ex T (\lambda (v: T).(eq T t0 (lift (S O) d
-v))))).(ex_ind T (\lambda (v: T).(eq T t0 (lift (S O) d v))) (or (\forall (w:
-T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex
-T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) (\lambda (x:
-T).(\lambda (H3: (eq T t0 (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in
-(let H4 \def H_x0 in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s
-k d) w t1 (lift (S O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S
-O) (s k d) v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead
-k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1)
-(lift (S O) d v))))) (\lambda (H5: ((\forall (w: T).(ex T (\lambda (v:
-T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))))))).(eq_ind_r T (lift (S O)
-d x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w
-(THead k t2 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t2
-t1) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v:
-T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v))))) (ex T
-(\lambda (v: T).(eq T (THead k (lift (S O) d x) t1) (lift (S O) d v))))
-(\lambda (w: T).(let H_x1 \def (H5 w) in (let H6 \def H_x1 in (ex_ind T
-(\lambda (v: T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))) (ex T (\lambda
-(v: T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v)))) (\lambda
-(x0: T).(\lambda (H7: (subst0 (s k d) w t1 (lift (S O) (s k d)
-x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x) t1)
-(lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift
-(S O) (s k d) x0)) (\lambda (t2: T).(subst0 d w (THead k (lift (S O) d x) t1)
-t2)) (subst0_snd k w (lift (S O) (s k d) x0) t1 d H7 (lift (S O) d x)) (lift
-(S O) d (THead k x x0)) (lift_head k x x0 (S O) d))))) H6))))) t0 H3))
-(\lambda (H5: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d)
-v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or
-(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O)
-d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v)))))
-(\lambda (x0: T).(\lambda (H6: (eq T t1 (lift (S O) (s k d) x0))).(eq_ind_r T
-(lift (S O) (s k d) x0) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda
-(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v:
-T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (eq_ind_r T (lift (S O) d x)
-(\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead
-k t2 (lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq
-T (THead k t2 (lift (S O) (s k d) x0)) (lift (S O) d v)))))) (or_intror
-(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x)
-(lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T
-(THead k (lift (S O) d x) (lift (S O) (s k d) x0)) (lift (S O) d v))))
-(ex_intro T (\lambda (v: T).(eq T (THead k (lift (S O) d x) (lift (S O) (s k
-d) x0)) (lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d
-x) (lift (S O) (s k d) x0)) (\lambda (t2: T).(eq T (THead k (lift (S O) d x)
-(lift (S O) (s k d) x0)) t2)) (refl_equal T (THead k (lift (S O) d x) (lift
-(S O) (s k d) x0))) (lift (S O) d (THead k x x0)) (lift_head k x x0 (S O)
-d)))) t0 H3) t1 H6))) H5)) H4))))) H2)) H1))))))))) t).
-
-theorem pr2_change:
- \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1:
-T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Bind b) v1) t1 t2) \to
-(\forall (v2: T).(pr2 (CHead c (Bind b) v2) t1 t2))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda
-(v1: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind
-b) v1) t1 t2)).(\lambda (v2: T).(insert_eq C (CHead c (Bind b) v1) (\lambda
-(c0: C).(pr2 c0 t1 t2)) (pr2 (CHead c (Bind b) v2) t1 t2) (\lambda (y:
-C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t:
-T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) v1)) \to (pr2 (CHead c (Bind
-b) v2) t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda
-(H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) v1))).(pr2_free
-(CHead c (Bind b) v2) t3 t4 H2)))))) (\lambda (c0: C).(\lambda (d:
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind
-Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3
-t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: (eq C c0
-(CHead c (Bind b) v1))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1
-(CHead d (Bind Abbr) u))) H2 (CHead c (Bind b) v1) H5) in (nat_ind (\lambda
-(n: nat).((getl n (CHead c (Bind b) v1) (CHead d (Bind Abbr) u)) \to ((subst0
-n u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t)))) (\lambda (H7: (getl O
-(CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda (H8: (subst0 O u t4
-t)).(let H9 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1]))
-(CHead d (Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d
-(Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u)
-H7))) in ((let H10 \def (f_equal C B (\lambda (e: C).(match e in C return
-(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u)
-(CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1
-(getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H11
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d
-(Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr)
-u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in
-(\lambda (H12: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H14 \def (eq_ind
-T u (\lambda (t0: T).(subst0 O t0 t4 t)) H8 v1 H11) in (let H15 \def
-(eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abbr))) H Abbr H12) in (eq_ind B
-Abbr (\lambda (b0: B).(pr2 (CHead c (Bind b0) v2) t3 t)) (let H16 \def (match
-(H15 (refl_equal B Abbr)) in False return (\lambda (_: False).(pr2 (CHead c
-(Bind Abbr) v2) t3 t)) with []) in H16) b H12)))))) H10)) H9)))) (\lambda
-(i0: nat).(\lambda (_: (((getl i0 (CHead c (Bind b) v1) (CHead d (Bind Abbr)
-u)) \to ((subst0 i0 u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))).(\lambda
-(H7: (getl (S i0) (CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda
-(H8: (subst0 (S i0) u t4 t)).(pr2_delta (CHead c (Bind b) v2) d u (S i0)
-(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c
-(CHead d (Bind Abbr) u) v1 i0 H7) v2) t3 t4 H3 t H8))))) i H6 H4)))))))))))))
-y t1 t2 H1))) H0)))))))).
-
-theorem pr3_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).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall
-(c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind
-b0) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x
-(THead (Bind b0) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b0) u1) t1 t2)))) (pr3
-(CHead c (Bind b0) u1) t1 (lift (S O) O x)))))))))) (\lambda (_: (not (eq B
-Abbr Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x:
-T).(\lambda (H0: (pr3 c (THead (Bind Abbr) u1 t1) x)).(let H1 \def
-(pr3_gen_abbr c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr)
-u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (or (ex3_2 T
-T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda
-(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1)
-t1 (lift (S O) O x))) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1
-t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
-Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda
-(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1)
-t1 (lift (S O) O x))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T x
-(THead (Bind Abbr) x0 x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: (pr3
-(CHead c (Bind Abbr) u1) t1 x1)).(or_introl (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
-(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S
-O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
-(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) x0 x1
-H3 H4 H5))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S
-O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x
-(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3
-(CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) H2)) H1)))))))) (\lambda (H:
-(not (eq B Abst Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1:
-T).(\lambda (x: T).(\lambda (_: (pr3 c (THead (Bind Abst) u1 t1) x)).(let H1
-\def (match (H (refl_equal B Abst)) in False return (\lambda (_: False).(or
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(pr3 (CHead c (Bind Abst) u1) t1 t2)))) (pr3 (CHead c
-(Bind Abst) u1) t1 (lift (S O) O x)))) with []) in H1))))))) (\lambda (_:
-(not (eq B Void Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1:
-T).(\lambda (x: T).(\lambda (H0: (pr3 c (THead (Bind Void) u1 t1) x)).(let H1
-\def (pr3_gen_void c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
-(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t2)))))) (pr3 (CHead c
-(Bind Void) u1) t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void)
-u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda
-(H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void)
-u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0)
-u) t1 t2))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x
-(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))
-(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead
-c (Bind b0) u) t1 t2))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1
-t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H3: (eq T x (THead (Bind Void) x0
-x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: ((\forall (b0: B).(\forall
-(u: T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(or_introl (ex3_2 T T (\lambda
-(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
-(CHead c (Bind Void) u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S
-O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
-(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 t2))) x0 x1
-H3 H4 (H5 Void u1)))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Void) u1) t1
-(lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1
-t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x)) H2)) H1)))))))) b).
-
-theorem pr3_iso_appls_abbr:
- \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c
-(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).(let u1 \def (THeads (Flat
-Appl) vs (TLRef i)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to
-(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) vs (lift (S i) O w))
-u2))))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda
-(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind
-(\lambda (t: TList).(let u1 \def (THeads (Flat Appl) t (TLRef i)) in (\forall
-(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to
-(pr3 c (THeads (Flat Appl) t (lift (S i) O w)) u2)))))) (\lambda (u2:
-T).(\lambda (H0: (pr3 c (TLRef i) u2)).(\lambda (H1: (((iso (TLRef i) u2) \to
-(\forall (P: Prop).P)))).(let H2 \def (pr3_gen_lref c u2 i H0) in (or_ind (eq
-T u2 (TLRef i)) (ex3_3 C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_:
-T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u:
-T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(v: T).(eq T u2 (lift (S i) O v)))))) (pr3 c (lift (S i) O w) u2) (\lambda
-(H3: (eq T u2 (TLRef i))).(let H4 \def (eq_ind T u2 (\lambda (t: T).((iso
-(TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H3) in (eq_ind_r T
-(TLRef i) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (H4 (iso_refl (TLRef
-i)) (pr3 c (lift (S i) O w) (TLRef i))) u2 H3))) (\lambda (H3: (ex3_3 C T T
-(\lambda (d0: C).(\lambda (u: T).(\lambda (_: T).(getl i c (CHead d0 (Bind
-Abbr) u))))) (\lambda (d0: C).(\lambda (u: T).(\lambda (v: T).(pr3 d0 u v))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T u2 (lift (S i) O
-v))))))).(ex3_3_ind C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_:
-T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u:
-T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(v: T).(eq T u2 (lift (S i) O v))))) (pr3 c (lift (S i) O w) u2) (\lambda
-(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H4: (getl i c (CHead x0
-(Bind Abbr) x1))).(\lambda (H5: (pr3 x0 x1 x2)).(\lambda (H6: (eq T u2 (lift
-(S i) O x2))).(let H7 \def (eq_ind T u2 (\lambda (t: T).((iso (TLRef i) t)
-\to (\forall (P: Prop).P))) H1 (lift (S i) O x2) H6) in (eq_ind_r T (lift (S
-i) O x2) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (let H8 \def (eq_ind C
-(CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H (CHead x0 (Bind
-Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1)
-H4)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow
-c0])) (CHead d (Bind Abbr) w) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d
-(Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) H4)) in ((let H10 \def (f_equal
-C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) w) (CHead
-x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind
-Abbr) x1) H4)) in (\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1
-(\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 w H10) in (let H13
-\def (eq_ind_r T x1 (\lambda (t: T).(pr3 x0 t x2)) H5 w H10) in (let H14 \def
-(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) w))) H12 d
-H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c0: C).(pr3 c0 w x2)) H13 d
-H11) in (pr3_lift c d (S i) O (getl_drop Abbr c d w i H14) w x2 H15)))))))
-H9))) u2 H6)))))))) H3)) H2))))) (\lambda (t: T).(\lambda (t0:
-TList).(\lambda (H0: ((\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (TLRef
-i)) u2) \to ((((iso (THeads (Flat Appl) t0 (TLRef i)) u2) \to (\forall (P:
-Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (lift (S i) O w))
-u2)))))).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads
-(Flat Appl) t0 (TLRef i))) u2)).(\lambda (H2: (((iso (THead (Flat Appl) t
-(THeads (Flat Appl) t0 (TLRef i))) u2) \to (\forall (P: Prop).P)))).(let H3
-\def (pr3_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) u2 H1) in (or3_ind
-(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3
-t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_:
-T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2)))) (ex4_4 T
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3
-c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i))
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t
-(THeads (Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (H4: (ex3_2 T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2:
-T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))))).(ex3_2_ind T T (\lambda
-(u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c
-(THeads (Flat Appl) t0 (TLRef i)) t2))) (pr3 c (THead (Flat Appl) t (THeads
-(Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H5: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c t
-x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(let H8 \def
-(eq_ind T u2 (\lambda (t1: T).((iso (THead (Flat Appl) t (THeads (Flat Appl)
-t0 (TLRef i))) t1) \to (\forall (P: Prop).P))) H2 (THead (Flat Appl) x0 x1)
-H5) in (eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(pr3 c (THead
-(Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) t1)) (H8 (iso_head t
-x0 (THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) (pr3 c (THead (Flat
-Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) (THead (Flat Appl) x0 x1)))
-u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2)))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t
-u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3
-c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i))
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
-z1 t2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O
-w))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
-T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H6: (pr3 c t
-x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind
-Abst) x0 x1))).(\lambda (H8: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c
-(Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) (THead (Flat Appl) t
-(THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t
-(THead (Bind Abst) x0 x1)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift
-(S i) O w))) c (pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead
-(Bind Abst) x0 x1) (H0 (THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso
-(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (P:
-Prop).(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H9 P)))) t Appl) (THead
-(Bind Abbr) t x1) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1))
-(THead (Bind Abbr) t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst)
-x0 x1)) (THead (Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1
-(pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t
-x1) c (pr3_head_12 c t x2 H6 (Bind Abbr) x1 x3 (H8 Abbr x2)) u2 H5))))))))))
-H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i))
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind
-B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1))))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
-(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift
-(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3
-(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat
-Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda
-(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not
-(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i))
-(THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c (THead (Bind x0) x5 (THead
-(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H8: (pr3 c t x4)).(\lambda
-(H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat
-Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Bind x0)
-x1 (THead (Flat Appl) (lift (S O) O t) x2)) (THead (Flat Appl) t (THeads
-(Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t (THead (Bind
-x0) x1 x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c
-(pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead (Bind x0) x1
-x2) (H0 (THead (Bind x0) x1 x2) H6 (\lambda (H11: (iso (THeads (Flat Appl) t0
-(TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (P:
-Prop).(iso_flats_lref_bind_false Appl x0 i x1 x2 t0 H11 P)))) t Appl) (THead
-(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr3_pr2 c (THead (Flat
-Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift
-(S O) O t) x2)) (pr2_free c (THead (Flat Appl) t (THead (Bind x0) x1 x2))
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr0_upsilon x0
-H5 t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2))))) (THead (Bind
-x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr3_head_12 c x1 x1
-(pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift (S O) O t) x2) (THead
-(Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead c (Bind x0) x1) (lift
-(S O) O t) (lift (S O) O x4) (pr3_lift (CHead c (Bind x0) x1) c (S O) O
-(drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H8) (Flat Appl) x2 x2
-(pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift (S O) O x4)) x2))))
-u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c (pr3_head_12
-c x1 x5 H9 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat
-Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10
-(lift (S O) O x4) Appl)) u2 H7)))))))))))))) H4)) H3)))))))) vs)))))).
-
-theorem pr3_iso_appl_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2:
-T).(\forall (t: T).(let u1 \def (THead (Flat Appl) v1 (THead (Bind b) v2 t))
-in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to
-(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) v2 (THead (Flat Appl)
-(lift (S O) O v1) t)) u2))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v1: T).(\lambda
-(v2: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H0: (pr3 c
-(THead (Flat Appl) v1 (THead (Bind b) v2 t)) u2)).(\lambda (H1: (((iso (THead
-(Flat Appl) v1 (THead (Bind b) v2 t)) u2) \to (\forall (P: Prop).P)))).(let
-H2 \def (pr3_gen_appl c v1 (THead (Bind b) v2 t) u2 H0) in (or3_ind (ex3_2 T
-T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda
-(t2: T).(pr3 c (THead (Bind b) v2 t) t2)))) (ex4_4 T T T T (\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1
-t2)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst))))))))
-(\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1
-z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
-T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat
-Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b0:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c (THead (Bind b) v2
-(THead (Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (H3: (ex3_2 T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda
-(t2: T).(pr3 c (THead (Bind b) v2 t) t2))))).(ex3_2_ind T T (\lambda (u3:
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c
-(THead (Bind b) v2 t) t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl)
-(lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq
-T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v1 x0)).(\lambda (_:
-(pr3 c (THead (Bind b) v2 t) x1)).(let H7 \def (eq_ind T u2 (\lambda (t0:
-T).((iso (THead (Flat Appl) v1 (THead (Bind b) v2 t)) t0) \to (\forall (P:
-Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl)
-x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S
-O) O v1) t)) t0)) (H7 (iso_head v1 x0 (THead (Bind b) v2 t) x1 (Flat Appl))
-(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead
-(Flat Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda
-(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1
-t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3)))))
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
-(THead (Bind b) v2 t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u:
-T).(pr3 (CHead c (Bind b0) u) z1 t2))))))) (pr3 c (THead (Bind b) v2 (THead
-(Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c (THead (Bind Abbr)
-x2 x3) u2)).(\lambda (H5: (pr3 c v1 x2)).(\lambda (H6: (pr3 c (THead (Bind b)
-v2 t) (THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b0: B).(\forall
-(u: T).(pr3 (CHead c (Bind b0) u) x1 x3))))).(pr3_t (THead (Bind Abbr) x2 x3)
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) c (let H_x \def
-(pr3_gen_bind b H c v2 t (THead (Bind Abst) x0 x1) H6) in (let H8 \def H_x in
-(or_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst)
-x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2))))
-(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind Abst) x0 x1))) (pr3 c
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind
-Abbr) x2 x3)) (\lambda (H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq
-T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3
-(CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2:
-T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3
-(CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl)
-(lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3)) (\lambda (x4: T).(\lambda
-(x5: T).(\lambda (H10: (eq T (THead (Bind Abst) x0 x1) (THead (Bind b) x4
-x5))).(\lambda (H11: (pr3 c v2 x4)).(\lambda (H12: (pr3 (CHead c (Bind b) v2)
-t x5)).(let H13 \def (f_equal T B (\lambda (e: T).(match e in T return
-(\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow
-Abst | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with
-[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (THead (Bind Abst)
-x0 x1) (THead (Bind b) x4 x5) H10) in ((let H14 \def (f_equal T T (\lambda
-(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0
-| (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind
-Abst) x0 x1) (THead (Bind b) x4 x5) H10) in ((let H15 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) \Rightarrow t0]))
-(THead (Bind Abst) x0 x1) (THead (Bind b) x4 x5) H10) in (\lambda (H16: (eq T
-x0 x4)).(\lambda (H17: (eq B Abst b)).(let H18 \def (eq_ind_r T x5 (\lambda
-(t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H12 x1 H15) in (let H19 \def
-(eq_ind_r T x4 (\lambda (t0: T).(pr3 c v2 t0)) H11 x0 H16) in (let H20 \def
-(eq_ind_r B b (\lambda (b0: B).(pr3 (CHead c (Bind b0) v2) t x1)) H18 Abst
-H17) in (let H21 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H
-Abst H17) in (eq_ind B Abst (\lambda (b0: B).(pr3 c (THead (Bind b0) v2
-(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3))) (let H22
-\def (match (H21 (refl_equal B Abst)) in False return (\lambda (_:
-False).(pr3 c (THead (Bind Abst) v2 (THead (Flat Appl) (lift (S O) O v1) t))
-(THead (Bind Abbr) x2 x3))) with []) in H22) b H17)))))))) H14)) H13)))))))
-H9)) (\lambda (H9: (pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind
-Abst) x0 x1)))).(pr3_t (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O
-x2) (lift (S O) O (THead (Bind Abst) x0 x1)))) (THead (Bind b) v2 (THead
-(Flat Appl) (lift (S O) O v1) t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift
-(S O) O v1) t) (THead (Flat Appl) (lift (S O) O x2) (lift (S O) O (THead
-(Bind Abst) x0 x1))) (Bind b) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O
-v1) (lift (S O) O x2) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop
-(Bind b) O c c (drop_refl c) v2) v1 x2 H5) t (lift (S O) O (THead (Bind Abst)
-x0 x1)) H9 Appl)) (THead (Bind Abbr) x2 x3) (eq_ind T (lift (S O) O (THead
-(Flat Appl) x2 (THead (Bind Abst) x0 x1))) (\lambda (t0: T).(pr3 c (THead
-(Bind b) v2 t0) (THead (Bind Abbr) x2 x3))) (pr3_sing c (THead (Bind Abbr) x2
-x1) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst)
-x0 x1)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2
-(THead (Bind Abst) x0 x1)))) (THead (Bind Abbr) x2 x1) (pr0_zeta b H (THead
-(Flat Appl) x2 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x1) (pr0_beta
-x0 x2 x2 (pr0_refl x2) x1 x1 (pr0_refl x1)) v2)) (THead (Bind Abbr) x2 x3)
-(pr3_head_12 c x2 x2 (pr3_refl c x2) (Bind Abbr) x1 x3 (H7 Abbr x2))) (THead
-(Flat Appl) (lift (S O) O x2) (lift (S O) O (THead (Bind Abst) x0 x1)))
-(lift_flat Appl x2 (THead (Bind Abst) x0 x1) (S O) O)))) H8))) u2 H4)))))))))
-H3)) (\lambda (H3: (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0)
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(_: T).(pr3 c v1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind
-B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 z1)))))))) (\lambda
-(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3:
-T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O)
-O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0)
-y2) z1 z2))))))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O
-v1) t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
-T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0
-Abst))).(\lambda (H5: (pr3 c (THead (Bind b) v2 t) (THead (Bind x0) x1
-x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O)
-O x4) x3)) u2)).(\lambda (H7: (pr3 c v1 x4)).(\lambda (H8: (pr3 c x1
-x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t (THead (Bind
-x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind b) v2 (THead
-(Flat Appl) (lift (S O) O v1) t)) c (let H_x \def (pr3_gen_bind b H c v2 t
-(THead (Bind x0) x1 x2) H5) in (let H10 \def H_x in (or_ind (ex3_2 T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_:
-T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) (pr3 (CHead c (Bind
-b) v2) t (lift (S O) O (THead (Bind x0) x1 x2))) (pr3 c (THead (Bind b) v2
-(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat
-Appl) (lift (S O) O x4) x3))) (\lambda (H11: (ex3_2 T T (\lambda (u3:
-T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda
-(t2: T).(pr3 (CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3:
-T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda
-(t2: T).(pr3 (CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead
-(Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat Appl)
-(lift (S O) O x4) x3))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H12: (eq
-T (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7))).(\lambda (H13: (pr3 c v2
-x6)).(\lambda (H14: (pr3 (CHead c (Bind b) v2) t x7)).(let H15 \def (f_equal
-T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _)
-\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) \Rightarrow (match
-k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in
-((let H16 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ t0
-_) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in
-((let H17 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow x2 | (TLRef _) \Rightarrow x2 | (THead _ _
-t0) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in
-(\lambda (H18: (eq T x1 x6)).(\lambda (H19: (eq B x0 b)).(let H20 \def
-(eq_ind_r T x7 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H14 x2 H17)
-in (let H21 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c v2 t0)) H13 x1 H18)
-in (let H22 \def (eq_ind B x0 (\lambda (b0: B).(pr3 (CHead c (Bind b0) x5) x2
-x3)) H9 b H19) in (let H23 \def (eq_ind B x0 (\lambda (b0: B).(not (eq B b0
-Abst))) H4 b H19) in (eq_ind_r B b (\lambda (b0: B).(pr3 c (THead (Bind b) v2
-(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind b0) x5 (THead (Flat
-Appl) (lift (S O) O x4) x3)))) (pr3_head_21 c v2 x5 (pr3_t x1 v2 c H21 x5 H8)
-(Bind b) (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat Appl) (lift (S
-O) O x4) x3) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O
-x4) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c
-(drop_refl c) v2) v1 x4 H7) t x3 (pr3_t x2 t (CHead c (Bind b) v2) H20 x3
-(pr3_pr3_pr3_t c v2 x1 H21 x2 x3 (Bind b) (pr3_pr3_pr3_t c x1 x5 H8 x2 x3
-(Bind b) H22))) Appl)) x0 H19)))))))) H16)) H15))))))) H11)) (\lambda (H11:
-(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind x0) x1 x2)))).(pr3_t
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead
-(Bind x0) x1 x2)))) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1)
-t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat
-Appl) (lift (S O) O x4) (lift (S O) O (THead (Bind x0) x1 x2))) (Bind b)
-(pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O x4) (pr3_lift
-(CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) v2)
-v1 x4 H7) t (lift (S O) O (THead (Bind x0) x1 x2)) H11 Appl)) (THead (Bind
-x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (eq_ind T (lift (S O) O
-(THead (Flat Appl) x4 (THead (Bind x0) x1 x2))) (\lambda (t0: T).(pr3 c
-(THead (Bind b) v2 t0) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O
-x4) x3)))) (pr3_sing c (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O
-x4) x2)) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x4 (THead (Bind
-x0) x1 x2)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl)
-x4 (THead (Bind x0) x1 x2)))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S
-O) O x4) x2)) (pr0_zeta b H (THead (Flat Appl) x4 (THead (Bind x0) x1 x2))
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr0_upsilon x0
-H4 x4 x4 (pr0_refl x4) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2)) v2)) (THead
-(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (pr3_head_12 c x1 x5
-H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat Appl)
-(lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H9 (lift (S
-O) O x4) Appl))) (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead
-(Bind x0) x1 x2))) (lift_flat Appl x4 (THead (Bind x0) x1 x2) (S O) O))))
-H10))) u2 H6))))))))))))) H3)) H2)))))))))).
-
-theorem pr3_iso_appls_appl_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (v: T).(\forall (u:
-T).(\forall (t: T).(\forall (vs: TList).(let u1 \def (THeads (Flat Appl) vs
-(THead (Flat Appl) v (THead (Bind b) u t))) in (\forall (c: C).(\forall (u2:
-T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c
-(THeads (Flat Appl) vs (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v)
-t))) u2)))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v: T).(\lambda
-(u: T).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0:
-TList).(let u1 \def (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind
-b) u t))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1
-u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead
-(Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))) (\lambda (c:
-C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) v (THead (Bind b)
-u t)) u2)).(\lambda (H1: (((iso (THead (Flat Appl) v (THead (Bind b) u t))
-u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b H v u t c u2 H0 H1)))))
-(\lambda (t0: T).(\lambda (t1: TList).(\lambda (H0: ((\forall (c: C).(\forall
-(u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u
-t))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind
-b) u t))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1
-(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))).(\lambda
-(c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t0 (THeads
-(Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t)))) u2)).(\lambda
-(H2: (((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v
-(THead (Bind b) u t)))) u2) \to (\forall (P: Prop).P)))).(let H3 \def
-(pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind
-b) u t))) u2 H1) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq
-T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead
-(Flat Appl) v (THead (Bind b) u t))) t2)))) (ex4_4 T T T T (\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
-Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0:
-B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2)))))))) (ex6_6 B T T T
-T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
-c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0)
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c
-(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat
-Appl) (lift (S O) O v) t)))) u2) (\lambda (H4: (ex3_2 T T (\lambda (u3:
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c
-(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t)))
-t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
-(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3)))
-(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
-Appl) v (THead (Bind b) u t))) t2))) (pr3 c (THead (Flat Appl) t0 (THeads
-(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t))))
-u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T u2 (THead (Flat
-Appl) x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat
-Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) x1)).(let H8 \def
-(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl)
-t1 (THead (Flat Appl) v (THead (Bind b) u t)))) t2) \to (\forall (P:
-Prop).P))) H2 (THead (Flat Appl) x0 x1) H5) in (eq_ind_r T (THead (Flat Appl)
-x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1
-(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) t2)) (H8
-(iso_head t0 x0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u
-t))) x1 (Flat Appl)) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1
-(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) (THead (Flat
-Appl) x0 x1))) u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
-Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0:
-B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))))).(ex4_4_ind T T
-T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c
-(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u3: T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
-Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0:
-B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))) (pr3 c (THead
-(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl)
-(lift (S O) O v) t)))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2:
-T).(\lambda (x3: T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3)
-u2)).(\lambda (H6: (pr3 c t0 x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t1
-(THead (Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0
-x1))).(\lambda (H8: ((\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind
-b0) u0) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0
-(THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v)
-t)))) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Flat
-Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S
-O) O v) t)))) c (pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u
-(THead (Flat Appl) (lift (S O) O v) t))) (THead (Bind Abst) x0 x1) (H0 c
-(THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso (THeads (Flat Appl) t1 (THead
-(Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 x1))).(\lambda (P:
-Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 (THead (Bind b) u t)
-t1 H9 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c (THead (Flat Appl)
-t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr2_free c (THead
-(Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr0_beta
-x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr)
-x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c t0 x2 H6 (Bind Abbr) x1 x3
-(H8 Abbr x2)) u2 H5)))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda
-(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
-(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0)
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind
-B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u
-t))) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind
-b0) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))) (pr3 c
-(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat
-Appl) (lift (S O) O v) t)))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda
-(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not
-(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t1 (THead (Flat
-Appl) v (THead (Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c
-(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda
-(H8: (pr3 c t0 x4)).(\lambda (H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c
-(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S
-O) O x4) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u
-(THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t (THead (Bind x0) x1 (THead
-(Flat Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl)
-t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t
-(THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0 (THeads
-(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c
-(pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl)
-(lift (S O) O v) t))) (THead (Bind x0) x1 x2) (H0 c (THead (Bind x0) x1 x2)
-H6 (\lambda (H11: (iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead
-(Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (P:
-Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind b) u t) t1
-H11 P)))) t0 Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0)
-x2)) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind
-x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat
-Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl)
-(lift (S O) O t0) x2)) (pr0_upsilon x0 H5 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl
-x1) x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S
-O) O x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat
-Appl) (lift (S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2)
-(pr3_head_12 (CHead c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4)
-(pr3_lift (CHead c (Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c
-(drop_refl c) x1) t0 x4 H8) (Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind
-x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5
-(THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat
-Appl) (lift (S O) O x4) x2)) c (pr3_head_12 c x1 x5 H9 (Bind x0) (THead (Flat
-Appl) (lift (S O) O x4) x2) (THead (Flat Appl) (lift (S O) O x4) x3)
-(pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 (lift (S O) O x4) Appl)) u2
-H7)))))))))))))) H4)) H3))))))))) vs)))))).
-
-theorem pr3_iso_appls_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (vs: TList).(\forall (u:
-T).(\forall (t: T).(let u1 \def (THeads (Flat Appl) vs (THead (Bind b) u t))
-in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to
-(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl)
-(lifts (S O) O vs) t)) u2))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (vs:
-TList).(tlist_ind_rew (\lambda (t: TList).(\forall (u: T).(\forall (t0:
-T).(let u1 \def (THeads (Flat Appl) t (THead (Bind b) u t0)) in (\forall (c:
-C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P:
-Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t)
-t0)) u2))))))))) (\lambda (u: T).(\lambda (t: T).(\lambda (c: C).(\lambda
-(u2: T).(\lambda (H0: (pr3 c (THead (Bind b) u t) u2)).(\lambda (_: (((iso
-(THead (Bind b) u t) u2) \to (\forall (P: Prop).P)))).H0)))))) (\lambda (ts:
-TList).(\lambda (t: T).(\lambda (H0: ((\forall (u: T).(\forall (t0:
-T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) ts (THead
-(Bind b) u t0)) u2) \to ((((iso (THeads (Flat Appl) ts (THead (Bind b) u t0))
-u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat
-Appl) (lifts (S O) O ts) t0)) u2))))))))).(\lambda (u: T).(\lambda (t0:
-T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THeads (Flat Appl)
-(TApp ts t) (THead (Bind b) u t0)) u2)).(\lambda (H2: (((iso (THeads (Flat
-Appl) (TApp ts t) (THead (Bind b) u t0)) u2) \to (\forall (P:
-Prop).P)))).(eq_ind_r TList (TApp (lifts (S O) O ts) (lift (S O) O t))
-(\lambda (t1: TList).(pr3 c (THead (Bind b) u (THeads (Flat Appl) t1 t0))
-u2)) (eq_ind_r T (THeads (Flat Appl) (lifts (S O) O ts) (THead (Flat Appl)
-(lift (S O) O t) t0)) (\lambda (t1: T).(pr3 c (THead (Bind b) u t1) u2)) (let
-H3 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind b) u t0))
-(\lambda (t1: T).(pr3 c t1 u2)) H1 (THeads (Flat Appl) ts (THead (Flat Appl)
-t (THead (Bind b) u t0))) (theads_tapp (Flat Appl) ts t (THead (Bind b) u
-t0))) in (let H4 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind
-b) u t0)) (\lambda (t1: T).((iso t1 u2) \to (\forall (P: Prop).P))) H2
-(THeads (Flat Appl) ts (THead (Flat Appl) t (THead (Bind b) u t0)))
-(theads_tapp (Flat Appl) ts t (THead (Bind b) u t0))) in (TList_ind (\lambda
-(t1: TList).(((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall
-(u3: T).((pr3 c0 (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to
-((((iso (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to (\forall (P:
-Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O
-t1) t2)) u3)))))))) \to ((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) t
-(THead (Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat
-Appl) t (THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c
-(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t1) (THead (Flat Appl)
-(lift (S O) O t) t0))) u2))))) (\lambda (_: ((\forall (u0: T).(\forall (t1:
-T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) TNil (THead
-(Bind b) u0 t1)) u3) \to ((((iso (THeads (Flat Appl) TNil (THead (Bind b) u0
-t1)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads
-(Flat Appl) (lifts (S O) O TNil) t1)) u3))))))))).(\lambda (H6: (pr3 c
-(THeads (Flat Appl) TNil (THead (Flat Appl) t (THead (Bind b) u t0)))
-u2)).(\lambda (H7: (((iso (THeads (Flat Appl) TNil (THead (Flat Appl) t
-(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b
-H t u t0 c u2 H6 H7)))) (\lambda (t1: T).(\lambda (ts0: TList).(\lambda (_:
-((((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall (u3: T).((pr3
-c0 (THeads (Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads
-(Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to
-(pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O ts0) t2))
-u3)))))))) \to ((pr3 c (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead
-(Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) ts0 (THead (Flat Appl) t
-(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead
-(Bind b) u (THeads (Flat Appl) (lifts (S O) O ts0) (THead (Flat Appl) (lift
-(S O) O t) t0))) u2)))))).(\lambda (H5: ((\forall (u0: T).(\forall (t2:
-T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) (TCons t1
-ts0) (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads (Flat Appl) (TCons t1
-ts0) (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0
-(THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O (TCons t1 ts0)) t2))
-u3))))))))).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t1 ts0) (THead
-(Flat Appl) t (THead (Bind b) u t0))) u2)).(\lambda (H7: (((iso (THeads (Flat
-Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) u2) \to
-(\forall (P: Prop).P)))).(H5 u (THead (Flat Appl) (lift (S O) O t) t0) c u2
-(pr3_iso_appls_appl_bind b H t u t0 (TCons t1 ts0) c u2 H6 H7) (\lambda (H8:
-(iso (THeads (Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl)
-(lift (S O) O t) t0))) u2)).(\lambda (P: Prop).(H7 (iso_trans (THeads (Flat
-Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) (THeads
-(Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl) (lift (S O) O
-t) t0))) (iso_head t1 t1 (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead
-(Bind b) u t0))) (THeads (Flat Appl) ts0 (THead (Bind b) u (THead (Flat Appl)
-(lift (S O) O t) t0))) (Flat Appl)) u2 H8) P)))))))))) ts H0 H3 H4))) (THeads
-(Flat Appl) (TApp (lifts (S O) O ts) (lift (S O) O t)) t0) (theads_tapp (Flat
-Appl) (lifts (S O) O ts) (lift (S O) O t) t0)) (lifts (S O) O (TApp ts t))
-(lifts_tapp (S O) O t ts))))))))))) vs))).
-
-theorem pr3_iso_beta:
- \forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 \def (THead (Flat
-Appl) v (THead (Bind Abst) w t)) in (\forall (c: C).(\forall (u2: T).((pr3 c
-u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind
-Abbr) v t) u2))))))))
-\def
- \lambda (v: T).(\lambda (w: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2:
-T).(\lambda (H: (pr3 c (THead (Flat Appl) v (THead (Bind Abst) w t))
-u2)).(\lambda (H0: (((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) u2)
-\to (\forall (P: Prop).P)))).(let H1 \def (pr3_gen_appl c v (THead (Bind
-Abst) w t) u2 H) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq
-T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) w t) t2))))
-(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2:
-T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst)
-w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind
-b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(_: T).(pr3 c v u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c
-(THead (Bind Abbr) v t) u2) (\lambda (H2: (ex3_2 T T (\lambda (u3:
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c
-(THead (Bind Abst) w t) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2:
-T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_:
-T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst)
-w t) t2))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H3: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v
-x0)).(\lambda (_: (pr3 c (THead (Bind Abst) w t) x1)).(let H6 \def (eq_ind T
-u2 (\lambda (t0: T).((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) t0)
-\to (\forall (P: Prop).P))) H0 (THead (Flat Appl) x0 x1) H3) in (eq_ind_r T
-(THead (Flat Appl) x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind Abbr) v t)
-t0)) (H6 (iso_head v x0 (THead (Bind Abst) w t) x1 (Flat Appl)) (pr3 c (THead
-(Bind Abbr) v t) (THead (Flat Appl) x0 x1))) u2 H3))))))) H2)) (\lambda (H2:
-(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2:
-T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst)
-w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind
-b) u) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2)))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v
-u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
-(u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Bind Abbr) v t)
-u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
-T).(\lambda (H3: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H4: (pr3 c v
-x2)).(\lambda (H5: (pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) x0
-x1))).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b)
-u) x1 x3))))).(let H7 \def (pr3_gen_abst c w t (THead (Bind Abst) x0 x1) H5)
-in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst)
-x0 x1) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w
-u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3
-(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda
-(x4: T).(\lambda (x5: T).(\lambda (H8: (eq T (THead (Bind Abst) x0 x1) (THead
-(Bind Abst) x4 x5))).(\lambda (H9: (pr3 c w x4)).(\lambda (H10: ((\forall (b:
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x5))))).(let H11 \def (f_equal
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0]))
-(THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in ((let H12 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0)
-\Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in
-(\lambda (H13: (eq T x0 x4)).(let H14 \def (eq_ind_r T x5 (\lambda (t0:
-T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) H10 x1
-H12) in (let H15 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c w t0)) H9 x0
-H13) in (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) v t) c
-(pr3_head_12 c v x2 H4 (Bind Abbr) t x3 (pr3_t x1 t (CHead c (Bind Abbr) x2)
-(H14 Abbr x2) x3 (H6 Abbr x2))) u2 H3))))) H11))))))) H7)))))))))) H2))
-(\lambda (H2: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead (Bind b) y1
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
-T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat
-Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
-(THead (Bind Abst) w t) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2))
-u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
-y2) z1 z2))))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: B).(\lambda
-(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
-T).(\lambda (H3: (not (eq B x0 Abst))).(\lambda (H4: (pr3 c (THead (Bind
-Abst) w t) (THead (Bind x0) x1 x2))).(\lambda (H5: (pr3 c (THead (Bind x0) x5
-(THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (_: (pr3 c v
-x4)).(\lambda (_: (pr3 c x1 x5)).(\lambda (H8: (pr3 (CHead c (Bind x0) x5) x2
-x3)).(let H9 \def (pr3_gen_abst c w t (THead (Bind x0) x1 x2) H4) in
-(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1
-x2) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w
-u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3
-(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda
-(x6: T).(\lambda (x7: T).(\lambda (H10: (eq T (THead (Bind x0) x1 x2) (THead
-(Bind Abst) x6 x7))).(\lambda (H11: (pr3 c w x6)).(\lambda (H12: ((\forall
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x7))))).(let H13 \def
-(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with
-[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
-\Rightarrow b | (Flat _) \Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead
-(Bind Abst) x6 x7) H10) in ((let H14 \def (f_equal T T (\lambda (e: T).(match
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _)
-\Rightarrow x1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind x0) x1 x2)
-(THead (Bind Abst) x6 x7) H10) in ((let H15 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 |
-(TLRef _) \Rightarrow x2 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind x0)
-x1 x2) (THead (Bind Abst) x6 x7) H10) in (\lambda (H16: (eq T x1
-x6)).(\lambda (H17: (eq B x0 Abst)).(let H18 \def (eq_ind_r T x7 (\lambda
-(t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0))))
-H12 x2 H15) in (let H19 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c w t0))
-H11 x1 H16) in (let H20 \def (eq_ind B x0 (\lambda (b: B).(pr3 (CHead c (Bind
-b) x5) x2 x3)) H8 Abst H17) in (let H21 \def (eq_ind B x0 (\lambda (b:
-B).(pr3 c (THead (Bind b) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2))
-H5 Abst H17) in (let H22 \def (eq_ind B x0 (\lambda (b: B).(not (eq B b
-Abst))) H3 Abst H17) in (let H23 \def (match (H22 (refl_equal B Abst)) in
-False return (\lambda (_: False).(pr3 c (THead (Bind Abbr) v t) u2)) with [])
-in H23))))))))) H14)) H13))))))) H9)))))))))))))) H2)) H1)))))))).
-
-theorem pr3_iso_appls_beta:
- \forall (us: TList).(\forall (v: T).(\forall (w: T).(\forall (t: T).(let u1
-\def (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) w t))) in
-(\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to
-(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) us (THead (Bind Abbr)
-v t)) u2)))))))))
-\def
- \lambda (us: TList).(TList_ind (\lambda (t: TList).(\forall (v: T).(\forall
-(w: T).(\forall (t0: T).(let u1 \def (THeads (Flat Appl) t (THead (Flat Appl)
-v (THead (Bind Abst) w t0))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1
-u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat
-Appl) t (THead (Bind Abbr) v t0)) u2)))))))))) (\lambda (v: T).(\lambda (w:
-T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H: (pr3 c
-(THead (Flat Appl) v (THead (Bind Abst) w t)) u2)).(\lambda (H0: (((iso
-(THead (Flat Appl) v (THead (Bind Abst) w t)) u2) \to (\forall (P:
-Prop).P)))).(pr3_iso_beta v w t c u2 H H0)))))))) (\lambda (t: T).(\lambda
-(t0: TList).(\lambda (H: ((\forall (v: T).(\forall (w: T).(\forall (t1:
-T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (THead
-(Flat Appl) v (THead (Bind Abst) w t1))) u2) \to ((((iso (THeads (Flat Appl)
-t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) u2) \to (\forall (P:
-Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))
-u2)))))))))).(\lambda (v: T).(\lambda (w: T).(\lambda (t1: T).(\lambda (c:
-C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) t (THeads (Flat
-Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) u2)).(\lambda (H1:
-(((iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Flat Appl) v
-(THead (Bind Abst) w t1)))) u2) \to (\forall (P: Prop).P)))).(let H2 \def
-(pr3_gen_appl c t (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind
-Abst) w t1))) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2:
-T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_:
-T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl)
-t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2)))) (ex4_4 T T T T
-(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c
-(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat
-Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b:
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
-(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2)
-(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
-(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3)))
-(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat
-Appl) v (THead (Bind Abst) w t1))) t2))))).(ex3_2_ind T T (\lambda (u3:
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c
-(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2)))
-(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1)))
-u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat
-Appl) x0 x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat
-Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) x1)).(let H7 \def
-(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t (THeads (Flat Appl)
-t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) t2) \to (\forall (P:
-Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl)
-x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0
-(THead (Bind Abbr) v t1))) t2)) (H7 (iso_head t x0 (THeads (Flat Appl) t0
-(THead (Flat Appl) v (THead (Bind Abst) w t1))) x1 (Flat Appl)) (pr3 c (THead
-(Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) (THead (Flat
-Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat
-Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b:
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T T T
-T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c
-(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat
-Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b:
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Flat
-Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c
-(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t x2)).(\lambda (H6: (pr3
-c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))
-(THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b: B).(\forall (u:
-T).(pr3 (CHead c (Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1)
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c
-(pr3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Flat Appl) t
-(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads
-(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind Abst) x0 x1) (H v w t1
-c (THead (Bind Abst) x0 x1) H6 (\lambda (H8: (iso (THeads (Flat Appl) t0
-(THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) x0
-x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1
-(THead (Bind Abst) w t1) t0 H8 P)))) t Appl) (THead (Bind Abbr) t x1)
-(pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Bind Abbr)
-t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead
-(Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 (pr0_refl x1))))) u2
-(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t x1) c (pr3_head_12 c t
-x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3:
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst)
-w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2)))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1
-z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat
-Appl) v (THead (Bind Abst) w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2))
-u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
-y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead
-(Bind Abbr) v t1))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2:
-T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq
-B x0 Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v
-(THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c
-(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda
-(H7: (pr3 c t x4)).(\lambda (H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c
-(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S
-O) O x4) x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr)
-v t1))) c (pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2))
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c
-(pr3_t (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Flat Appl) t
-(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads
-(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind x0) x1 x2) (H v w t1 c
-(THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads (Flat Appl) t0 (THead
-(Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda
-(P: Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind Abst)
-w t1) t0 H10 P)))) t Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O)
-O t) x2)) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead
-(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr2_free c (THead
-(Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl)
-(lift (S O) O t) x2)) (pr0_upsilon x0 H4 t t (pr0_refl t) x1 x1 (pr0_refl x1)
-x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O
-x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl)
-(lift (S O) O t) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12
-(CHead c (Bind x0) x1) (lift (S O) O t) (lift (S O) O x4) (pr3_lift (CHead c
-(Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H7)
-(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift
-(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S
-O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c
-(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2)
-(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5)
-x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))))))) us).
-
-theorem csuba_gen_abst_rev:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c
-(CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H:
-(csuba g c (CHead d1 (Bind Abst) u))).(let H0 \def (match H in csuba return
-(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 c)
-\to ((eq C c1 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq C c
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) with
-[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1:
-(eq C (CSort n) (CHead d1 (Bind Abst) u))).(eq_ind C (CSort n) (\lambda (c0:
-C).((eq C (CSort n) (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq
-C c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda
-(H2: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H3 \def (eq_ind C (CSort
-n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abst)
-u) H2) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) H3))) c H0 H1))) | (csuba_head
-c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) c)).(\lambda
-(H2: (eq C (CHead c2 k u0) (CHead d1 (Bind Abst) u))).(eq_ind C (CHead c1 k
-u0) (\lambda (c0: C).((eq C (CHead c2 k u0) (CHead d1 (Bind Abst) u)) \to
-((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda (H3: (eq C (CHead c2 k
-u0) (CHead d1 (Bind Abst) u))).(let H4 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
-(CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) H3)
-in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
-(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0]))
-(CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) in ((let H6 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1
-(Bind Abst) u) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind Abst)) \to
-((eq T u0 u) \to ((csuba g c1 c0) \to (ex2 C (\lambda (d2: C).(eq C (CHead c1
-k u0) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
-(\lambda (H7: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda (k0:
-K).((eq T u0 u) \to ((csuba g c1 d1) \to (ex2 C (\lambda (d2: C).(eq C (CHead
-c1 k0 u0) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))))
-(\lambda (H8: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((csuba g c1 d1) \to
-(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda (H9: (csuba g c1
-d1)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) u) (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1
-(Bind Abst) u)) H9)) u0 (sym_eq T u0 u H8))) k (sym_eq K k (Bind Abst) H7)))
-c2 (sym_eq C c2 d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a
-H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t)
-c)).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst)
-u))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2
-(Bind Abbr) u0) (CHead d1 (Bind Abst) u)) \to ((csuba g c1 c2) \to ((arity g
-c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c0
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) (\lambda
-(H5: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst) u))).(let H6 \def
-(eq_ind C (CHead c2 (Bind Abbr) u0) (\lambda (e: C).(match e in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (CHead d1 (Bind Abst) u) H5) in (False_ind ((csuba g
-c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C
-(\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1)))))) H6))) c H3 H4 H0 H1 H2)))]) in (H0
-(refl_equal C c) (refl_equal C (CHead d1 (Bind Abst) u)))))))).
-
-theorem csuba_gen_void_rev:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c
-(CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind
-Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H:
-(csuba g c (CHead d1 (Bind Void) u))).(let H0 \def (match H in csuba return
-(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 c)
-\to ((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c
-(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) with
-[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1:
-(eq C (CSort n) (CHead d1 (Bind Void) u))).(eq_ind C (CSort n) (\lambda (c0:
-C).((eq C (CSort n) (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq
-C c0 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda
-(H2: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H3 \def (eq_ind C (CSort
-n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Void)
-u) H2) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind
-Void) u))) (\lambda (d2: C).(csuba g d2 d1))) H3))) c H0 H1))) | (csuba_head
-c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) c)).(\lambda
-(H2: (eq C (CHead c2 k u0) (CHead d1 (Bind Void) u))).(eq_ind C (CHead c1 k
-u0) (\lambda (c0: C).((eq C (CHead c2 k u0) (CHead d1 (Bind Void) u)) \to
-((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Void)
-u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda (H3: (eq C (CHead c2 k
-u0) (CHead d1 (Bind Void) u))).(let H4 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
-(CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3)
-in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
-(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0]))
-(CHead c2 k u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1
-(Bind Void) u) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind Void)) \to
-((eq T u0 u) \to ((csuba g c1 c0) \to (ex2 C (\lambda (d2: C).(eq C (CHead c1
-k u0) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
-(\lambda (H7: (eq K k (Bind Void))).(eq_ind K (Bind Void) (\lambda (k0:
-K).((eq T u0 u) \to ((csuba g c1 d1) \to (ex2 C (\lambda (d2: C).(eq C (CHead
-c1 k0 u0) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))))
-(\lambda (H8: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((csuba g c1 d1) \to
-(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) t) (CHead d2 (Bind Void)
-u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda (H9: (csuba g c1
-d1)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u) (CHead d2
-(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1
-(Bind Void) u)) H9)) u0 (sym_eq T u0 u H8))) k (sym_eq K k (Bind Void) H7)))
-c2 (sym_eq C c2 d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a
-H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t)
-c)).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void)
-u))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2
-(Bind Abbr) u0) (CHead d1 (Bind Void) u)) \to ((csuba g c1 c2) \to ((arity g
-c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c0
-(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) (\lambda
-(H5: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void) u))).(let H6 \def
-(eq_ind C (CHead c2 (Bind Abbr) u0) (\lambda (e: C).(match e in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (CHead d1 (Bind Void) u) H5) in (False_ind ((csuba g
-c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C
-(\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u)))
-(\lambda (d2: C).(csuba g d2 d1)))))) H6))) c H3 H4 H0 H1 H2)))]) in (H0
-(refl_equal C c) (refl_equal C (CHead d1 (Bind Void) u)))))))).
-
-theorem csuba_gen_abbr_rev:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g c
-(CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-a))))))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
-(H: (csuba g c (CHead d1 (Bind Abbr) u1))).(let H0 \def (match H in csuba
-return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C
-c0 c) \to ((eq C c1 (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2:
-C).(eq C c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead
-d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))))))))) with [(csuba_sort n) \Rightarrow (\lambda
-(H0: (eq C (CSort n) c)).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr)
-u1))).(eq_ind C (CSort n) (\lambda (c0: C).((eq C (CSort n) (CHead d1 (Bind
-Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))
-(\lambda (H2: (eq C (CSort n) (CHead d1 (Bind Abbr) u1))).(let H3 \def
-(eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
-False])) I (CHead d1 (Bind Abbr) u1) H2) in (False_ind (or (ex2 C (\lambda
-(d2: C).(eq C (CSort n) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g
-d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C
-(CSort n) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) H3))) c H0 H1))) | (csuba_head
-c1 c2 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) c)).(\lambda
-(H2: (eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1))).(eq_ind C (CHead c1 k
-u) (\lambda (c0: C).((eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1)) \to
-((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))))))
-(\lambda (H3: (eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1))).(let H4 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u)
-(CHead d1 (Bind Abbr) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e:
-C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
-(CHead _ k0 _) \Rightarrow k0])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3)
-in ((let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0]))
-(CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in (eq_ind C d1 (\lambda (c0:
-C).((eq K k (Bind Abbr)) \to ((eq T u u1) \to ((csuba g c1 c0) \to (or (ex2 C
-(\lambda (d2: C).(eq C (CHead c1 k u) (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(eq C (CHead c1 k u) (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))))
-(\lambda (H7: (eq K k (Bind Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0:
-K).((eq T u u1) \to ((csuba g c1 d1) \to (or (ex2 C (\lambda (d2: C).(eq C
-(CHead c1 k0 u) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C
-(CHead c1 k0 u) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))))))) (\lambda (H8: (eq T u
-u1)).(eq_ind T u1 (\lambda (t: T).((csuba g c1 d1) \to (or (ex2 C (\lambda
-(d2: C).(eq C (CHead c1 (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(eq C (CHead c1 (Bind Abbr) t) (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-a)))))))) (\lambda (H9: (csuba g c1 d1)).(or_introl (ex2 C (\lambda (d2:
-C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C
-(\lambda (d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind Abbr) u1))
-H9))) u (sym_eq T u u1 H8))) k (sym_eq K k (Bind Abbr) H7))) c2 (sym_eq C c2
-d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u H2)
-\Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) c)).(\lambda (H4:
-(eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1))).(eq_ind C (CHead
-c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2 (Bind Abbr) u) (CHead d1
-(Bind Abbr) u1)) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to
-((arity g c2 u a) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g a0)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 a0)))))))))))
-(\lambda (H5: (eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1))).(let
-H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c2
-(Bind Abbr) u) (CHead d1 (Bind Abbr) u1) H5) in ((let H7 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind Abbr) u)
-(CHead d1 (Bind Abbr) u1) H5) in (eq_ind C d1 (\lambda (c0: C).((eq T u u1)
-\to ((csuba g c1 c0) \to ((arity g c1 t (asucc g a)) \to ((arity g c0 u a)
-\to (or (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
-A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a0: A).(arity g d1 u1 a0))))))))))) (\lambda (H8: (eq T u u1)).(eq_ind T u1
-(\lambda (t0: T).((csuba g c1 d1) \to ((arity g c1 t (asucc g a)) \to ((arity
-g d1 t0 a) \to (or (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t)
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst)
-t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
-A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a0: A).(arity g d1 u1 a0)))))))))) (\lambda (H9: (csuba g c1 d1)).(\lambda
-(H10: (arity g c1 t (asucc g a))).(\lambda (H11: (arity g d1 u1
-a)).(or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
-A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a0: A).(arity g d1 u1 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g
-a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1
-a0)))) c1 t a (refl_equal C (CHead c1 (Bind Abst) t)) H9 H10 H11))))) u
-(sym_eq T u u1 H8))) c2 (sym_eq C c2 d1 H7))) H6))) c H3 H4 H0 H1 H2)))]) in
-(H0 (refl_equal C c) (refl_equal C (CHead d1 (Bind Abbr) u1)))))))).
-
-theorem csuba_gen_flat_rev:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall
-(f: F).((csuba g c (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
-(f: F).(\lambda (H: (csuba g c (CHead d1 (Flat f) u1))).(let H0 \def (match H
-in csuba return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0
-c1)).((eq C c0 c) \to ((eq C c1 (CHead d1 (Flat f) u1)) \to (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) with [(csuba_sort n)
-\Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1: (eq C (CSort n)
-(CHead d1 (Flat f) u1))).(eq_ind C (CSort n) (\lambda (c0: C).((eq C (CSort
-n) (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(eq C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba
-g d2 d1)))))) (\lambda (H2: (eq C (CSort n) (CHead d1 (Flat f) u1))).(let H3
-\def (eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
-False])) I (CHead d1 (Flat f) u1) H2) in (False_ind (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))) H3))) c H0 H1))) | (csuba_head c1 c2 H0
-k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) c)).(\lambda (H2: (eq C
-(CHead c2 k u) (CHead d1 (Flat f) u1))).(eq_ind C (CHead c1 k u) (\lambda
-(c0: C).((eq C (CHead c2 k u) (CHead d1 (Flat f) u1)) \to ((csuba g c1 c2)
-\to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Flat f)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (H3:
-(eq C (CHead c2 k u) (CHead d1 (Flat f) u1))).(let H4 \def (f_equal C T
-(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Flat
-f) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return
-(\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow
-k0])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u) (CHead d1
-(Flat f) u1) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Flat f)) \to ((eq
-T u u1) \to ((csuba g c1 c0) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(eq C (CHead c1 k u) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1)))))))) (\lambda (H7: (eq K k (Flat f))).(eq_ind K
-(Flat f) (\lambda (k0: K).((eq T u u1) \to ((csuba g c1 d1) \to (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k0 u) (CHead d2 (Flat f)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (H8:
-(eq T u u1)).(eq_ind T u1 (\lambda (t: T).((csuba g c1 d1) \to (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Flat f) t) (CHead d2 (Flat
-f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (\lambda (H9:
-(csuba g c1 d1)).(ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(eq C
-(CHead c1 (Flat f) u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1))) c1 u1 (refl_equal C (CHead c1 (Flat f) u1)) H9)) u
-(sym_eq T u u1 H8))) k (sym_eq K k (Flat f) H7))) c2 (sym_eq C c2 d1 H6)))
-H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u H2) \Rightarrow
-(\lambda (H3: (eq C (CHead c1 (Bind Abst) t) c)).(\lambda (H4: (eq C (CHead
-c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(eq_ind C (CHead c1 (Bind Abst) t)
-(\lambda (c0: C).((eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1)) \to
-((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u a) \to
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Flat f)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) (\lambda (H5:
-(eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind
-C (CHead c2 (Bind Abbr) u) (\lambda (e: C).(match e in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
-k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat
-_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind ((csuba
-g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u a) \to (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) t) (CHead d2
-(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H6)))
-c H3 H4 H0 H1 H2)))]) in (H0 (refl_equal C c) (refl_equal C (CHead d1 (Flat
-f) u1))))))))).
-
-theorem csuba_gen_bind_rev:
- \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall
-(v1: T).((csuba g c2 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2:
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2)))))
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))))))
-\def
- \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda
-(v1: T).(\lambda (H: (csuba g c2 (CHead e1 (Bind b1) v1))).(let H0 \def
-(match H in csuba return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csuba
-? c c0)).((eq C c c2) \to ((eq C c0 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2
-e1)))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n)
-c2)).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind b1) v1))).(eq_ind C (CSort
-n) (\lambda (c: C).((eq C (CSort n) (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2
-e1))))))) (\lambda (H2: (eq C (CSort n) (CHead e1 (Bind b1) v1))).(let H3
-\def (eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
-False])) I (CHead e1 (Bind b1) v1) H2) in (False_ind (ex2_3 B C T (\lambda
-(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 (Bind b2)
-v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))
-H3))) c2 H0 H1))) | (csuba_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C
-(CHead c1 k u) c2)).(\lambda (H2: (eq C (CHead c0 k u) (CHead e1 (Bind b1)
-v1))).(eq_ind C (CHead c1 k u) (\lambda (c: C).((eq C (CHead c0 k u) (CHead
-e1 (Bind b1) v1)) \to ((csuba g c1 c0) \to (ex2_3 B C T (\lambda (b2:
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2)))))
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))))
-(\lambda (H3: (eq C (CHead c0 k u) (CHead e1 (Bind b1) v1))).(let H4 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u)
-(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e:
-C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
-(CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u) (CHead e1 (Bind b1) v1) H3)
-in ((let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c]))
-(CHead c0 k u) (CHead e1 (Bind b1) v1) H3) in (eq_ind C e1 (\lambda (c:
-C).((eq K k (Bind b1)) \to ((eq T u v1) \to ((csuba g c1 c) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k u)
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csuba g e2 e1))))))))) (\lambda (H7: (eq K k (Bind b1))).(eq_ind K (Bind
-b1) (\lambda (k0: K).((eq T u v1) \to ((csuba g c1 e1) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k0 u)
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csuba g e2 e1)))))))) (\lambda (H8: (eq T u v1)).(eq_ind T v1 (\lambda
-(t: T).((csuba g c1 e1) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
-C).(\lambda (v2: T).(eq C (CHead c1 (Bind b1) t) (CHead e2 (Bind b2) v2)))))
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))
-(\lambda (H9: (csuba g c1 e1)).(let H10 \def (eq_ind T u (\lambda (t: T).(eq
-C (CHead c1 k t) c2)) H1 v1 H8) in (let H11 \def (eq_ind K k (\lambda (k0:
-K).(eq C (CHead c1 k0 v1) c2)) H10 (Bind b1) H7) in (let H12 \def (eq_ind_r C
-c2 (\lambda (c: C).(csuba g c (CHead e1 (Bind b1) v1))) H (CHead c1 (Bind b1)
-v1) H11) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
-(v2: T).(eq C (CHead c1 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda
-(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))) b1 c1 v1
-(refl_equal C (CHead c1 (Bind b1) v1)) H9))))) u (sym_eq T u v1 H8))) k
-(sym_eq K k (Bind b1) H7))) c0 (sym_eq C c0 e1 H6))) H5)) H4))) c2 H1 H2
-H0))) | (csuba_abst c1 c0 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C
-(CHead c1 (Bind Abst) t) c2)).(\lambda (H4: (eq C (CHead c0 (Bind Abbr) u)
-(CHead e1 (Bind b1) v1))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c:
-C).((eq C (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1)) \to ((csuba g c1
-c0) \to ((arity g c1 t (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2
-e1)))))))))) (\lambda (H5: (eq C (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1)
-v1))).(let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
-(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0]))
-(CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
-Abbr])])) (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H8
-\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C)
-with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0
-(Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in (eq_ind C e1 (\lambda (c:
-C).((eq B Abbr b1) \to ((eq T u v1) \to ((csuba g c1 c) \to ((arity g c1 t
-(asucc g a)) \to ((arity g c u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda
-(e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2)
-v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2
-e1))))))))))) (\lambda (H9: (eq B Abbr b1)).(eq_ind B Abbr (\lambda (_:
-B).((eq T u v1) \to ((csuba g c1 e1) \to ((arity g c1 t (asucc g a)) \to
-((arity g e1 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
-(v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) v2))))) (\lambda
-(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))))) (\lambda
-(H10: (eq T u v1)).(eq_ind T v1 (\lambda (t0: T).((csuba g c1 e1) \to ((arity
-g c1 t (asucc g a)) \to ((arity g e1 t0 a) \to (ex2_3 B C T (\lambda (b2:
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2
-(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g
-e2 e1))))))))) (\lambda (H11: (csuba g c1 e1)).(\lambda (_: (arity g c1 t
-(asucc g a))).(\lambda (_: (arity g e1 v1 a)).(let H14 \def (eq_ind_r C c2
-(\lambda (c: C).(csuba g c (CHead e1 (Bind b1) v1))) H (CHead c1 (Bind Abst)
-t) H3) in (let H15 \def (eq_ind_r B b1 (\lambda (b: B).(csuba g (CHead c1
-(Bind Abst) t) (CHead e1 (Bind b) v1))) H14 Abbr H9) in (ex2_3_intro B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind
-Abst) t) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
-C).(\lambda (_: T).(csuba g e2 e1)))) Abst c1 t (refl_equal C (CHead c1 (Bind
-Abst) t)) H11)))))) u (sym_eq T u v1 H10))) b1 H9)) c0 (sym_eq C c0 e1 H8)))
-H7)) H6))) c2 H3 H4 H0 H1 H2)))]) in (H0 (refl_equal C c2) (refl_equal C
-(CHead e1 (Bind b1) v1))))))))).
-
-theorem csuba_clear_trans:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c2 c1) \to
-(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1))
-(\lambda (e2: C).(clear c2 e2))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c2
-c1)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear
-c0 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c
-e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n)
-e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e2 e1))
-(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4:
-C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c4
-e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c3
-e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2:
-(clear (CHead c4 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c4 k0 u)
-e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear
-(CHead c3 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c4 (Bind
-b) u) e1)).(eq_ind_r C (CHead c4 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda
-(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind b) u) e2))))
-(ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind b) u))) (\lambda
-(e2: C).(clear (CHead c3 (Bind b) u) e2)) (CHead c3 (Bind b) u) (csuba_head g
-c3 c4 H0 (Bind b) u) (clear_bind b c3 u)) e1 (clear_gen_bind b c4 e1 u H3))))
-(\lambda (f: F).(\lambda (H3: (clear (CHead c4 (Flat f) u) e1)).(let H4 \def
-(H1 e1 (clear_gen_flat f c4 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g
-e2 e1)) (\lambda (e2: C).(clear c3 e2)) (ex2 C (\lambda (e2: C).(csuba g e2
-e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f) u) e2))) (\lambda (x:
-C).(\lambda (H5: (csuba g x e1)).(\lambda (H6: (clear c3 x)).(ex_intro2 C
-(\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f)
-u) e2)) x H5 (clear_flat c3 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3:
-C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall
-(e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda
-(e2: C).(clear c3 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H2:
-(arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u
-a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c4 (Bind Abbr) u)
-e1)).(eq_ind_r C (CHead c4 (Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda
-(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind Abst) t)
-e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind Abbr) u)))
-(\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) e2)) (CHead c3 (Bind Abst)
-t) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abst c3 t)) e1
-(clear_gen_bind Abbr c4 e1 u H4))))))))))))) c2 c1 H)))).
-
-theorem csuba_drop_abst_rev:
- \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i
-O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g
-c2 c1) \to (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1))))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1:
-C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g:
-G).(\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda (d2: C).(drop n O c2
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))))))
-(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1
-(CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0:
-(csuba g c2 c1)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c2 c)) H0
-(CHead d1 (Bind Abst) u) (drop_gen_refl c1 (CHead d1 (Bind Abst) u) H)) in
-(let H_x \def (csuba_gen_abst_rev g d1 c2 u H1) in (let H2 \def H_x in
-(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x: C).(\lambda (H3:
-(eq C c2 (CHead x (Bind Abst) u))).(\lambda (H4: (csuba g x d1)).(eq_ind_r C
-(CHead x (Bind Abst) u) (\lambda (c: C).(ex2 C (\lambda (d2: C).(drop O O c
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (ex_intro2 C
-(\lambda (d2: C).(drop O O (CHead x (Bind Abst) u) (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1)) x (drop_refl (CHead x (Bind Abst) u)) H4)
-c2 H3)))) H2))))))))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1:
-C).(\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abst) u))
-\to (\forall (g: G).(\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda
-(d2: C).(drop n O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
-d1)))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1:
-C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall
-(g: G).(\forall (c2: C).((csuba g c2 c) \to (ex2 C (\lambda (d2: C).(drop (S
-n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))))))))
-(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n)
-O (CSort n0) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2:
-C).(\lambda (_: (csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind
-Abst) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (ex2 C (\lambda (d2:
-C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
-d1))) (\lambda (H2: (eq C (CHead d1 (Bind Abst) u) (CSort n0))).(\lambda (_:
-(eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (match H2 in eq
-return (\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c (CSort n0)) \to (ex2
-C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1)))))) with [refl_equal \Rightarrow (\lambda (H5: (eq C
-(CHead d1 (Bind Abst) u) (CSort n0))).(let H6 \def (eq_ind C (CHead d1 (Bind
-Abst) u) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with
-[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n0)
-H5) in (False_ind (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) H6)))]) in (H5 (refl_equal C
-(CSort n0))))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u)
-H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u:
-T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall
-(c2: C).((csuba g c2 c) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead
-d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))))))))).(\lambda (k:
-K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n)
-O (CHead c k t) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2:
-C).(\lambda (H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: K).((csuba
-g c2 (CHead c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abst) u)) \to
-(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda
-(d2: C).(csuba g d2 d1)))))) (\lambda (b: B).(\lambda (H3: (csuba g c2 (CHead
-c (Bind b) t))).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abst)
-u))).(B_ind (\lambda (b0: B).((csuba g c2 (CHead c (Bind b0) t)) \to ((drop
-(r (Bind b0) n) O c (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2:
-C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
-d1)))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) t))).(\lambda (H6:
-(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abst) u))).(let H_x \def
-(csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C
-(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba
-g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
-C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g c t a))))) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (H8: (ex2 C
-(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba
-g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t)))
-(\lambda (d2: C).(csuba g d2 c)) (ex2 C (\lambda (d2: C).(drop (S n) O c2
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x:
-C).(\lambda (H9: (eq C c2 (CHead x (Bind Abbr) t))).(\lambda (H10: (csuba g x
-c)).(eq_ind_r C (CHead x (Bind Abbr) t) (\lambda (c0: C).(ex2 C (\lambda (d2:
-C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
-d1)))) (let H11 \def (H c d1 u H6 g x H10) in (ex2_ind C (\lambda (d2:
-C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))
-(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x0: C).(\lambda (H12:
-(drop n O x (CHead x0 (Bind Abst) u))).(\lambda (H13: (csuba g x0 d1)).(let
-H14 \def (refl_equal nat (r (Bind Abst) n)) in (let H15 \def (eq_ind nat n
-(\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abst) u))) H12 (r (Bind Abst)
-n) H14) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr)
-t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop
-(Bind Abbr) n x (CHead x0 (Bind Abst) u) H15 t) H13)))))) H11)) c2 H9))))
-H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) (ex2 C
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2:
-A).(\lambda (H9: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H10: (csuba g
-x0 c)).(\lambda (_: (arity g x0 x1 (asucc g x2))).(\lambda (_: (arity g c t
-x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c0: C).(ex2 C (\lambda
-(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g
-d2 d1)))) (let H13 \def (H c d1 u H6 g x0 H10) in (ex2_ind C (\lambda (d2:
-C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))
-(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x: C).(\lambda
-(H14: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H15: (csuba g x
-d1)).(let H16 \def (refl_equal nat (r (Bind Abst) n)) in (let H17 \def
-(eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x (Bind Abst) u))) H14
-(r (Bind Abst) n) H16) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead
-x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
-d1)) x (drop_drop (Bind Abst) n x0 (CHead x (Bind Abst) u) H17 x1) H15))))))
-H13)) c2 H9)))))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind
-Abst) t))).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst)
-u))).(let H_x \def (csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in
-(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2:
-C).(csuba g d2 c)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x: C).(\lambda (H8:
-(eq C c2 (CHead x (Bind Abst) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C
-(CHead x (Bind Abst) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n)
-O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H10
-\def (H c d1 u H6 g x H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead
-d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda
-(d2: C).(csuba g d2 d1))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead
-x0 (Bind Abst) u))).(\lambda (H12: (csuba g x0 d1)).(let H13 \def (refl_equal
-nat (r (Bind Abst) n)) in (let H14 \def (eq_ind nat n (\lambda (n0:
-nat).(drop n0 O x (CHead x0 (Bind Abst) u))) H11 (r (Bind Abst) n) H13) in
-(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop (Bind Abst)
-n x (CHead x0 (Bind Abst) u) H14 t) H12)))))) H10)) c2 H8)))) H7)))))
-(\lambda (H5: (csuba g c2 (CHead c (Bind Void) t))).(\lambda (H6: (drop (r
-(Bind Void) n) O c (CHead d1 (Bind Abst) u))).(let H_x \def
-(csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda
-(d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: C).(csuba g d2 c))
-(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda
-(d2: C).(csuba g d2 d1))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x
-(Bind Void) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C (CHead x (Bind
-Void) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H10 \def (H c d1 u
-H6 g x H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(drop (S n) O
-(CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g
-d2 d1))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead x0 (Bind Abst)
-u))).(\lambda (H12: (csuba g x0 d1)).(let H13 \def (refl_equal nat (r (Bind
-Abst) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x
-(CHead x0 (Bind Abst) u))) H11 (r (Bind Abst) n) H13) in (ex_intro2 C
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop (Bind Void) n x (CHead
-x0 (Bind Abst) u) H14 t) H12)))))) H10)) c2 H8)))) H7))))) b H3 H4))))
-(\lambda (f: F).(\lambda (H3: (csuba g c2 (CHead c (Flat f) t))).(\lambda
-(H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abst) u))).(let H_x \def
-(csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T
-(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 c))) (ex2 C (\lambda (d2: C).(drop (S n)
-O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda
-(x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f)
-x1))).(\lambda (H7: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Flat f) x1)
-(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H8 \def (H0 d1 u H4 g x0
-H7) in (ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(drop (S n) O
-(CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g
-d2 d1))) (\lambda (x: C).(\lambda (H9: (drop (S n) O x0 (CHead x (Bind Abst)
-u))).(\lambda (H10: (csuba g x d1)).(ex_intro2 C (\lambda (d2: C).(drop (S n)
-O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g
-d2 d1)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u) H9 x1) H10))))
-H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abst) u) t n
-H1)))))))))))) c1)))) i).
-
-theorem csuba_drop_abbr_rev:
- \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i
-O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba
-g c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-a)))))))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1:
-C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g:
-G).(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop n
-O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1:
-T).(\lambda (H: (drop O O c1 (CHead d1 (Bind Abbr) u1))).(\lambda (g:
-G).(\lambda (c2: C).(\lambda (H0: (csuba g c2 c1)).(let H1 \def (eq_ind C c1
-(\lambda (c: C).(csuba g c2 c)) H0 (CHead d1 (Bind Abbr) u1) (drop_gen_refl
-c1 (CHead d1 (Bind Abbr) u1) H)) in (let H_x \def (csuba_gen_abbr_rev g d1 c2
-u1 H1) in (let H2 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H3:
-(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2:
-C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O
-O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x
-(Bind Abbr) u1))).(\lambda (H5: (csuba g x d1)).(eq_ind_r C (CHead x (Bind
-Abbr) u1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-a))))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abbr)
-u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x (Bind
-Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2:
-C).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1)) x (drop_refl (CHead x (Bind Abbr) u1)) H5)) c2
-H4)))) H3)) (\lambda (H3: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H4: (eq C c2 (CHead x0 (Bind
-Abst) x1))).(\lambda (H5: (csuba g x0 d1)).(\lambda (H6: (arity g x0 x1
-(asucc g x2))).(\lambda (H7: (arity g d1 u1 x2)).(eq_ind_r C (CHead x0 (Bind
-Abst) x1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-a))))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Abst)
-x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind
-Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abst) x1)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_refl (CHead x0 (Bind Abst) x1)) H5
-H6 H7)) c2 H4)))))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H:
-((\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop n O c1 (CHead d1
-(Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c1) \to
-(or (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-a))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1:
-C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to (\forall
-(g: G).(\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop
-(S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
-c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))))))))))) (\lambda (n0: nat).(\lambda (d1:
-C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) (CHead d1 (Bind
-Abbr) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (csuba g c2 (CSort
-n0))).(and3_ind (eq C (CHead d1 (Bind Abbr) u1) (CSort n0)) (eq nat (S n) O)
-(eq nat O O) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))
-(\lambda (H2: (eq C (CHead d1 (Bind Abbr) u1) (CSort n0))).(\lambda (_: (eq
-nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (match H2 in eq return
-(\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c (CSort n0)) \to (or (ex2 C
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))) with
-[refl_equal \Rightarrow (\lambda (H5: (eq C (CHead d1 (Bind Abbr) u1) (CSort
-n0))).(let H6 \def (eq_ind C (CHead d1 (Bind Abbr) u1) (\lambda (e: C).(match
-e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
-(CHead _ _ _) \Rightarrow True])) I (CSort n0) H5) in (False_ind (or (ex2 C
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) H6)))]) in (H5
-(refl_equal C (CSort n0))))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind
-Abbr) u1) H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1:
-C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to (\forall
-(g: G).(\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop
-(S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
-c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))))))))))))).(\lambda (k: K).(\lambda (t:
-T).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H1: (drop (S n) O (CHead c k
-t) (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2:
-(csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: K).((csuba g c2 (CHead c k0
-t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))) (\lambda (b:
-B).(\lambda (H3: (csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop (r
-(Bind b) n) O c (CHead d1 (Bind Abbr) u1))).(B_ind (\lambda (b0: B).((csuba g
-c2 (CHead c (Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind
-Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-a))))))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) t))).(\lambda (H6:
-(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def
-(csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C
-(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba
-g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
-C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g c t a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind
-Abbr) t))) (\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq
-C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x:
-C).(\lambda (H9: (eq C c2 (CHead x (Bind Abbr) t))).(\lambda (H10: (csuba g x
-c)).(eq_ind_r C (CHead x (Bind Abbr) t) (\lambda (c0: C).(or (ex2 C (\lambda
-(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba
-g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))))) (let H11 \def (H c d1 u1 H6 g
-x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or
-(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n
-O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind
-C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind
-Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
-(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x0:
-C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14:
-(csuba g x0 d1)).(let H15 \def (refl_equal nat (r (Bind Abst) n)) in (let H16
-\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr)
-u1))) H13 (r (Bind Abst) n) H15) in (or_introl (ex2 C (\lambda (d2: C).(drop
-(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop (Bind Abbr)
-n x (CHead x0 (Bind Abbr) u1) H16 t) H14))))))) H12)) (\lambda (H12: (ex4_3 C
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or (ex2 C (\lambda (d2: C).(drop
-(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x
-(CHead x0 (Bind Abst) x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15:
-(arity g x0 x1 (asucc g x2))).(\lambda (H16: (arity g d1 u1 x2)).(let H17
-\def (refl_equal nat (r (Bind Abst) n)) in (let H18 \def (eq_ind nat n
-(\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abst) x1))) H13 (r (Bind
-Abst) n) H17) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x
-(Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
-(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex4_3_intro C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x
-(Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Abbr) n
-x (CHead x0 (Bind Abst) x1) H18 t) H14 H15 H16))))))))))) H12)) H11)) c2
-H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc
-g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a))))
-(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2
-(CHead x0 (Bind Abst) x1))).(\lambda (H10: (csuba g x0 c)).(\lambda (_:
-(arity g x0 x1 (asucc g x2))).(\lambda (_: (arity g c t x2)).(eq_ind_r C
-(CHead x0 (Bind Abst) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop
-(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
-c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in
-(or_ind (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))))) (\lambda (H14: (ex2 C (\lambda (d2: C).(drop n
-O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind
-C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind
-Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
-(CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x:
-C).(\lambda (H15: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H16:
-(csuba g x d1)).(let H17 \def (refl_equal nat (r (Bind Abst) n)) in (let H18
-\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x (Bind Abbr)
-u1))) H15 (r (Bind Abst) n) H17) in (or_introl (ex2 C (\lambda (d2: C).(drop
-(S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind
-Abst) n x0 (CHead x (Bind Abbr) u1) H18 x1) H16))))))) H14)) (\lambda (H14:
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5:
-A).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Abst) x4))).(\lambda (H16:
-(csuba g x3 d1)).(\lambda (H17: (arity g x3 x4 (asucc g x5))).(\lambda (H18:
-(arity g d1 u1 x5)).(let H19 \def (refl_equal nat (r (Bind Abst) n)) in (let
-H20 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x3 (Bind Abst)
-x4))) H15 (r (Bind Abst) n) H19) in (or_intror (ex2 C (\lambda (d2: C).(drop
-(S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
-n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x3 x4 x5
-(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Abst) x4) H20 x1) H16 H17
-H18))))))))))) H14)) H13)) c2 H9)))))))) H8)) H7))))) (\lambda (H5: (csuba g
-c2 (CHead c (Bind Abst) t))).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead
-d1 (Bind Abbr) u1))).(let H_x \def (csuba_gen_abst_rev g c c2 t H5) in (let
-H7 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst)
-t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n)
-O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead
-d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x
-(Bind Abst) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C (CHead x (Bind
-Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda
-(d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n
-O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead
-x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
-n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H11:
-(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr)
-u1))).(\lambda (H13: (csuba g x0 d1)).(let H14 \def (refl_equal nat (r (Bind
-Abst) n)) in (let H15 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x
-(CHead x0 (Bind Abbr) u1))) H12 (r (Bind Abst) n) H14) in (or_introl (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O
-(CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g
-d2 d1)) x0 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abbr) u1) H15 t)
-H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t)
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x
-(Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x0: C).(\lambda (x1:
-T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst)
-x1))).(\lambda (H13: (csuba g x0 d1)).(\lambda (H14: (arity g x0 x1 (asucc g
-x2))).(\lambda (H15: (arity g d1 u1 x2)).(let H16 \def (refl_equal nat (r
-(Bind Abst) n)) in (let H17 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O
-x (CHead x0 (Bind Abst) x1))) H12 (r (Bind Abst) n) H16) in (or_intror (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))) x0 x1 x2 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abst) x1) H17 t)
-H13 H14 H15))))))))))) H11)) H10)) c2 H8)))) H7))))) (\lambda (H5: (csuba g
-c2 (CHead c (Bind Void) t))).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead
-d1 (Bind Abbr) u1))).(let H_x \def (csuba_gen_void_rev g c c2 t H5) in (let
-H7 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void)
-t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n)
-O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead
-d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x
-(Bind Void) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C (CHead x (Bind
-Void) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda
-(d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n
-O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead
-x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
-n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H11:
-(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2:
-C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr)
-u1))).(\lambda (H13: (csuba g x0 d1)).(let H14 \def (refl_equal nat (r (Bind
-Abst) n)) in (let H15 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x
-(CHead x0 (Bind Abbr) u1))) H12 (r (Bind Abst) n) H14) in (or_introl (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O
-(CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g
-d2 d1)) x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abbr) u1) H15 t)
-H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t)
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x
-(Bind Void) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x0: C).(\lambda (x1:
-T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst)
-x1))).(\lambda (H13: (csuba g x0 d1)).(\lambda (H14: (arity g x0 x1 (asucc g
-x2))).(\lambda (H15: (arity g d1 u1 x2)).(let H16 \def (refl_equal nat (r
-(Bind Abst) n)) in (let H17 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O
-x (CHead x0 (Bind Abst) x1))) H12 (r (Bind Abst) n) H16) in (or_intror (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))) x0 x1 x2 (drop_drop (Bind Void) n x (CHead x0 (Bind Abst) x1) H17 t)
-H13 H14 H15))))))))))) H11)) H10)) c2 H8)))) H7))))) b H3 H4)))) (\lambda (f:
-F).(\lambda (H3: (csuba g c2 (CHead c (Flat f) t))).(\lambda (H4: (drop (r
-(Flat f) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def (csuba_gen_flat_rev
-g c c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T (\lambda (d2:
-C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O
-c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead
-d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6:
-(eq C c2 (CHead x0 (Flat f) x1))).(\lambda (H7: (csuba g x0 c)).(eq_ind_r C
-(CHead x0 (Flat f) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S
-n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3
-C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))))) (let H8 \def (H0 d1 u1 H4 g x0 H7) in (or_ind
-(ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or (ex2 C
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S
-n) O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1)))).(ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S
-n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))
-(\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abbr)
-u1))).(\lambda (H11: (csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop
-(S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Flat f) n
-x0 (CHead x (Bind Abbr) u1) H10 x1) H11))))) H9)) (\lambda (H9: (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(drop (S n) O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or (ex2 C (\lambda (d2: C).(drop
-(S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))
-(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H10: (drop (S n)
-O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H11: (csuba g x2 d1)).(\lambda
-(H12: (arity g x2 x3 (asucc g x4))).(\lambda (H13: (arity g d1 u1
-x4)).(or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1)
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
-(Flat f) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))) x2 x3 x4 (drop_drop (Flat f) n x0 (CHead x2 (Bind
-Abst) x3) H10 x1) H11 H12 H13))))))))) H9)) H8)) c2 H6))))) H5)))))) k H2
-(drop_gen_drop k c (CHead d1 (Bind Abbr) u1) t n H1)))))))))))) c1)))) i).
-
-theorem csuba_getl_abst_rev:
- \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall
-(i: nat).((getl i c1 (CHead d1 (Bind Abst) u)) \to (\forall (c2: C).((csuba g
-c2 c1) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda
-(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u))).(let H0 \def
-(getl_gen_all c1 (CHead d1 (Bind Abst) u) i H) in (ex2_ind C (\lambda (e:
-C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u)))
-(\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda (d2: C).(getl i c2
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda (x:
-C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind
-Abst) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1
-(Bind Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda
-(d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
-d1)))))))) (\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda
-(H4: (clear (CSort n) (CHead d1 (Bind Abst) u))).(clear_gen_sort (CHead d1
-(Bind Abst) u) n H4 (\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda
-(d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
-d1))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0
-(CHead d1 (Bind Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (ex2 C
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3:
-(drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1
-(Bind Abst) u))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to
-((clear (CHead x0 k0 t) (CHead d1 (Bind Abst) u)) \to (\forall (c2:
-C).((csuba g c2 c1) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) (\lambda (b: B).(\lambda
-(H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0
-(Bind b) t) (CHead d1 (Bind Abst) u))).(let H7 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 |
-(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t)
-(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in ((let H8 \def
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 in K
-return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow Abst])])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t)
-(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in ((let H9 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind
-Abst) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u)
-t H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda
-(c2: C).(\lambda (H12: (csuba g c2 c1)).(let H13 \def (eq_ind_r T t (\lambda
-(t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def
-(eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u))) H13 Abst
-H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c
-(Bind Abst) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abst_rev i c1 d1 u
-H15 g c2 H12) in (ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(getl i
-c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda
-(x1: C).(\lambda (H17: (drop i O c2 (CHead x1 (Bind Abst) u))).(\lambda (H18:
-(csuba g x1 d1)).(ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x1 (getl_intro i c2 (CHead x1
-(Bind Abst) u) (CHead x1 (Bind Abst) u) H17 (clear_bind Abst x1 u)) H18))))
-H16)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead
-x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind
-Abst) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c
-(CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) \to (ex2 C
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop n
-O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (ex2 C
-(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
-C).(csuba g d2 d1)))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead
-x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g c2 x1)).(let H10
-\def (eq_ind C x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead x0 (Flat f) t)
-(drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def (clear_flat x0
-(CHead d1 (Bind Abst) u) (clear_gen_flat f x0 (CHead d1 (Bind Abst) u) t H6)
-f t) in (let H11 \def (csuba_clear_trans g (CHead x0 (Flat f) t) c2 H10
-(CHead d1 (Bind Abst) u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g e2
-(CHead d1 (Bind Abst) u))) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda
-(d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
-d1))) (\lambda (x2: C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abst)
-u))).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abst_rev g d1 x2
-u H12) in (let H14 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2:
-C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
-(\lambda (x3: C).(\lambda (H15: (eq C x2 (CHead x3 (Bind Abst) u))).(\lambda
-(H16: (csuba g x3 d1)).(let H17 \def (eq_ind C x2 (\lambda (c: C).(clear c2
-c)) H13 (CHead x3 (Bind Abst) u) H15) in (ex_intro2 C (\lambda (d2: C).(getl
-O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x3
-(getl_intro O c2 (CHead x3 (Bind Abst) u) c2 (drop_refl c2) H17) H16)))))
-H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1:
-C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1)
-\to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda
-(d2: C).(csuba g d2 d1))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O
-x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g c2
-x1)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B
-C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind
-b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead
-x0 (Flat f) t))))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x2: B).(\lambda (x3:
-C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2)
-x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 \def
-(csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C
-(\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) (\lambda (e2:
-C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x5: C).(\lambda (H15:
-(csuba g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: (clear c2 x5)).(let H_x
-\def (csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let H17 \def H_x in
-(ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csuba g e2 x3)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x6: B).(\lambda (x7:
-C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6)
-x8))).(\lambda (H19: (csuba g x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c:
-C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13
-x7 H19) in (ex2_ind C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(getl (S n) c2
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x9:
-C).(\lambda (H22: (getl n x7 (CHead x9 (Bind Abst) u))).(\lambda (H23: (csuba
-g x9 d1)).(ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20
-(CHead x9 (Bind Abst) u) n H22) H23)))) H21)))))))) H17)))))) H14)))))))
-H11)))))))) i) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))).
-
-theorem csuba_getl_abbr_rev:
- \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall
-(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba
-g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda
-(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u1))).(let H0 \def
-(getl_gen_all c1 (CHead d1 (Bind Abbr) u1) i H) in (ex2_ind C (\lambda (e:
-C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u1)))
-(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))))))) (\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2:
-(clear x (CHead d1 (Bind Abbr) u1))).(C_ind (\lambda (c: C).((drop i O c1 c)
-\to ((clear c (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2
-c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))))) (\lambda
-(n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n)
-(CHead d1 (Bind Abbr) u1))).(clear_gen_sort (CHead d1 (Bind Abbr) u1) n H4
-(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear
-x0 (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 c1) \to (or
-(ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))))))))))).(\lambda (k: K).(\lambda
-(t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear
-(CHead x0 k t) (CHead d1 (Bind Abbr) u1))).(K_ind (\lambda (k0: K).((drop i O
-c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind Abbr) u1))
-\to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i
-c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))))))))) (\lambda (b: B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b)
-t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 (Bind Abbr)
-u1))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c]))
-(CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead
-d1 (Bind Abbr) u1) t H6)) in ((let H8 \def (f_equal C B (\lambda (e:
-C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr |
-(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with
-[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d1 (Bind
-Abbr) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr)
-u1) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e in C return
-(\lambda (_: C).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t0)
-\Rightarrow t0])) (CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b) t)
-(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) in (\lambda (H10: (eq B
-Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: C).(\lambda (H12: (csuba
-g c2 c1)).(let H13 \def (eq_ind_r T t (\lambda (t0: T).(drop i O c1 (CHead x0
-(Bind b) t0))) H5 u1 H9) in (let H14 \def (eq_ind_r B b (\lambda (b0:
-B).(drop i O c1 (CHead x0 (Bind b0) u1))) H13 Abbr H10) in (let H15 \def
-(eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c (Bind Abbr) u1))) H14 d1
-H11) in (let H16 \def (csuba_drop_abbr_rev i c1 d1 u1 H15 g c2 H12) in
-(or_ind (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or (ex2 C
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H17: (ex2 C (\lambda
-(d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(getl i c2
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))))) (\lambda (x1: C).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abbr)
-u1))).(\lambda (H19: (csuba g x1 d1)).(or_introl (ex2 C (\lambda (d2:
-C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x1 (getl_intro i c2
-(CHead x1 (Bind Abbr) u1) (CHead x1 (Bind Abbr) u1) H18 (clear_bind Abbr x1
-u1)) H19))))) H17)) (\lambda (H17: (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x1:
-C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1
-(Bind Abst) x2))).(\lambda (H19: (csuba g x1 d1)).(\lambda (H20: (arity g x1
-x2 (asucc g x3))).(\lambda (H21: (arity g d1 u1 x3)).(or_intror (ex2 C
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x1 x2 x3
-(getl_intro i c2 (CHead x1 (Bind Abst) x2) (CHead x1 (Bind Abst) x2) H18
-(clear_bind Abst x1 x2)) H19 H20 H21))))))))) H17)) H16)))))))))) H8))
-H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f)
-t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr)
-u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0
-(Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda
-(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i
-c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))))))))) (nat_ind (\lambda (n: nat).(\forall (x1:
-C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1)
-\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))))) (\lambda
-(x1: C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2:
-C).(\lambda (H9: (csuba g c2 x1)).(let H10 \def (eq_ind C x1 (\lambda (c:
-C).(csuba g c2 c)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat
-f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abbr) u1)
-(clear_gen_flat f x0 (CHead d1 (Bind Abbr) u1) t H6) f t) in (let H11 \def
-(csuba_clear_trans g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u1)
-H_y) in (ex2_ind C (\lambda (e2: C).(csuba g e2 (CHead d1 (Bind Abbr) u1)))
-(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))
-(\lambda (x2: C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abbr)
-u1))).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abbr_rev g d1 x2
-u1 H12) in (let H14 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C x2
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a))))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H15:
-(ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2:
-C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead
-x3 (Bind Abbr) u1))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C
-x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) u1) H16) in
-(or_introl (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C
-(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abbr) u1) c2
-(drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(eq C x2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or (ex2 C (\lambda (d2: C).(getl
-O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H16:
-(eq C x2 (CHead x3 (Bind Abst) x4))).(\lambda (H17: (csuba g x3 d1)).(\lambda
-(H18: (arity g x3 x4 (asucc g x5))).(\lambda (H19: (arity g d1 u1 x5)).(let
-H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst)
-x4) H16) in (or_intror (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O
-c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))) x3 x4 x5 (getl_intro O c2 (CHead x3 (Bind Abst)
-x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) H14)))))) H11))))))))
-(\lambda (n: nat).(\lambda (H8: ((\forall (x1: C).((drop n O x1 (CHead x0
-(Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or (ex2 C (\lambda
-(d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n
-c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S
-n) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g c2
-x1)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B
-C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind
-b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead
-x0 (Flat f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))
-(\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1
-(CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f)
-t))).(let H14 \def (csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4)
-H12) in (ex2_ind C (\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4)))
-(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))))) (\lambda (x5: C).(\lambda (H15: (csuba g x5 (CHead x3 (Bind x2)
-x4))).(\lambda (H16: (clear c2 x5)).(let H_x \def (csuba_gen_bind_rev g x2 x3
-x5 x4 H15) in (let H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda
-(e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_:
-B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 x3)))) (or (ex2 C (\lambda
-(d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g
-d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl
-(S n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x6: B).(\lambda (x7:
-C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6)
-x8))).(\lambda (H19: (csuba g x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c:
-C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13
-x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or
-(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H22:
-(ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2:
-C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S
-n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))))) (\lambda (x9: C).(\lambda (H23: (getl n x7
-(CHead x9 (Bind Abbr) u1))).(\lambda (H24: (csuba g x9 d1)).(or_introl (ex2 C
-(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(getl (S n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2:
-C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) u1) n H23)
-H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
-u1 a)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))
-(\lambda (x9: C).(\lambda (x10: T).(\lambda (x11: A).(\lambda (H23: (getl n
-x7 (CHead x9 (Bind Abst) x10))).(\lambda (H24: (csuba g x9 d1)).(\lambda
-(H25: (arity g x9 x10 (asucc g x11))).(\lambda (H26: (arity g d1 u1
-x11)).(or_intror (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr)
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
-(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S
-n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a)))) x9 x10 x11 (getl_clear_bind x6 c2 x7 x8 H20
-(CHead x9 (Bind Abst) x10) n H23) H24 H25 H26))))))))) H22)) H21))))))))
-H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) x H1 H2))))
-H0))))))).
-
-theorem sn3_gen_bind:
- \forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c
-(THead (Bind b) u t)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t))))))
-\def
- \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
-(sn3 c (THead (Bind b) u t))).(insert_eq T (THead (Bind b) u t) (\lambda (t0:
-T).(sn3 c t0)) (land (sn3 c u) (sn3 (CHead c (Bind b) u) t)) (\lambda (y:
-T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T y (THead
-(Bind b) u t0)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t0)))) (unintro
-T u (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind b) t0 x)) \to
-(land (sn3 c t0) (sn3 (CHead c (Bind b) t0) x))))) (sn3_ind c (\lambda (t0:
-T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Bind b) x x0)) \to
-(land (sn3 c x) (sn3 (CHead c (Bind b) x) x0)))))) (\lambda (t1: T).(\lambda
-(H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3
-c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2)
-\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall
-(x0: T).((eq T t2 (THead (Bind b) x x0)) \to (land (sn3 c x) (sn3 (CHead c
-(Bind b) x) x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T
-t1 (THead (Bind b) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0:
-T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c
-t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Bind b) x1
-x2)) \to (land (sn3 c x1) (sn3 (CHead c (Bind b) x1) x2))))))))) H2 (THead
-(Bind b) x x0) H3) in (let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall
-(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to
-(sn3 c t2))))) H1 (THead (Bind b) x x0) H3) in (conj (sn3 c x) (sn3 (CHead c
-(Bind b) x) x0) (sn3_sing c x (\lambda (t2: T).(\lambda (H6: (((eq T x t2)
-\to (\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x t2)).(let H8 \def (H4
-(THead (Bind b) t2 x0) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind
-b) t2 x0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x |
-(TLRef _) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind b) x
-x0) (THead (Bind b) t2 x0) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0:
-T).(pr3 c x t0)) H7 x H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0:
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H6 x H9) in (H11 (refl_equal T
-x) P)))))) (pr3_head_12 c x t2 H7 (Bind b) x0 x0 (pr3_refl (CHead c (Bind b)
-t2) x0)) t2 x0 (refl_equal T (THead (Bind b) t2 x0))) in (and_ind (sn3 c t2)
-(sn3 (CHead c (Bind b) t2) x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda
-(_: (sn3 (CHead c (Bind b) t2) x0)).H9)) H8)))))) (sn3_sing (CHead c (Bind b)
-x) x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P:
-Prop).P)))).(\lambda (H7: (pr3 (CHead c (Bind b) x) x0 t2)).(let H8 \def (H4
-(THead (Bind b) x t2) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind
-b) x t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind b) x
-x0) (THead (Bind b) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0:
-T).(pr3 (CHead c (Bind b) x) x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T
-t2 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in
-(H11 (refl_equal T x0) P)))))) (pr3_head_12 c x x (pr3_refl c x) (Bind b) x0
-t2 H7) x t2 (refl_equal T (THead (Bind b) x t2))) in (and_ind (sn3 c x) (sn3
-(CHead c (Bind b) x) t2) (sn3 (CHead c (Bind b) x) t2) (\lambda (_: (sn3 c
-x)).(\lambda (H10: (sn3 (CHead c (Bind b) x) t2)).H10)) H8))))))))))))))) y
-H0))))) H))))).
-
-theorem sn3_gen_head:
- \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c
-(THead k u t)) \to (sn3 c u)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (c: C).(\forall (u:
-T).(\forall (t: T).((sn3 c (THead k0 u t)) \to (sn3 c u)))))) (\lambda (b:
-B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
-(Bind b) u t))).(let H_x \def (sn3_gen_bind b c u t H) in (let H0 \def H_x in
-(and_ind (sn3 c u) (sn3 (CHead c (Bind b) u) t) (sn3 c u) (\lambda (H1: (sn3
-c u)).(\lambda (_: (sn3 (CHead c (Bind b) u) t)).H1)) H0)))))))) (\lambda (f:
-F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
-(Flat f) u t))).(let H_x \def (sn3_gen_flat f c u t H) in (let H0 \def H_x in
-(and_ind (sn3 c u) (sn3 c t) (sn3 c u) (\lambda (H1: (sn3 c u)).(\lambda (_:
-(sn3 c t)).H1)) H0)))))))) k).
-
-theorem sn3_gen_cflat:
- \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 (CHead
-c (Flat f) u) t) \to (sn3 c t)))))
-\def
- \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
-(sn3 (CHead c (Flat f) u) t)).(sn3_ind (CHead c (Flat f) u) (\lambda (t0:
-T).(sn3 c t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1
-t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to
-(sn3 (CHead c (Flat f) u) t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T
-t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to
-(sn3 c t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H2: (((eq T t1 t2)
-\to (\forall (P: Prop).P)))).(\lambda (H3: (pr3 c t1 t2)).(H1 t2 H2
-(pr3_cflat c t1 t2 H3 f u))))))))) t H))))).
-
-theorem sn3_cflat:
- \forall (c: C).(\forall (t: T).((sn3 c t) \to (\forall (f: F).(\forall (u:
-T).(sn3 (CHead c (Flat f) u) t)))))
-\def
- \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(\lambda (f:
-F).(\lambda (u: T).(sn3_ind c (\lambda (t0: T).(sn3 (CHead c (Flat f) u) t0))
-(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
-(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
-(sn3 (CHead c (Flat f) u) t2)))))).(sn3_pr2_intro (CHead c (Flat f) u) t1
-(\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) \to (\forall (P:
-Prop).P)))).(\lambda (H3: (pr2 (CHead c (Flat f) u) t1 t2)).(H1 t2 H2
-(pr3_pr2 c t1 t2 (pr2_gen_cflat f c u t1 t2 H3)))))))))) t H))))).
-
-theorem sn3_shift:
- \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c
-(THead (Bind b) v t)) \to (sn3 (CHead c (Bind b) v) t)))))
-\def
- \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
-(sn3 c (THead (Bind b) v t))).(let H_x \def (sn3_gen_bind b c v t H) in (let
-H0 \def H_x in (and_ind (sn3 c v) (sn3 (CHead c (Bind b) v) t) (sn3 (CHead c
-(Bind b) v) t) (\lambda (_: (sn3 c v)).(\lambda (H2: (sn3 (CHead c (Bind b)
-v) t)).H2)) H0))))))).
-
-theorem sn3_change:
- \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1:
-T).(\forall (t: T).((sn3 (CHead c (Bind b) v1) t) \to (\forall (v2: T).(sn3
-(CHead c (Bind b) v2) t)))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda
-(v1: T).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c (Bind b) v1) t)).(\lambda
-(v2: T).(sn3_ind (CHead c (Bind b) v1) (\lambda (t0: T).(sn3 (CHead c (Bind
-b) v2) t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2)
-\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to (sn3
-(CHead c (Bind b) v1) t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1
-t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to
-(sn3 (CHead c (Bind b) v2) t2)))))).(sn3_pr2_intro (CHead c (Bind b) v2) t1
-(\lambda (t2: T).(\lambda (H3: (((eq T t1 t2) \to (\forall (P:
-Prop).P)))).(\lambda (H4: (pr2 (CHead c (Bind b) v2) t1 t2)).(H2 t2 H3
-(pr3_pr2 (CHead c (Bind b) v1) t1 t2 (pr2_change b H c v2 t1 t2 H4
-v1)))))))))) t H0))))))).
-
-theorem sn3_cpr3_trans:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
-(k: K).(\forall (t: T).((sn3 (CHead c k u1) t) \to (sn3 (CHead c k u2)
-t)))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
-u2)).(\lambda (k: K).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c k u1)
-t)).(sn3_ind (CHead c k u1) (\lambda (t0: T).(sn3 (CHead c k u2) t0))
-(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
-(P: Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u1)
-t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
-Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u2)
-t2)))))).(sn3_sing (CHead c k u2) t1 (\lambda (t2: T).(\lambda (H3: (((eq T
-t1 t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 (CHead c k u2) t1
-t2)).(H2 t2 H3 (pr3_pr3_pr3_t c u1 u2 H t1 t2 k H4))))))))) t H0))))))).
-
-theorem sn3_bind:
- \forall (b: B).(\forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t:
-T).((sn3 (CHead c (Bind b) u) t) \to (sn3 c (THead (Bind b) u t)))))))
-\def
- \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c
-u)).(sn3_ind c (\lambda (t: T).(\forall (t0: T).((sn3 (CHead c (Bind b) t)
-t0) \to (sn3 c (THead (Bind b) t t0))))) (\lambda (t1: T).(\lambda (_:
-((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1
-t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to
-(\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (t: T).((sn3 (CHead c
-(Bind b) t2) t) \to (sn3 c (THead (Bind b) t2 t))))))))).(\lambda (t:
-T).(\lambda (H2: (sn3 (CHead c (Bind b) t1) t)).(sn3_ind (CHead c (Bind b)
-t1) (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (\lambda (t2:
-T).(\lambda (H3: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
-Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind b)
-t1) t3)))))).(\lambda (H4: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
-Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 c (THead (Bind b)
-t1 t3))))))).(sn3_sing c (THead (Bind b) t1 t2) (\lambda (t3: T).(\lambda
-(H5: (((eq T (THead (Bind b) t1 t2) t3) \to (\forall (P: Prop).P)))).(\lambda
-(H6: (pr3 c (THead (Bind b) t1 t2) t3)).(let H_x \def (bind_dec_not b Abst)
-in (let H7 \def H_x in (or_ind (eq B b Abst) (not (eq B b Abst)) (sn3 c t3)
-(\lambda (H8: (eq B b Abst)).(let H9 \def (eq_ind B b (\lambda (b0: B).(pr3 c
-(THead (Bind b0) t1 t2) t3)) H6 Abst H8) in (let H10 \def (eq_ind B b
-(\lambda (b0: B).((eq T (THead (Bind b0) t1 t2) t3) \to (\forall (P:
-Prop).P))) H5 Abst H8) in (let H11 \def (eq_ind B b (\lambda (b0: B).(\forall
-(t4: T).((((eq T t2 t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind
-b0) t1) t2 t4) \to (sn3 c (THead (Bind b0) t1 t4)))))) H4 Abst H8) in (let
-H12 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T t2 t4) \to
-(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) t2 t4) \to (sn3
-(CHead c (Bind b0) t1) t4))))) H3 Abst H8) in (let H13 \def (eq_ind B b
-(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P)))
-\to ((pr3 c t1 t4) \to (\forall (t0: T).((sn3 (CHead c (Bind b0) t4) t0) \to
-(sn3 c (THead (Bind b0) t4 t0)))))))) H1 Abst H8) in (let H14 \def
-(pr3_gen_abst c t1 t2 t3 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4:
-T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall
-(u0: T).(pr3 (CHead c (Bind b0) u0) t2 t4))))) (sn3 c t3) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H15: (eq T t3 (THead (Bind Abst) x0
-x1))).(\lambda (H16: (pr3 c t1 x0)).(\lambda (H17: ((\forall (b0: B).(\forall
-(u0: T).(pr3 (CHead c (Bind b0) u0) t2 x1))))).(let H18 \def (eq_ind T t3
-(\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) t0) \to (\forall (P:
-Prop).P))) H10 (THead (Bind Abst) x0 x1) H15) in (eq_ind_r T (THead (Bind
-Abst) x0 x1) (\lambda (t0: T).(sn3 c t0)) (let H_x0 \def (term_dec t1 x0) in
-(let H19 \def H_x0 in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P:
-Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H20: (eq T t1 x0)).(let
-H21 \def (eq_ind_r T x0 (\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2)
-(THead (Bind Abst) t0 x1)) \to (\forall (P: Prop).P))) H18 t1 H20) in (let
-H22 \def (eq_ind_r T x0 (\lambda (t0: T).(pr3 c t1 t0)) H16 t1 H20) in
-(eq_ind T t1 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t0 x1))) (let H_x1
-\def (term_dec t2 x1) in (let H23 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2
-x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abst) t1 x1)) (\lambda
-(H24: (eq T t2 x1)).(let H25 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T
-(THead (Bind Abst) t1 t2) (THead (Bind Abst) t1 t0)) \to (\forall (P:
-Prop).P))) H21 t2 H24) in (let H26 \def (eq_ind_r T x1 (\lambda (t0:
-T).(\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) t2 t0))))
-H17 t2 H24) in (eq_ind T t2 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t1
-t0))) (H25 (refl_equal T (THead (Bind Abst) t1 t2)) (sn3 c (THead (Bind Abst)
-t1 t2))) x1 H24)))) (\lambda (H24: (((eq T t2 x1) \to (\forall (P:
-Prop).P)))).(H11 x1 H24 (H17 Abst t1))) H23))) x0 H20)))) (\lambda (H20:
-(((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x1 \def (term_dec t2 x1)
-in (let H21 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P:
-Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H22: (eq T t2 x1)).(let
-H23 \def (eq_ind_r T x1 (\lambda (t0: T).(\forall (b0: B).(\forall (u0:
-T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) H17 t2 H22) in (eq_ind T t2 (\lambda
-(t0: T).(sn3 c (THead (Bind Abst) x0 t0))) (H13 x0 H20 H16 t2 (sn3_cpr3_trans
-c t1 x0 H16 (Bind Abst) t2 (sn3_sing (CHead c (Bind Abst) t1) t2 H12))) x1
-H22))) (\lambda (H22: (((eq T t2 x1) \to (\forall (P: Prop).P)))).(H13 x0 H20
-H16 x1 (sn3_cpr3_trans c t1 x0 H16 (Bind Abst) x1 (H12 x1 H22 (H17 Abst
-t1))))) H21)))) H19))) t3 H15))))))) H14)))))))) (\lambda (H8: (not (eq B b
-Abst))).(let H_x0 \def (pr3_gen_bind b H8 c t1 t2 t3 H6) in (let H9 \def H_x0
-in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
-b) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
-T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) t2 t4)))) (pr3 (CHead c (Bind
-b) t1) t2 (lift (S O) O t3)) (sn3 c t3) (\lambda (H10: (ex3_2 T T (\lambda
-(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3
-(CHead c (Bind b) t1) t2 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
-(t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_:
-T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 (CHead c (Bind b)
-t1) t2 t4))) (sn3 c t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq
-T t3 (THead (Bind b) x0 x1))).(\lambda (H12: (pr3 c t1 x0)).(\lambda (H13:
-(pr3 (CHead c (Bind b) t1) t2 x1)).(let H14 \def (eq_ind T t3 (\lambda (t0:
-T).((eq T (THead (Bind b) t1 t2) t0) \to (\forall (P: Prop).P))) H5 (THead
-(Bind b) x0 x1) H11) in (eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0:
-T).(sn3 c t0)) (let H_x1 \def (term_dec t1 x0) in (let H15 \def H_x1 in
-(or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: Prop).P)) (sn3 c (THead
-(Bind b) x0 x1)) (\lambda (H16: (eq T t1 x0)).(let H17 \def (eq_ind_r T x0
-(\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t0 x1)) \to
-(\forall (P: Prop).P))) H14 t1 H16) in (let H18 \def (eq_ind_r T x0 (\lambda
-(t0: T).(pr3 c t1 t0)) H12 t1 H16) in (eq_ind T t1 (\lambda (t0: T).(sn3 c
-(THead (Bind b) t0 x1))) (let H_x2 \def (term_dec t2 x1) in (let H19 \def
-H_x2 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: Prop).P)) (sn3 c
-(THead (Bind b) t1 x1)) (\lambda (H20: (eq T t2 x1)).(let H21 \def (eq_ind_r
-T x1 (\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t1 t0))
-\to (\forall (P: Prop).P))) H17 t2 H20) in (let H22 \def (eq_ind_r T x1
-(\lambda (t0: T).(pr3 (CHead c (Bind b) t1) t2 t0)) H13 t2 H20) in (eq_ind T
-t2 (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (H21 (refl_equal T (THead
-(Bind b) t1 t2)) (sn3 c (THead (Bind b) t1 t2))) x1 H20)))) (\lambda (H20:
-(((eq T t2 x1) \to (\forall (P: Prop).P)))).(H4 x1 H20 H13)) H19))) x0
-H16)))) (\lambda (H16: (((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x2
-\def (term_dec t2 x1) in (let H17 \def H_x2 in (or_ind (eq T t2 x1) ((eq T t2
-x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind b) x0 x1)) (\lambda (H18:
-(eq T t2 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t0: T).(pr3 (CHead c
-(Bind b) t1) t2 t0)) H13 t2 H18) in (eq_ind T t2 (\lambda (t0: T).(sn3 c
-(THead (Bind b) x0 t0))) (H1 x0 H16 H12 t2 (sn3_cpr3_trans c t1 x0 H12 (Bind
-b) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3))) x1 H18))) (\lambda (H18: (((eq
-T t2 x1) \to (\forall (P: Prop).P)))).(H1 x0 H16 H12 x1 (sn3_cpr3_trans c t1
-x0 H12 (Bind b) x1 (H3 x1 H18 H13)))) H17)))) H15))) t3 H11))))))) H10))
-(\lambda (H10: (pr3 (CHead c (Bind b) t1) t2 (lift (S O) O
-t3))).(sn3_gen_lift (CHead c (Bind b) t1) t3 (S O) O (sn3_pr3_trans (CHead c
-(Bind b) t1) t2 (sn3_pr2_intro (CHead c (Bind b) t1) t2 (\lambda (t0:
-T).(\lambda (H11: (((eq T t2 t0) \to (\forall (P: Prop).P)))).(\lambda (H12:
-(pr2 (CHead c (Bind b) t1) t2 t0)).(H3 t0 H11 (pr3_pr2 (CHead c (Bind b) t1)
-t2 t0 H12)))))) (lift (S O) O t3) H10) c (drop_drop (Bind b) O c c (drop_refl
-c) t1))) H9)))) H7)))))))))) t H2)))))) u H)))).
-
-theorem sn3_beta:
- \forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c (THead (Bind Abbr) v
-t)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead
-(Bind Abst) w t))))))))
-\def
- \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
-(Bind Abbr) v t))).(insert_eq T (THead (Bind Abbr) v t) (\lambda (t0: T).(sn3
-c t0)) (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead
-(Bind Abst) w t))))) (\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t
-(\lambda (t0: T).((eq T y (THead (Bind Abbr) v t0)) \to (\forall (w: T).((sn3
-c w) \to (sn3 c (THead (Flat Appl) v (THead (Bind Abst) w t0))))))) (unintro
-T v (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind Abbr) t0 x)) \to
-(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) t0 (THead (Bind
-Abst) w x)))))))) (sn3_ind c (\lambda (t0: T).(\forall (x: T).(\forall (x0:
-T).((eq T t0 (THead (Bind Abbr) x x0)) \to (\forall (w: T).((sn3 c w) \to
-(sn3 c (THead (Flat Appl) x (THead (Bind Abst) w x0))))))))) (\lambda (t1:
-T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
-Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
-(\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Bind Abbr) x x0)) \to
-(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst)
-w x0))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1
-(THead (Bind Abbr) x x0))).(\lambda (w: T).(\lambda (H4: (sn3 c w)).(let H5
-\def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to
-(\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (\forall (x1: T).(\forall (x2:
-T).((eq T t2 (THead (Bind Abbr) x1 x2)) \to (\forall (w0: T).((sn3 c w0) \to
-(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) w0 x2)))))))))))) H2 (THead
-(Bind Abbr) x x0) H3) in (let H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall
-(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to
-(sn3 c t2))))) H1 (THead (Bind Abbr) x x0) H3) in (sn3_ind c (\lambda (t0:
-T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t0 x0)))) (\lambda (t2:
-T).(\lambda (H7: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
-Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H8: ((\forall
-(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to
-(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t3 x0)))))))).(sn3_pr2_intro c
-(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (\lambda (t3: T).(\lambda
-(H9: (((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t3) \to (\forall
-(P: Prop).P)))).(\lambda (H10: (pr2 c (THead (Flat Appl) x (THead (Bind Abst)
-t2 x0)) t3)).(let H11 \def (pr2_gen_appl c x (THead (Bind Abst) t2 x0) t3
-H10) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) t2 x0) t4))))
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(eq T (THead (Bind Abst) t2 x0) (THead (Bind Abst) y1 z1)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
-(Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
-b) u) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c t3)
-(\lambda (H12: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) t2 x0)
-t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) t2 x0) t4))) (sn3
-c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H13: (eq T t3 (THead (Flat
-Appl) x1 x2))).(\lambda (H14: (pr2 c x x1)).(\lambda (H15: (pr2 c (THead
-(Bind Abst) t2 x0) x2)).(let H16 \def (eq_ind T t3 (\lambda (t0: T).((eq T
-(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P:
-Prop).P))) H9 (THead (Flat Appl) x1 x2) H13) in (eq_ind_r T (THead (Flat
-Appl) x1 x2) (\lambda (t0: T).(sn3 c t0)) (let H17 \def (pr2_gen_abst c t2 x0
-x2 H15) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead
-(Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t2 u2)))
-(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead
-c (Bind b) u) x0 t4))))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3:
-T).(\lambda (x4: T).(\lambda (H18: (eq T x2 (THead (Bind Abst) x3
-x4))).(\lambda (H19: (pr2 c t2 x3)).(\lambda (H20: ((\forall (b: B).(\forall
-(u: T).(pr2 (CHead c (Bind b) u) x0 x4))))).(let H21 \def (eq_ind T x2
-(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0))
-(THead (Flat Appl) x1 t0)) \to (\forall (P: Prop).P))) H16 (THead (Bind Abst)
-x3 x4) H18) in (eq_ind_r T (THead (Bind Abst) x3 x4) (\lambda (t0: T).(sn3 c
-(THead (Flat Appl) x1 t0))) (let H_x \def (term_dec t2 x3) in (let H22 \def
-H_x in (or_ind (eq T t2 x3) ((eq T t2 x3) \to (\forall (P: Prop).P)) (sn3 c
-(THead (Flat Appl) x1 (THead (Bind Abst) x3 x4))) (\lambda (H23: (eq T t2
-x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: T).((eq T (THead (Flat Appl)
-x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) x1 (THead (Bind Abst) t0
-x4))) \to (\forall (P: Prop).P))) H21 t2 H23) in (let H25 \def (eq_ind_r T x3
-(\lambda (t0: T).(pr2 c t2 t0)) H19 t2 H23) in (eq_ind T t2 (\lambda (t0:
-T).(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t0 x4)))) (let H_x0 \def
-(term_dec x x1) in (let H26 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to
-(\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t2
-x4))) (\lambda (H27: (eq T x x1)).(let H28 \def (eq_ind_r T x1 (\lambda (t0:
-T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (THead (Flat Appl)
-t0 (THead (Bind Abst) t2 x4))) \to (\forall (P: Prop).P))) H24 x H27) in (let
-H29 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H27) in (eq_ind
-T x (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) t2
-x4)))) (let H_x1 \def (term_dec x0 x4) in (let H30 \def H_x1 in (or_ind (eq T
-x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x
-(THead (Bind Abst) t2 x4))) (\lambda (H31: (eq T x0 x4)).(let H32 \def
-(eq_ind_r T x4 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind
-Abst) t2 x0)) (THead (Flat Appl) x (THead (Bind Abst) t2 t0))) \to (\forall
-(P: Prop).P))) H28 x0 H31) in (let H33 \def (eq_ind_r T x4 (\lambda (t0:
-T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0
-H31) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead
-(Bind Abst) t2 t0)))) (H32 (refl_equal T (THead (Flat Appl) x (THead (Bind
-Abst) t2 x0))) (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)))) x4
-H31)))) (\lambda (H31: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead
-(Bind Abbr) x x4) (\lambda (H32: (eq T (THead (Bind Abbr) x x0) (THead (Bind
-Abbr) x x4))).(\lambda (P: Prop).(let H33 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
-Abbr) x x0) (THead (Bind Abbr) x x4) H32) in (let H34 \def (eq_ind_r T x4
-(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H31 x0 H33) in
-(let H35 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u:
-T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H33) in (H34 (refl_equal T x0)
-P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4)
-(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead
-(Bind Abbr) x x4)) t2 (sn3_sing c t2 H7))) H30))) x1 H27)))) (\lambda (H27:
-(((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind Abbr) x1 x4)
-(\lambda (H28: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x1
-x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _)
-\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0)
-(THead (Bind Abbr) x1 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
-Abbr) x x0) (THead (Bind Abbr) x1 x4) H28) in (\lambda (H31: (eq T x
-x1)).(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall
-(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (let H33 \def
-(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
-H27 x H31) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14
-x H31) in (H33 (refl_equal T x) P)))))) H29)))) (pr3_head_12 c x x1 (pr3_pr2
-c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20
-Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) t2 (sn3_sing c t2
-H7))) H26))) x3 H23)))) (\lambda (H23: (((eq T t2 x3) \to (\forall (P:
-Prop).P)))).(let H_x0 \def (term_dec x x1) in (let H24 \def H_x0 in (or_ind
-(eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl)
-x1 (THead (Bind Abst) x3 x4))) (\lambda (H25: (eq T x x1)).(let H26 \def
-(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H25) in (eq_ind T x
-(\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4))))
-(let H_x1 \def (term_dec x0 x4) in (let H27 \def H_x1 in (or_ind (eq T x0 x4)
-((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x (THead
-(Bind Abst) x3 x4))) (\lambda (H28: (eq T x0 x4)).(let H29 \def (eq_ind_r T
-x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-x0 t0)))) H20 x0 H28) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat
-Appl) x (THead (Bind Abst) x3 t0)))) (H8 x3 H23 (pr3_pr2 c t2 x3 H19)) x4
-H28))) (\lambda (H28: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead
-(Bind Abbr) x x4) (\lambda (H29: (eq T (THead (Bind Abbr) x x0) (THead (Bind
-Abbr) x x4))).(\lambda (P: Prop).(let H30 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
-Abbr) x x0) (THead (Bind Abbr) x x4) H29) in (let H31 \def (eq_ind_r T x4
-(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H28 x0 H30) in
-(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u:
-T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (H31 (refl_equal T x0)
-P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4)
-(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead
-(Bind Abbr) x x4)) x3 (H7 x3 H23 (pr3_pr2 c t2 x3 H19)))) H27))) x1 H25)))
-(\lambda (H25: (((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind
-Abbr) x1 x4) (\lambda (H26: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr)
-x1 x4))).(\lambda (P: Prop).(let H27 \def (f_equal T T (\lambda (e: T).(match
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _)
-\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0)
-(THead (Bind Abbr) x1 x4) H26) in ((let H28 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
-Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in (\lambda (H29: (eq T x
-x1)).(let H30 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall
-(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (let H31 \def
-(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
-H25 x H29) in (let H32 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14
-x H29) in (H31 (refl_equal T x) P)))))) H27)))) (pr3_head_12 c x x1 (pr3_pr2
-c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20
-Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) x3 (H7 x3 H23
-(pr3_pr2 c t2 x3 H19)))) H24)))) H22))) x2 H18))))))) H17)) t3 H13)))))))
-H12)) (\lambda (H12: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind Abst) t2 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
-Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-z1 t4))))))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
-T).(\lambda (x4: T).(\lambda (H13: (eq T (THead (Bind Abst) t2 x0) (THead
-(Bind Abst) x1 x2))).(\lambda (H14: (eq T t3 (THead (Bind Abbr) x3
-x4))).(\lambda (H15: (pr2 c x x3)).(\lambda (H16: ((\forall (b: B).(\forall
-(u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H17 \def (eq_ind T t3
-(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0)
-\to (\forall (P: Prop).P))) H9 (THead (Bind Abbr) x3 x4) H14) in (eq_ind_r T
-(THead (Bind Abbr) x3 x4) (\lambda (t0: T).(sn3 c t0)) (let H18 \def (f_equal
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) \Rightarrow t0]))
-(THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in ((let H19 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0)
-\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in
-(\lambda (_: (eq T t2 x1)).(let H21 \def (eq_ind_r T x2 (\lambda (t0:
-T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t0 x4)))) H16 x0
-H19) in (let H_x \def (term_dec x x3) in (let H22 \def H_x in (or_ind (eq T x
-x3) ((eq T x x3) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abbr) x3 x4))
-(\lambda (H23: (eq T x x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0:
-T).(pr2 c x t0)) H15 x H23) in (eq_ind T x (\lambda (t0: T).(sn3 c (THead
-(Bind Abbr) t0 x4))) (let H_x0 \def (term_dec x0 x4) in (let H25 \def H_x0 in
-(or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead
-(Bind Abbr) x x4)) (\lambda (H26: (eq T x0 x4)).(let H27 \def (eq_ind_r T x4
-(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0
-t0)))) H21 x0 H26) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Bind Abbr)
-x t0))) (sn3_sing c (THead (Bind Abbr) x x0) H6) x4 H26))) (\lambda (H26:
-(((eq T x0 x4) \to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x x4)
-(\lambda (H27: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x
-x4))).(\lambda (P: Prop).(let H28 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
-\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0)
-(THead (Bind Abbr) x x4) H27) in (let H29 \def (eq_ind_r T x4 (\lambda (t0:
-T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H26 x0 H28) in (let H30 \def
-(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c
-(Bind b) u) x0 t0)))) H21 x0 H28) in (H29 (refl_equal T x0) P)))))) (pr3_pr2
-c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4
-(Bind Abbr) (H21 Abbr x))))) H25))) x3 H23))) (\lambda (H23: (((eq T x x3)
-\to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x3 x4) (\lambda (H24: (eq
-T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x3 x4))).(\lambda (P:
-Prop).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x |
-(THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr)
-x3 x4) H24) in ((let H26 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
-\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0)
-(THead (Bind Abbr) x3 x4) H24) in (\lambda (H27: (eq T x x3)).(let H28 \def
-(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c
-(Bind b) u) x0 t0)))) H21 x0 H26) in (let H29 \def (eq_ind_r T x3 (\lambda
-(t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H23 x H27) in (let H30
-\def (eq_ind_r T x3 (\lambda (t0: T).(pr2 c x t0)) H15 x H27) in (H29
-(refl_equal T x) P)))))) H25)))) (pr3_head_12 c x x3 (pr3_pr2 c x x3 H15)
-(Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x3) x0 x4 (H21 Abbr x3)))))
-H22)))))) H18)) t3 H14)))))))))) H12)) (\lambda (H12: (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
-y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3)
-(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda
-(x5: T).(\lambda (x6: T).(\lambda (H13: (not (eq B x1 Abst))).(\lambda (H14:
-(eq T (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3))).(\lambda (H15: (eq
-T t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
-(_: (pr2 c x x5)).(\lambda (H17: (pr2 c x2 x6)).(\lambda (H18: (pr2 (CHead c
-(Bind x1) x6) x3 x4)).(let H19 \def (eq_ind T t3 (\lambda (t0: T).((eq T
-(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P:
-Prop).P))) H9 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))
-H15) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5)
-x4)) (\lambda (t0: T).(sn3 c t0)) (let H20 \def (f_equal T B (\lambda (e:
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst |
-(TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
-Abst])])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in ((let H21
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _)
-\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in
-((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
-t0) \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14)
-in (\lambda (H23: (eq T t2 x2)).(\lambda (H24: (eq B Abst x1)).(let H25 \def
-(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H18 x0
-H22) in (let H26 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H17 t2
-H23) in (let H27 \def (eq_ind_r B x1 (\lambda (b: B).(pr2 (CHead c (Bind b)
-x6) x0 x4)) H25 Abst H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b:
-B).(not (eq B b Abst))) H13 Abst H24) in (eq_ind B Abst (\lambda (b: B).(sn3
-c (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (let H29
-\def (match (H28 (refl_equal B Abst)) in False return (\lambda (_:
-False).(sn3 c (THead (Bind Abst) x6 (THead (Flat Appl) (lift (S O) O x5)
-x4)))) with []) in H29) x1 H24)))))))) H21)) H20)) t3 H15)))))))))))))) H12))
-H11))))))))) w H4))))))))))) y H0))))) H)))).
-
-theorem nf3_appl_abbr:
- \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c
-(CHead d (Bind Abbr) w)) \to (\forall (v: T).((sn3 c (THead (Flat Appl) v
-(lift (S i) O w))) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda
-(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (v: T).(\lambda (H0: (sn3 c
-(THead (Flat Appl) v (lift (S i) O w)))).(insert_eq T (THead (Flat Appl) v
-(lift (S i) O w)) (\lambda (t: T).(sn3 c t)) (sn3 c (THead (Flat Appl) v
-(TLRef i))) (\lambda (y: T).(\lambda (H1: (sn3 c y)).(unintro T v (\lambda
-(t: T).((eq T y (THead (Flat Appl) t (lift (S i) O w))) \to (sn3 c (THead
-(Flat Appl) t (TLRef i))))) (sn3_ind c (\lambda (t: T).(\forall (x: T).((eq T
-t (THead (Flat Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x
-(TLRef i)))))) (\lambda (t1: T).(\lambda (H2: ((\forall (t2: T).((((eq T t1
-t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
-t2)))))).(\lambda (H3: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
-Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).((eq T t2 (THead (Flat
-Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x (TLRef
-i)))))))))).(\lambda (x: T).(\lambda (H4: (eq T t1 (THead (Flat Appl) x (lift
-(S i) O w)))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2:
-T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall
-(x0: T).((eq T t2 (THead (Flat Appl) x0 (lift (S i) O w))) \to (sn3 c (THead
-(Flat Appl) x0 (TLRef i))))))))) H3 (THead (Flat Appl) x (lift (S i) O w))
-H4) in (let H6 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: T).((((eq T t
-t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (sn3 c t2))))) H2
-(THead (Flat Appl) x (lift (S i) O w)) H4) in (sn3_pr2_intro c (THead (Flat
-Appl) x (TLRef i)) (\lambda (t2: T).(\lambda (H7: (((eq T (THead (Flat Appl)
-x (TLRef i)) t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr2 c (THead
-(Flat Appl) x (TLRef i)) t2)).(let H9 \def (pr2_gen_appl c x (TLRef i) t2 H8)
-in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
-T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))
-(sn3 c t2) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T
-T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H12: (pr2 c
-x x0)).(\lambda (H13: (pr2 c (TLRef i) x1)).(let H14 \def (eq_ind T t2
-(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P:
-Prop).P))) H7 (THead (Flat Appl) x0 x1) H11) in (eq_ind_r T (THead (Flat
-Appl) x0 x1) (\lambda (t: T).(sn3 c t)) (let H15 \def (pr2_gen_lref c x1 i
-H13) in (or_ind (eq T x1 (TLRef i)) (ex2_2 C T (\lambda (d0: C).(\lambda (u:
-T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq
-T x1 (lift (S i) O u))))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda (H16:
-(eq T x1 (TLRef i))).(let H17 \def (eq_ind T x1 (\lambda (t: T).((eq T (THead
-(Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P:
-Prop).P))) H14 (TLRef i) H16) in (eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c
-(THead (Flat Appl) x0 t))) (let H_x \def (term_dec x x0) in (let H18 \def H_x
-in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c (THead
-(Flat Appl) x0 (TLRef i))) (\lambda (H19: (eq T x x0)).(let H20 \def
-(eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead
-(Flat Appl) t (TLRef i))) \to (\forall (P: Prop).P))) H17 x H19) in (let H21
-\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H19) in (eq_ind T x
-(\lambda (t: T).(sn3 c (THead (Flat Appl) t (TLRef i)))) (H20 (refl_equal T
-(THead (Flat Appl) x (TLRef i))) (sn3 c (THead (Flat Appl) x (TLRef i)))) x0
-H19)))) (\lambda (H19: (((eq T x x0) \to (\forall (P: Prop).P)))).(H5 (THead
-(Flat Appl) x0 (lift (S i) O w)) (\lambda (H20: (eq T (THead (Flat Appl) x
-(lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)))).(\lambda (P:
-Prop).(let H21 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x |
-(THead _ t _) \Rightarrow t])) (THead (Flat Appl) x (lift (S i) O w)) (THead
-(Flat Appl) x0 (lift (S i) O w)) H20) in (let H22 \def (eq_ind_r T x0
-(\lambda (t: T).((eq T x t) \to (\forall (P0: Prop).P0))) H19 x H21) in (let
-H23 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H21) in (H22
-(refl_equal T x) P)))))) (pr3_pr2 c (THead (Flat Appl) x (lift (S i) O w))
-(THead (Flat Appl) x0 (lift (S i) O w)) (pr2_head_1 c x x0 H12 (Flat Appl)
-(lift (S i) O w))) x0 (refl_equal T (THead (Flat Appl) x0 (lift (S i) O
-w))))) H18))) x1 H16))) (\lambda (H16: (ex2_2 C T (\lambda (d0: C).(\lambda
-(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
-T).(eq T x1 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda
-(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
-T).(eq T x1 (lift (S i) O u)))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda
-(x2: C).(\lambda (x3: T).(\lambda (H17: (getl i c (CHead x2 (Bind Abbr)
-x3))).(\lambda (H18: (eq T x1 (lift (S i) O x3))).(let H19 \def (eq_ind T x1
-(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0
-t)) \to (\forall (P: Prop).P))) H14 (lift (S i) O x3) H18) in (eq_ind_r T
-(lift (S i) O x3) (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H20
-\def (eq_ind C (CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H
-(CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2
-(Bind Abbr) x3) H17)) in (let H21 \def (f_equal C C (\lambda (e: C).(match e
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
-\Rightarrow c0])) (CHead d (Bind Abbr) w) (CHead x2 (Bind Abbr) x3)
-(getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 (Bind Abbr) x3) H17)) in
-((let H22 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
-C).T) with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d
-(Bind Abbr) w) (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w)
-i H (CHead x2 (Bind Abbr) x3) H17)) in (\lambda (H23: (eq C d x2)).(let H24
-\def (eq_ind_r T x3 (\lambda (t: T).(getl i c (CHead x2 (Bind Abbr) t))) H20
-w H22) in (eq_ind T w (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 (lift (S
-i) O t)))) (let H25 \def (eq_ind_r C x2 (\lambda (c0: C).(getl i c (CHead c0
-(Bind Abbr) w))) H24 d H23) in (let H_x \def (term_dec x x0) in (let H26 \def
-H_x in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c
-(THead (Flat Appl) x0 (lift (S i) O w))) (\lambda (H27: (eq T x x0)).(let H28
-\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H27) in (eq_ind T x
-(\lambda (t: T).(sn3 c (THead (Flat Appl) t (lift (S i) O w)))) (sn3_sing c
-(THead (Flat Appl) x (lift (S i) O w)) H6) x0 H27))) (\lambda (H27: (((eq T x
-x0) \to (\forall (P: Prop).P)))).(H6 (THead (Flat Appl) x0 (lift (S i) O w))
-(\lambda (H28: (eq T (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat
-Appl) x0 (lift (S i) O w)))).(\lambda (P: Prop).(let H29 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t]))
-(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O
-w)) H28) in (let H30 \def (eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to
-(\forall (P0: Prop).P0))) H27 x H29) in (let H31 \def (eq_ind_r T x0 (\lambda
-(t: T).(pr2 c x t)) H12 x H29) in (H30 (refl_equal T x) P)))))) (pr3_pr2 c
-(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O
-w)) (pr2_head_1 c x x0 H12 (Flat Appl) (lift (S i) O w))))) H26)))) x3
-H22)))) H21))) x1 H18)))))) H16)) H15)) t2 H11))))))) H10)) (\lambda (H10:
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1
-t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
-b) u) z1 t3))))))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2:
-T).(\lambda (x3: T).(\lambda (H11: (eq T (TLRef i) (THead (Bind Abst) x0
-x1))).(\lambda (H12: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c
-x x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b)
-u) x1 x3))))).(let H15 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat
-Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 (THead (Bind Abbr) x2
-x3) H12) in (eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t))
-(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0
-x1) H11) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H16)) t2
-H12)))))))))) H10)) (\lambda (H10: (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i)
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind
-B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
-T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
-(sn3 c t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
-T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0
-Abst))).(\lambda (H12: (eq T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda
-(H13: (eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4)
-x3)))).(\lambda (_: (pr2 c x x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_:
-(pr2 (CHead c (Bind x0) x5) x2 x3)).(let H17 \def (eq_ind T t2 (\lambda (t:
-T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7
-(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) H13) in
-(eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))
-(\lambda (t: T).(sn3 c t)) (let H18 \def (eq_ind T (TLRef i) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
-(THead (Bind x0) x1 x2) H12) in (False_ind (sn3 c (THead (Bind x0) x5 (THead
-(Flat Appl) (lift (S O) O x4) x3))) H18)) t2 H13)))))))))))))) H10))
-H9))))))))))))) y H1)))) H0))))))).
-
-theorem sn3_appl_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u:
-T).((sn3 c u) \to (\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) u)
-(THead (Flat Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v
-(THead (Bind b) u t))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda
-(u: T).(\lambda (H0: (sn3 c u)).(sn3_ind c (\lambda (t: T).(\forall (t0:
-T).(\forall (v: T).((sn3 (CHead c (Bind b) t) (THead (Flat Appl) (lift (S O)
-O v) t0)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t t0)))))))
-(\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
-(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
-(\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) t2) (THead (Flat
-Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t2
-t))))))))))).(\lambda (t: T).(\lambda (v: T).(\lambda (H3: (sn3 (CHead c
-(Bind b) t1) (THead (Flat Appl) (lift (S O) O v) t))).(insert_eq T (THead
-(Flat Appl) (lift (S O) O v) t) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1)
-t0)) (sn3 c (THead (Flat Appl) v (THead (Bind b) t1 t))) (\lambda (y:
-T).(\lambda (H4: (sn3 (CHead c (Bind b) t1) y)).(unintro T t (\lambda (t0:
-T).((eq T y (THead (Flat Appl) (lift (S O) O v) t0)) \to (sn3 c (THead (Flat
-Appl) v (THead (Bind b) t1 t0))))) (unintro T v (\lambda (t0: T).(\forall (x:
-T).((eq T y (THead (Flat Appl) (lift (S O) O t0) x)) \to (sn3 c (THead (Flat
-Appl) t0 (THead (Bind b) t1 x)))))) (sn3_ind (CHead c (Bind b) t1) (\lambda
-(t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat Appl) (lift
-(S O) O x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) t1 x0)))))))
-(\lambda (t2: T).(\lambda (H5: ((\forall (t3: T).((((eq T t2 t3) \to (\forall
-(P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind
-b) t1) t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t2 t3) \to (\forall
-(P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (\forall (x:
-T).(\forall (x0: T).((eq T t3 (THead (Flat Appl) (lift (S O) O x) x0)) \to
-(sn3 c (THead (Flat Appl) x (THead (Bind b) t1 x0))))))))))).(\lambda (x:
-T).(\lambda (x0: T).(\lambda (H7: (eq T t2 (THead (Flat Appl) (lift (S O) O
-x) x0))).(let H8 \def (eq_ind T t2 (\lambda (t0: T).(\forall (t3: T).((((eq T
-t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t0 t3) \to
-(\forall (x1: T).(\forall (x2: T).((eq T t3 (THead (Flat Appl) (lift (S O) O
-x1) x2)) \to (sn3 c (THead (Flat Appl) x1 (THead (Bind b) t1 x2)))))))))) H6
-(THead (Flat Appl) (lift (S O) O x) x0) H7) in (let H9 \def (eq_ind T t2
-(\lambda (t0: T).(\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P)))
-\to ((pr3 (CHead c (Bind b) t1) t0 t3) \to (sn3 (CHead c (Bind b) t1) t3)))))
-H5 (THead (Flat Appl) (lift (S O) O x) x0) H7) in (sn3_pr2_intro c (THead
-(Flat Appl) x (THead (Bind b) t1 x0)) (\lambda (t3: T).(\lambda (H10: (((eq T
-(THead (Flat Appl) x (THead (Bind b) t1 x0)) t3) \to (\forall (P:
-Prop).P)))).(\lambda (H11: (pr2 c (THead (Flat Appl) x (THead (Bind b) t1
-x0)) t3)).(let H12 \def (pr2_gen_appl c x (THead (Bind b) t1 x0) t3 H11) in
-(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat
-Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_:
-T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
-Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0)
-u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
-b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2)))))))) (sn3 c t3)
-(\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0)
-t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))) (sn3 c
-t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H14: (eq T t3 (THead (Flat
-Appl) x1 x2))).(\lambda (H15: (pr2 c x x1)).(\lambda (H16: (pr2 c (THead
-(Bind b) t1 x0) x2)).(let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T
-(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P)))
-H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat Appl) x1 x2)
-(\lambda (t0: T).(sn3 c t0)) (let H_x \def (pr3_gen_bind b H c t1 x0 x2) in
-(let H18 \def (H_x (pr3_pr2 c (THead (Bind b) t1 x0) x2 H16)) in (or_ind
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
-T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4)))) (pr3 (CHead c (Bind
-b) t1) x0 (lift (S O) O x2)) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (H19:
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
-T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 t4)))) (\lambda
-(u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3
-(CHead c (Bind b) t1) x0 t4))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda
-(x3: T).(\lambda (x4: T).(\lambda (H20: (eq T x2 (THead (Bind b) x3
-x4))).(\lambda (H21: (pr3 c t1 x3)).(\lambda (H22: (pr3 (CHead c (Bind b) t1)
-x0 x4)).(let H23 \def (eq_ind T x2 (\lambda (t0: T).((eq T (THead (Flat Appl)
-x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 t0)) \to (\forall (P:
-Prop).P))) H17 (THead (Bind b) x3 x4) H20) in (eq_ind_r T (THead (Bind b) x3
-x4) (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 t0))) (let H_x0 \def
-(term_dec t1 x3) in (let H24 \def H_x0 in (or_ind (eq T t1 x3) ((eq T t1 x3)
-\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind b) x3
-x4))) (\lambda (H25: (eq T t1 x3)).(let H26 \def (eq_ind_r T x3 (\lambda (t0:
-T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1
-(THead (Bind b) t0 x4))) \to (\forall (P: Prop).P))) H23 t1 H25) in (let H27
-\def (eq_ind_r T x3 (\lambda (t0: T).(pr3 c t1 t0)) H21 t1 H25) in (eq_ind T
-t1 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 (THead (Bind b) t0 x4))))
-(let H_x1 \def (term_dec x0 x4) in (let H28 \def H_x1 in (or_ind (eq T x0 x4)
-((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead
-(Bind b) t1 x4))) (\lambda (H29: (eq T x0 x4)).(let H30 \def (eq_ind_r T x4
-(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead
-(Flat Appl) x1 (THead (Bind b) t1 t0))) \to (\forall (P: Prop).P))) H26 x0
-H29) in (let H31 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b)
-t1) x0 t0)) H22 x0 H29) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat
-Appl) x1 (THead (Bind b) t1 t0)))) (let H_x2 \def (term_dec x x1) in (let H32
-\def H_x2 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3
-c (THead (Flat Appl) x1 (THead (Bind b) t1 x0))) (\lambda (H33: (eq T x
-x1)).(let H34 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl)
-x (THead (Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to
-(\forall (P: Prop).P))) H30 x H33) in (let H35 \def (eq_ind_r T x1 (\lambda
-(t0: T).(pr2 c x t0)) H15 x H33) in (eq_ind T x (\lambda (t0: T).(sn3 c
-(THead (Flat Appl) t0 (THead (Bind b) t1 x0)))) (H34 (refl_equal T (THead
-(Flat Appl) x (THead (Bind b) t1 x0))) (sn3 c (THead (Flat Appl) x (THead
-(Bind b) t1 x0)))) x1 H33)))) (\lambda (H33: (((eq T x x1) \to (\forall (P:
-Prop).P)))).(H8 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H34: (eq T
-(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
-x0))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
-lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
-t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
-(lift (S O) O x1) x0) H34) in (let H36 \def (eq_ind_r T x1 (\lambda (t0:
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H33 x (lift_inj x x1 (S O) O
-H35)) in (let H37 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
-(lift_inj x x1 (S O) O H35)) in (H36 (refl_equal T x) P)))))) (pr3_flat
-(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
-(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
-(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl) x1 x0
-(refl_equal T (THead (Flat Appl) (lift (S O) O x1) x0)))) H32))) x4 H29))))
-(\lambda (H29: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H8 (THead (Flat
-Appl) (lift (S O) O x1) x4) (\lambda (H30: (eq T (THead (Flat Appl) (lift (S
-O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P:
-Prop).(let H31 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
-\to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
-lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
-t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
-(lift (S O) O x1) x4) H30) in ((let H32 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H30) in
-(\lambda (H33: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H34 \def
-(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0)))
-H29 x0 H32) in (let H35 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T (THead
-(Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 (THead (Bind b)
-t1 t0))) \to (\forall (P0: Prop).P0))) H26 x0 H32) in (let H36 \def (eq_ind_r
-T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) H22 x0 H32) in (let
-H37 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead
-(Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to (\forall
-(P0: Prop).P0))) H35 x (lift_inj x x1 (S O) O H33)) in (let H38 \def
-(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O
-H33)) in (H34 (refl_equal T x0) P)))))))) H31)))) (pr3_flat (CHead c (Bind b)
-t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S
-O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15))
-x0 x4 H22 Appl) x1 x4 (refl_equal T (THead (Flat Appl) (lift (S O) O x1)
-x4)))) H28))) x3 H25)))) (\lambda (H25: (((eq T t1 x3) \to (\forall (P:
-Prop).P)))).(H2 x3 H25 H21 x4 x1 (sn3_cpr3_trans c t1 x3 H21 (Bind b) (THead
-(Flat Appl) (lift (S O) O x1) x4) (let H_x1 \def (term_dec x0 x4) in (let H26
-\def H_x1 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P))
-(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x4)) (\lambda
-(H27: (eq T x0 x4)).(let H28 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead
-c (Bind b) t1) x0 t0)) H22 x0 H27) in (eq_ind T x0 (\lambda (t0: T).(sn3
-(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) t0))) (let H_x2
-\def (term_dec x x1) in (let H29 \def H_x2 in (or_ind (eq T x x1) ((eq T x
-x1) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl)
-(lift (S O) O x1) x0)) (\lambda (H30: (eq T x x1)).(let H31 \def (eq_ind_r T
-x1 (\lambda (t0: T).(pr2 c x t0)) H15 x H30) in (eq_ind T x (\lambda (t0:
-T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0)))
-(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9)
-x1 H30))) (\lambda (H30: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9
-(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H31: (eq T (THead (Flat
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
-x0))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
-lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
-t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
-(lift (S O) O x1) x0) H31) in (let H33 \def (eq_ind_r T x1 (\lambda (t0:
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H30 x (lift_inj x x1 (S O) O
-H32)) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
-(lift_inj x x1 (S O) O H32)) in (H33 (refl_equal T x) P)))))) (pr3_flat
-(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
-(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
-(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl)))
-H29))) x4 H27))) (\lambda (H27: (((eq T x0 x4) \to (\forall (P:
-Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x1) x4) (\lambda (H28: (eq T
-(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
-x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
-lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
-t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
-(lift (S O) O x1) x4) H28) in ((let H30 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H28) in
-(\lambda (H31: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H32 \def
-(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0)))
-H27 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c
-(Bind b) t1) x0 t0)) H22 x0 H30) in (let H34 \def (eq_ind_r T x1 (\lambda
-(t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H31)) in (H32 (refl_equal
-T x0) P)))))) H29)))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift
-(S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c
-c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x4 H22 Appl))) H26))))))
-H24))) x2 H20))))))) H19)) (\lambda (H19: (pr3 (CHead c (Bind b) t1) x0 (lift
-(S O) O x2))).(sn3_gen_lift (CHead c (Bind b) t1) (THead (Flat Appl) x1 x2)
-(S O) O (eq_ind_r T (THead (Flat Appl) (lift (S O) O x1) (lift (S O) (s (Flat
-Appl) O) x2)) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) t0)) (sn3_pr3_trans
-(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x0) (let H_x0 \def
-(term_dec x x1) in (let H20 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to
-(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S
-O) O x1) x0)) (\lambda (H21: (eq T x x1)).(let H22 \def (eq_ind_r T x1
-(\lambda (t0: T).(pr2 c x t0)) H15 x H21) in (eq_ind T x (\lambda (t0:
-T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0)))
-(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9)
-x1 H21))) (\lambda (H21: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9
-(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H22: (eq T (THead (Flat
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
-x0))).(\lambda (P: Prop).(let H23 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
-lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (TLRef _) \Rightarrow
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
-t4))]) in lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (THead _ t0 _)
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
-(lift (S O) O x1) x0) H22) in (let H24 \def (eq_ind_r T x1 (\lambda (t0:
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H21 x (lift_inj x x1 (S O) O
-H23)) in (let H25 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
-(lift_inj x x1 (S O) O H23)) in (H24 (refl_equal T x) P)))))) (pr3_flat
-(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
-(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
-(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl)))
-H20))) (THead (Flat Appl) (lift (S O) O x1) (lift (S O) O x2)) (pr3_thin_dx
-(CHead c (Bind b) t1) x0 (lift (S O) O x2) H19 (lift (S O) O x1) Appl)) (lift
-(S O) O (THead (Flat Appl) x1 x2)) (lift_head (Flat Appl) x1 x2 (S O) O)) c
-(drop_drop (Bind b) O c c (drop_refl c) t1))) H18))) t3 H14))))))) H13))
-(\lambda (H13: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
-T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0:
-T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))))).(ex4_4_ind T T T T (\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
-b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4:
-T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) z1 t4)))))))
-(sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
-T).(\lambda (H14: (eq T (THead (Bind b) t1 x0) (THead (Bind Abst) x1
-x2))).(\lambda (H15: (eq T t3 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c
-x x3)).(\lambda (H17: ((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind
-b0) u0) x2 x4))))).(let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T (THead
-(Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) H10
-(THead (Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4)
-(\lambda (t0: T).(sn3 c t0)) (let H19 \def (f_equal T B (\lambda (e:
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
-(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
-b])])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in ((let H20
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _)
-\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in
-((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
-t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14)
-in (\lambda (_: (eq T t1 x1)).(\lambda (H23: (eq B b Abst)).(let H24 \def
-(eq_ind_r T x2 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead
-c (Bind b0) u0) t0 x4)))) H17 x0 H21) in (let H25 \def (eq_ind B b (\lambda
-(b0: B).((eq T (THead (Flat Appl) x (THead (Bind b0) t1 x0)) (THead (Bind
-Abbr) x3 x4)) \to (\forall (P: Prop).P))) H18 Abst H23) in (let H26 \def
-(eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T (THead (Flat Appl)
-(lift (S O) O x) x0) t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind
-b0) t1) (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (sn3 (CHead c (Bind
-b0) t1) t4))))) H9 Abst H23) in (let H27 \def (eq_ind B b (\lambda (b0:
-B).(\forall (t4: T).((((eq T (THead (Flat Appl) (lift (S O) O x) x0) t4) \to
-(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) (THead (Flat Appl)
-(lift (S O) O x) x0) t4) \to (\forall (x5: T).(\forall (x6: T).((eq T t4
-(THead (Flat Appl) (lift (S O) O x5) x6)) \to (sn3 c (THead (Flat Appl) x5
-(THead (Bind b0) t1 x6)))))))))) H8 Abst H23) in (let H28 \def (eq_ind B b
-(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P)))
-\to ((pr3 c t1 t4) \to (\forall (t0: T).(\forall (v0: T).((sn3 (CHead c (Bind
-b0) t4) (THead (Flat Appl) (lift (S O) O v0) t0)) \to (sn3 c (THead (Flat
-Appl) v0 (THead (Bind b0) t4 t0)))))))))) H2 Abst H23) in (let H29 \def
-(eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H Abst H23) in (let H30
-\def (match (H29 (refl_equal B Abst)) in False return (\lambda (_:
-False).(sn3 c (THead (Bind Abbr) x3 x4))) with []) in H30)))))))))) H20))
-H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 B T T T T T (\lambda (b0:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b)
-t1 x0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
-t3 (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
-y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1
-z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0
-Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b0) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b0:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2))))))) (sn3 c t3) (\lambda (x1:
-B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
-T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15: (eq T
-(THead (Bind b) t1 x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T t3
-(THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
-(H17: (pr2 c x x5)).(\lambda (H18: (pr2 c x2 x6)).(\lambda (H19: (pr2 (CHead
-c (Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t0: T).((eq T
-(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P)))
-H10 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) H16) in
-(eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))
-(\lambda (t0: T).(sn3 c t0)) (let H21 \def (f_equal T B (\lambda (e:
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
-(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
-b])])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in ((let H22 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _)
-\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in
-((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
-t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in
-(\lambda (H24: (eq T t1 x2)).(\lambda (H25: (eq B b x1)).(let H26 \def
-(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H19 x0
-H23) in (let H27 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H18 t1
-H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b0: B).(pr2 (CHead c (Bind b0)
-x6) x0 x4)) H26 b H25) in (eq_ind B b (\lambda (b0: B).(sn3 c (THead (Bind
-b0) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (sn3_pr3_trans c (THead
-(Bind b) t1 (THead (Flat Appl) (lift (S O) O x5) x4)) (sn3_bind b c t1
-(sn3_sing c t1 H1) (THead (Flat Appl) (lift (S O) O x5) x4) (let H_x \def
-(term_dec x x5) in (let H29 \def H_x in (or_ind (eq T x x5) ((eq T x x5) \to
-(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S
-O) O x5) x4)) (\lambda (H30: (eq T x x5)).(let H31 \def (eq_ind_r T x5
-(\lambda (t0: T).(pr2 c x t0)) H17 x H30) in (eq_ind T x (\lambda (t0:
-T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x4))) (let
-H_x0 \def (term_dec x0 x4) in (let H32 \def H_x0 in (or_ind (eq T x0 x4) ((eq
-T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat
-Appl) (lift (S O) O x) x4)) (\lambda (H33: (eq T x0 x4)).(let H34 \def
-(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0
-H33) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat
-Appl) (lift (S O) O x) t0))) (sn3_sing (CHead c (Bind b) t1) (THead (Flat
-Appl) (lift (S O) O x) x0) H9) x4 H33))) (\lambda (H33: (((eq T x0 x4) \to
-(\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x) x4) (\lambda
-(H34: (eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
-(S O) O x) x4))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x) x4) H34) in
-(let H36 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0:
-Prop).P0))) H33 x0 H35) in (let H37 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2
-(CHead c (Bind b) x6) x0 t0)) H28 x0 H35) in (H36 (refl_equal T x0) P))))))
-(pr3_pr3_pr3_t c t1 x6 (pr3_pr2 c t1 x6 H27) (THead (Flat Appl) (lift (S O) O
-x) x0) (THead (Flat Appl) (lift (S O) O x) x4) (Bind b) (pr3_pr2 (CHead c
-(Bind b) x6) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
-(S O) O x) x4) (pr2_thin_dx (CHead c (Bind b) x6) x0 x4 H28 (lift (S O) O x)
-Appl))))) H32))) x5 H30))) (\lambda (H30: (((eq T x x5) \to (\forall (P:
-Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x5) x4) (\lambda (H31: (eq T
-(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5)
-x4))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
-(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
-lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (TLRef _) \Rightarrow
-((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
-t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
-(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
-(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
-t4))]) in lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (THead _ t0 _)
-\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
-(lift (S O) O x5) x4) H31) in ((let H33 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
-Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) x4) H31) in
-(\lambda (H34: (eq T (lift (S O) O x) (lift (S O) O x5))).(let H35 \def
-(eq_ind_r T x5 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
-H30 x (lift_inj x x5 (S O) O H34)) in (let H36 \def (eq_ind_r T x5 (\lambda
-(t0: T).(pr2 c x t0)) H17 x (lift_inj x x5 (S O) O H34)) in (let H37 \def
-(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0
-H33) in (H35 (refl_equal T x) P)))))) H32)))) (pr3_pr3_pr3_t c t1 x6 (pr3_pr2
-c t1 x6 H27) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
-(S O) O x5) x4) (Bind b) (pr3_flat (CHead c (Bind b) x6) (lift (S O) O x)
-(lift (S O) O x5) (pr3_lift (CHead c (Bind b) x6) c (S O) O (drop_drop (Bind
-b) O c c (drop_refl c) x6) x x5 (pr3_pr2 c x x5 H17)) x0 x4 (pr3_pr2 (CHead c
-(Bind b) x6) x0 x4 H28) Appl)))) H29)))) (THead (Bind b) x6 (THead (Flat
-Appl) (lift (S O) O x5) x4)) (pr3_pr2 c (THead (Bind b) t1 (THead (Flat Appl)
-(lift (S O) O x5) x4)) (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O
-x5) x4)) (pr2_head_1 c t1 x6 H27 (Bind b) (THead (Flat Appl) (lift (S O) O
-x5) x4)))) x1 H25))))))) H22)) H21)) t3 H16)))))))))))))) H13))
-H12)))))))))))))) y H4))))) H3))))))) u H0))))).
-
-theorem sn3_appl_beta:
- \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((sn3 c
-(THead (Flat Appl) u (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w)
-\to (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind Abst) w
-t))))))))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
-(sn3 c (THead (Flat Appl) u (THead (Bind Abbr) v t)))).(\lambda (w:
-T).(\lambda (H0: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THead (Bind
-Abbr) v t) H) in (let H1 \def H_x in (and_ind (sn3 c u) (sn3 c (THead (Bind
-Abbr) v t)) (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind
-Abst) w t)))) (\lambda (H2: (sn3 c u)).(\lambda (H3: (sn3 c (THead (Bind
-Abbr) v t))).(sn3_appl_appl v (THead (Bind Abst) w t) c (sn3_beta c v t H3 w
-H0) u H2 (\lambda (u2: T).(\lambda (H4: (pr3 c (THead (Flat Appl) v (THead
-(Bind Abst) w t)) u2)).(\lambda (H5: (((iso (THead (Flat Appl) v (THead (Bind
-Abst) w t)) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat
-Appl) u (THead (Bind Abbr) v t)) H (THead (Flat Appl) u u2) (pr3_thin_dx c
-(THead (Bind Abbr) v t) u2 (pr3_iso_beta v w t c u2 H4 H5) u Appl))))))))
-H1))))))))).
-
-theorem sn3_appls_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u:
-T).((sn3 c u) \to (\forall (vs: TList).(\forall (t: T).((sn3 (CHead c (Bind
-b) u) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to (sn3 c (THeads (Flat
-Appl) vs (THead (Bind b) u t))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda
-(u: T).(\lambda (H0: (sn3 c u)).(\lambda (vs: TList).(TList_ind (\lambda (t:
-TList).(\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) (lifts
-(S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u t0))))))
-(\lambda (t: T).(\lambda (H1: (sn3 (CHead c (Bind b) u) t)).(sn3_bind b c u
-H0 t H1))) (\lambda (v: T).(\lambda (vs0: TList).(TList_ind (\lambda (t:
-TList).(((\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl)
-(lifts (S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u
-t0)))))) \to (\forall (t0: T).((sn3 (CHead c (Bind b) u) (THead (Flat Appl)
-(lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t) t0))) \to (sn3 c
-(THead (Flat Appl) v (THeads (Flat Appl) t (THead (Bind b) u t0))))))))
-(\lambda (_: ((\forall (t: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl)
-(lifts (S O) O TNil) t)) \to (sn3 c (THeads (Flat Appl) TNil (THead (Bind b)
-u t))))))).(\lambda (t: T).(\lambda (H2: (sn3 (CHead c (Bind b) u) (THead
-(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O TNil)
-t)))).(sn3_appl_bind b H c u H0 t v H2)))) (\lambda (t: T).(\lambda (t0:
-TList).(\lambda (_: ((((\forall (t1: T).((sn3 (CHead c (Bind b) u) (THeads
-(Flat Appl) (lifts (S O) O t0) t1)) \to (sn3 c (THeads (Flat Appl) t0 (THead
-(Bind b) u t1)))))) \to (\forall (t1: T).((sn3 (CHead c (Bind b) u) (THead
-(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t0) t1))) \to
-(sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (THead (Bind b) u
-t1))))))))).(\lambda (H2: ((\forall (t1: T).((sn3 (CHead c (Bind b) u)
-(THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) \to (sn3 c (THeads
-(Flat Appl) (TCons t t0) (THead (Bind b) u t1))))))).(\lambda (t1:
-T).(\lambda (H3: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O
-v) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))).(let H_x \def
-(sn3_gen_flat Appl (CHead c (Bind b) u) (lift (S O) O v) (THeads (Flat Appl)
-(lifts (S O) O (TCons t t0)) t1) H3) in (let H4 \def H_x in (and_ind (sn3
-(CHead c (Bind b) u) (lift (S O) O v)) (sn3 (CHead c (Bind b) u) (THeads
-(Flat Appl) (lifts (S O) O (TCons t t0)) t1)) (sn3 c (THead (Flat Appl) v
-(THeads (Flat Appl) (TCons t t0) (THead (Bind b) u t1)))) (\lambda (H5: (sn3
-(CHead c (Bind b) u) (lift (S O) O v))).(\lambda (H6: (sn3 (CHead c (Bind b)
-u) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1))).(let H_y \def
-(sn3_gen_lift (CHead c (Bind b) u) v (S O) O H5 c) in (sn3_appl_appls t
-(THead (Bind b) u t1) t0 c (H2 t1 H6) v (H_y (drop_drop (Bind b) O c c
-(drop_refl c) u)) (\lambda (u2: T).(\lambda (H7: (pr3 c (THeads (Flat Appl)
-(TCons t t0) (THead (Bind b) u t1)) u2)).(\lambda (H8: (((iso (THeads (Flat
-Appl) (TCons t t0) (THead (Bind b) u t1)) u2) \to (\forall (P:
-Prop).P)))).(let H9 \def (pr3_iso_appls_bind b H (TCons t t0) u t1 c u2 H7
-H8) in (sn3_pr3_trans c (THead (Flat Appl) v (THead (Bind b) u (THeads (Flat
-Appl) (lifts (S O) O (TCons t t0)) t1))) (sn3_appl_bind b H c u H0 (THeads
-(Flat Appl) (lifts (S O) O (TCons t t0)) t1) v H3) (THead (Flat Appl) v u2)
-(pr3_flat c v v (pr3_refl c v) (THead (Bind b) u (THeads (Flat Appl) (lifts
-(S O) O (TCons t t0)) t1)) u2 H9 Appl)))))))))) H4))))))))) vs0))) vs)))))).
-
-theorem sn3_appls_beta:
- \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (us: TList).((sn3 c
-(THeads (Flat Appl) us (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c
-w) \to (sn3 c (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst)
-w t))))))))))
-\def
- \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (us:
-TList).(TList_ind (\lambda (t0: TList).((sn3 c (THeads (Flat Appl) t0 (THead
-(Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat
-Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) (\lambda (H:
-(sn3 c (THead (Bind Abbr) v t))).(\lambda (w: T).(\lambda (H0: (sn3 c
-w)).(sn3_beta c v t H w H0)))) (\lambda (u: T).(\lambda (us0:
-TList).(TList_ind (\lambda (t0: TList).((((sn3 c (THeads (Flat Appl) t0
-(THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads
-(Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3
-c (THead (Flat Appl) u (THeads (Flat Appl) t0 (THead (Bind Abbr) v t)))) \to
-(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat
-Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))) (\lambda (_:
-(((sn3 c (THeads (Flat Appl) TNil (THead (Bind Abbr) v t))) \to (\forall (w:
-T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) TNil (THead (Flat Appl) v (THead
-(Bind Abst) w t))))))))).(\lambda (H0: (sn3 c (THead (Flat Appl) u (THeads
-(Flat Appl) TNil (THead (Bind Abbr) v t))))).(\lambda (w: T).(\lambda (H1:
-(sn3 c w)).(sn3_appl_beta c u v t H0 w H1))))) (\lambda (t0: T).(\lambda (t1:
-TList).(\lambda (_: (((((sn3 c (THeads (Flat Appl) t1 (THead (Bind Abbr) v
-t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) t1 (THead
-(Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 c (THead (Flat Appl) u
-(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)))) \to (\forall (w: T).((sn3 c
-w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) t1 (THead (Flat Appl)
-v (THead (Bind Abst) w t))))))))))).(\lambda (H0: (((sn3 c (THeads (Flat
-Appl) (TCons t0 t1) (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w)
-\to (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead
-(Bind Abst) w t))))))))).(\lambda (H1: (sn3 c (THead (Flat Appl) u (THeads
-(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t))))).(\lambda (w:
-T).(\lambda (H2: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THeads
-(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t)) H1) in (let H3 \def H_x in
-(and_ind (sn3 c u) (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Bind
-Abbr) v t))) (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1)
-(THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H4: (sn3 c
-u)).(\lambda (H5: (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr)
-v t)))).(sn3_appl_appls t0 (THead (Flat Appl) v (THead (Bind Abst) w t)) t1 c
-(H0 H5 w H2) u H4 (\lambda (u2: T).(\lambda (H6: (pr3 c (THeads (Flat Appl)
-(TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w t))) u2)).(\lambda
-(H7: (((iso (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead
-(Bind Abst) w t))) u2) \to (\forall (P: Prop).P)))).(let H8 \def
-(pr3_iso_appls_beta (TCons t0 t1) v w t c u2 H6 H7) in (sn3_pr3_trans c
-(THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v
-t))) H1 (THead (Flat Appl) u u2) (pr3_thin_dx c (THeads (Flat Appl) (TCons t0
-t1) (THead (Bind Abbr) v t)) u2 H8 u Appl))))))))) H3)))))))))) us0))) us)))).
-
-theorem sn3_appls_abbr:
- \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c
-(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).((sn3 c (THeads (Flat Appl)
-vs (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda
-(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind
-(\lambda (t: TList).((sn3 c (THeads (Flat Appl) t (lift (S i) O w))) \to (sn3
-c (THeads (Flat Appl) t (TLRef i))))) (\lambda (H0: (sn3 c (lift (S i) O
-w))).(let H_y \def (sn3_gen_lift c w (S i) O H0 d (getl_drop Abbr c d w i H))
-in (sn3_abbr c d w i H H_y))) (\lambda (v: T).(\lambda (vs0:
-TList).(TList_ind (\lambda (t: TList).((((sn3 c (THeads (Flat Appl) t (lift
-(S i) O w))) \to (sn3 c (THeads (Flat Appl) t (TLRef i))))) \to ((sn3 c
-(THead (Flat Appl) v (THeads (Flat Appl) t (lift (S i) O w)))) \to (sn3 c
-(THead (Flat Appl) v (THeads (Flat Appl) t (TLRef i))))))) (\lambda (_:
-(((sn3 c (THeads (Flat Appl) TNil (lift (S i) O w))) \to (sn3 c (THeads (Flat
-Appl) TNil (TLRef i)))))).(\lambda (H1: (sn3 c (THead (Flat Appl) v (THeads
-(Flat Appl) TNil (lift (S i) O w))))).(nf3_appl_abbr c d w i H v H1)))
-(\lambda (t: T).(\lambda (t0: TList).(\lambda (_: (((((sn3 c (THeads (Flat
-Appl) t0 (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) t0 (TLRef i)))))
-\to ((sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (lift (S i) O w))))
-\to (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (TLRef
-i)))))))).(\lambda (H1: (((sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i)
-O w))) \to (sn3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)))))).(\lambda
-(H2: (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (lift (S i)
-O w))))).(let H_x \def (sn3_gen_flat Appl c v (THeads (Flat Appl) (TCons t
-t0) (lift (S i) O w)) H2) in (let H3 \def H_x in (and_ind (sn3 c v) (sn3 c
-(THeads (Flat Appl) (TCons t t0) (lift (S i) O w))) (sn3 c (THead (Flat Appl)
-v (THeads (Flat Appl) (TCons t t0) (TLRef i)))) (\lambda (H4: (sn3 c
-v)).(\lambda (H5: (sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i) O
-w)))).(sn3_appl_appls t (TLRef i) t0 c (H1 H5) v H4 (\lambda (u2: T).(\lambda
-(H6: (pr3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2)).(\lambda (H7:
-(((iso (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2) \to (\forall (P:
-Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) v (THeads (Flat Appl) (TCons
-t t0) (lift (S i) O w))) H2 (THead (Flat Appl) v u2) (pr3_thin_dx c (THeads
-(Flat Appl) (TCons t t0) (lift (S i) O w)) u2 (pr3_iso_appls_abbr c d w i H
-(TCons t t0) u2 H6 H7) v Appl)))))))) H3)))))))) vs0))) vs)))))).
-
-theorem sn3_gen_def:
- \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
-(CHead d (Bind Abbr) v)) \to ((sn3 c (TLRef i)) \to (sn3 d v))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
-(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 c (TLRef
-i))).(sn3_gen_lift c v (S i) O (sn3_pr3_trans c (TLRef i) H0 (lift (S i) O v)
-(pr3_pr2 c (TLRef i) (lift (S i) O v) (pr2_delta c d v i H (TLRef i) (TLRef
-i) (pr0_refl (TLRef i)) (lift (S i) O v) (subst0_lref v i)))) d (getl_drop
-Abbr c d v i H))))))).
-
-theorem sn3_cdelta:
- \forall (v: T).(\forall (t: T).(\forall (i: nat).(((\forall (w: T).(ex T
-(\lambda (u: T).(subst0 i w t u))))) \to (\forall (c: C).(\forall (d:
-C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to (sn3 d v))))))))
-\def
- \lambda (v: T).(\lambda (t: T).(\lambda (i: nat).(\lambda (H: ((\forall (w:
-T).(ex T (\lambda (u: T).(subst0 i w t u)))))).(let H_x \def (H v) in (let H0
-\def H_x in (ex_ind T (\lambda (u: T).(subst0 i v t u)) (\forall (c:
-C).(\forall (d: C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to
-(sn3 d v))))) (\lambda (x: T).(\lambda (H1: (subst0 i v t x)).(subst0_ind
-(\lambda (n: nat).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_: T).(\forall
-(c: C).(\forall (d: C).((getl n c (CHead d (Bind Abbr) t0)) \to ((sn3 c t1)
-\to (sn3 d t0))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (c:
-C).(\lambda (d: C).(\lambda (H2: (getl i0 c (CHead d (Bind Abbr)
-v0))).(\lambda (H3: (sn3 c (TLRef i0))).(sn3_gen_def c d v0 i0 H2 H3)))))))
-(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0:
-nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c:
-C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to
-(sn3 d v0))))))).(\lambda (t0: T).(\lambda (k: K).(\lambda (c: C).(\lambda
-(d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) v0))).(\lambda (H5: (sn3
-c (THead k u1 t0))).(let H_y \def (sn3_gen_head k c u1 t0 H5) in (H3 c d H4
-H_y)))))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: T).(\lambda
-(t1: T).(\lambda (i0: nat).(\lambda (H2: (subst0 (s k i0) v0 t1 t2)).(\lambda
-(H3: ((\forall (c: C).(\forall (d: C).((getl (s k i0) c (CHead d (Bind Abbr)
-v0)) \to ((sn3 c t1) \to (sn3 d v0))))))).(\lambda (u: T).(\lambda (c:
-C).(\lambda (d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr)
-v0))).(\lambda (H5: (sn3 c (THead k u t1))).(K_ind (\lambda (k0: K).((subst0
-(s k0 i0) v0 t1 t2) \to (((\forall (c0: C).(\forall (d0: C).((getl (s k0 i0)
-c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0)))))) \to ((sn3
-c (THead k0 u t1)) \to (sn3 d v0))))) (\lambda (b: B).(\lambda (_: (subst0 (s
-(Bind b) i0) v0 t1 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0:
-C).((getl (s (Bind b) i0) c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to
-(sn3 d0 v0))))))).(\lambda (H8: (sn3 c (THead (Bind b) u t1))).(let H_x0 \def
-(sn3_gen_bind b c u t1 H8) in (let H9 \def H_x0 in (and_ind (sn3 c u) (sn3
-(CHead c (Bind b) u) t1) (sn3 d v0) (\lambda (_: (sn3 c u)).(\lambda (H11:
-(sn3 (CHead c (Bind b) u) t1)).(H7 (CHead c (Bind b) u) d (getl_clear_bind b
-(CHead c (Bind b) u) c u (clear_bind b c u) (CHead d (Bind Abbr) v0) i0 H4)
-H11))) H9))))))) (\lambda (f: F).(\lambda (_: (subst0 (s (Flat f) i0) v0 t1
-t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: C).((getl (s (Flat f) i0)
-c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0))))))).(\lambda
-(H8: (sn3 c (THead (Flat f) u t1))).(let H_x0 \def (sn3_gen_flat f c u t1 H8)
-in (let H9 \def H_x0 in (and_ind (sn3 c u) (sn3 c t1) (sn3 d v0) (\lambda (_:
-(sn3 c u)).(\lambda (H11: (sn3 c t1)).(H7 c d H4 H11))) H9))))))) k H2 H3
-H5))))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda
-(i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c:
-C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to
-(sn3 d v0))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: ((\forall (c: C).(\forall (d:
-C).((getl (s k i0) c (CHead d (Bind Abbr) v0)) \to ((sn3 c t1) \to (sn3 d
-v0))))))).(\lambda (c: C).(\lambda (d: C).(\lambda (H6: (getl i0 c (CHead d
-(Bind Abbr) v0))).(\lambda (H7: (sn3 c (THead k u1 t1))).(let H_y \def
-(sn3_gen_head k c u1 t1 H7) in (H3 c d H6 H_y))))))))))))))))) i v t x H1)))
-H0)))))).
-
-inductive csubn: C \to (C \to Prop) \def
-| csubn_sort: \forall (n: nat).(csubn (CSort n) (CSort n))
-| csubn_head: \forall (c1: C).(\forall (c2: C).((csubn c1 c2) \to (\forall
-(k: K).(\forall (v: T).(csubn (CHead c1 k v) (CHead c2 k v))))))
-| csubn_abst: \forall (c1: C).(\forall (c2: C).((csubn c1 c2) \to (\forall
-(v: T).(\forall (w: T).((sn3 c2 w) \to (csubn (CHead c1 (Bind Abst) v) (CHead
-c2 (Bind Abbr) w))))))).
-
-theorem csubc_csuba:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (csuba
-g c1 c2))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1
-c2)).(csubc_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda
-(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda
-(_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda
-(v: T).(csuba_head g c3 c4 H1 k v))))))) (\lambda (c3: C).(\lambda (c4:
-C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (v:
-T).(\lambda (a: A).(\lambda (H2: (sc3 g (asucc g a) c3 v)).(\lambda (w:
-T).(\lambda (H3: (sc3 g a c4 w)).(csuba_abst g c3 c4 H1 v a (sc3_arity_gen g
-c3 v (asucc g a) H2) w (sc3_arity_gen g c4 w a H3))))))))))) c1 c2 H)))).
-
-theorem csubc_csubn:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (csubn
-c1 c2))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1
-c2)).(csubc_ind g (\lambda (c: C).(\lambda (c0: C).(csubn c c0))) (\lambda
-(n: nat).(csubn_sort n)) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_:
-(csubc g c3 c4)).(\lambda (H1: (csubn c3 c4)).(\lambda (k: K).(\lambda (v:
-T).(csubn_head c3 c4 H1 k v))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda
-(_: (csubc g c3 c4)).(\lambda (H1: (csubn c3 c4)).(\lambda (v: T).(\lambda
-(a: A).(\lambda (_: (sc3 g (asucc g a) c3 v)).(\lambda (w: T).(\lambda (H3:
-(sc3 g a c4 w)).(csubn_abst c3 c4 H1 v w (sc3_sn3 g a c4 w H3))))))))))) c1
-c2 H)))).
-
-theorem ceq_arity_trans:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((ceqc g c2 c1) \to
-(\forall (t: T).(\forall (a: A).((arity g c1 t a) \to (arity g c2 t a)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (ceqc g c2
-c1)).(\lambda (t: T).(\lambda (a: A).(\lambda (H0: (arity g c1 t a)).(let H1
-\def H in (or_ind (csubc g c2 c1) (csubc g c1 c2) (arity g c2 t a) (\lambda
-(H2: (csubc g c2 c1)).(csuba_arity_rev g c1 t a H0 c2 (csubc_csuba g c2 c1
-H2))) (\lambda (H2: (csubc g c1 c2)).(csuba_arity g c1 t a H0 c2 (csubc_csuba
-g c1 c2 H2))) H1)))))))).
-
projects / helm.git / blobdiff