X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fspare.ma;h=f40cf08fecd59a2717d8d4609aadfecd91ae2ffd;hb=076f639446efce8d8cf83dcf7ca40b4376fc8c36;hp=70b94347b334b6a1f0fb901ec1a997d90ea53c43;hpb=fdda444a05fe4c68c925cd94e4e3a38c93d0c35f;p=helm.git

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/spare.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/spare.ma
index 70b94347b..f40cf08fe 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/spare.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/spare.ma
@@ -14,105 +14,529 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/spare".
+include "LambdaDelta-1/theory.ma".
 
-include "theory.ma".
+definition cbk:
+ C \to nat
+\def
+ let rec cbk (c: C) on c: nat \def (match c with [(CSort m) \Rightarrow m | 
+(CHead c0 _ _) \Rightarrow (cbk c0)]) in cbk.
+
+definition app1:
+ C \to (T \to T)
+\def
+ let rec app1 (c: C) on c: (T \to T) \def (\lambda (t: T).(match c with 
+[(CSort _) \Rightarrow t | (CHead c0 k u) \Rightarrow (app1 c0 (THead k u 
+t))])) in app1.
+
+theorem lifts_inj:
+ \forall (xs: TList).(\forall (ts: TList).(\forall (h: nat).(\forall (d: 
+nat).((eq TList (lifts h d xs) (lifts h d ts)) \to (eq TList xs ts)))))
+\def
+ \lambda (xs: TList).(TList_ind (\lambda (t: TList).(\forall (ts: 
+TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t) (lifts h 
+d ts)) \to (eq TList t ts)))))) (\lambda (ts: TList).(TList_ind (\lambda (t: 
+TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d TNil) (lifts 
+h d t)) \to (eq TList TNil t))))) (\lambda (_: nat).(\lambda (_: 
+nat).(\lambda (H: (eq TList TNil TNil)).H))) (\lambda (t: T).(\lambda (t0: 
+TList).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq TList TNil 
+(lifts h d t0)) \to (eq TList TNil t0)))))).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H0: (eq TList TNil (TCons (lift h d t) (lifts h d t0)))).(let 
+H1 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList return 
+(\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _) 
+\Rightarrow False])) I (TCons (lift h d t) (lifts h d t0)) H0) in (False_ind 
+(eq TList TNil (TCons t t0)) H1)))))))) ts)) (\lambda (t: T).(\lambda (t0: 
+TList).(\lambda (H: ((\forall (ts: TList).(\forall (h: nat).(\forall (d: 
+nat).((eq TList (lifts h d t0) (lifts h d ts)) \to (eq TList t0 
+ts))))))).(\lambda (ts: TList).(TList_ind (\lambda (t1: TList).(\forall (h: 
+nat).(\forall (d: nat).((eq TList (lifts h d (TCons t t0)) (lifts h d t1)) 
+\to (eq TList (TCons t t0) t1))))) (\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H0: (eq TList (TCons (lift h d t) (lifts h d t0)) TNil)).(let 
+H1 \def (eq_ind TList (TCons (lift h d t) (lifts h d t0)) (\lambda (ee: 
+TList).(match ee in TList return (\lambda (_: TList).Prop) with [TNil 
+\Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H0) in (False_ind 
+(eq TList (TCons t t0) TNil) H1))))) (\lambda (t1: T).(\lambda (t2: 
+TList).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq TList (TCons 
+(lift h d t) (lifts h d t0)) (lifts h d t2)) \to (eq TList (TCons t t0) 
+t2)))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq TList (TCons 
+(lift h d t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d t2)))).(let H2 
+\def (f_equal TList T (\lambda (e: TList).(match e in TList return (\lambda 
+(_: TList).T) with [TNil \Rightarrow ((let rec lref_map (f: ((nat \to nat))) 
+(d0: nat) (t3: T) on t3: T \def (match t3 with [(TSort n) \Rightarrow (TSort 
+n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i 
+| false \Rightarrow (f i)])) | (THead k u t4) \Rightarrow (THead k (lref_map 
+f d0 u) (lref_map f (s k d0) t4))]) in lref_map) (\lambda (x: nat).(plus x 
+h)) d t) | (TCons t3 _) \Rightarrow t3])) (TCons (lift h d t) (lifts h d t0)) 
+(TCons (lift h d t1) (lifts h d t2)) H1) in ((let H3 \def (f_equal TList 
+TList (\lambda (e: TList).(match e in TList return (\lambda (_: TList).TList) 
+with [TNil \Rightarrow ((let rec lifts (h0: nat) (d0: nat) (ts0: TList) on 
+ts0: TList \def (match ts0 with [TNil \Rightarrow TNil | (TCons t3 ts1) 
+\Rightarrow (TCons (lift h0 d0 t3) (lifts h0 d0 ts1))]) in lifts) h d t0) | 
+(TCons _ t3) \Rightarrow t3])) (TCons (lift h d t) (lifts h d t0)) (TCons 
+(lift h d t1) (lifts h d t2)) H1) in (\lambda (H4: (eq T (lift h d t) (lift h 
+d t1))).(eq_ind T t (\lambda (t3: T).(eq TList (TCons t t0) (TCons t3 t2))) 
+(f_equal2 T TList TList TCons t t t0 t2 (refl_equal T t) (H t2 h d H3)) t1 
+(lift_inj t t1 h d H4)))) H2)))))))) ts))))) xs).
+
+theorem nfs2_tapp:
+ \forall (c: C).(\forall (t: T).(\forall (ts: TList).((nfs2 c (TApp ts t)) 
+\to (land (nfs2 c ts) (nf2 c t)))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (ts: TList).(TList_ind (\lambda (t0: 
+TList).((nfs2 c (TApp t0 t)) \to (land (nfs2 c t0) (nf2 c t)))) (\lambda (H: 
+(land (nf2 c t) True)).(let H0 \def H in (and_ind (nf2 c t) True (land True 
+(nf2 c t)) (\lambda (H1: (nf2 c t)).(\lambda (_: True).(conj True (nf2 c t) I 
+H1))) H0))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (((nfs2 c 
+(TApp t1 t)) \to (land (nfs2 c t1) (nf2 c t))))).(\lambda (H0: (land (nf2 c 
+t0) (nfs2 c (TApp t1 t)))).(let H1 \def H0 in (and_ind (nf2 c t0) (nfs2 c 
+(TApp t1 t)) (land (land (nf2 c t0) (nfs2 c t1)) (nf2 c t)) (\lambda (H2: 
+(nf2 c t0)).(\lambda (H3: (nfs2 c (TApp t1 t))).(let H_x \def (H H3) in (let 
+H4 \def H_x in (and_ind (nfs2 c t1) (nf2 c t) (land (land (nf2 c t0) (nfs2 c 
+t1)) (nf2 c t)) (\lambda (H5: (nfs2 c t1)).(\lambda (H6: (nf2 c t)).(conj 
+(land (nf2 c t0) (nfs2 c t1)) (nf2 c t) (conj (nf2 c t0) (nfs2 c t1) H2 H5) 
+H6))) H4))))) H1)))))) ts))).
+
+theorem pc3_nf2_unfold:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c 
+t2) \to (pr3 c t1 t2)))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 
+t2)).(\lambda (H0: (nf2 c t2)).(let H1 \def H in (ex2_ind T (\lambda (t: 
+T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pr3 c t1 t2) (\lambda (x: 
+T).(\lambda (H2: (pr3 c t1 x)).(\lambda (H3: (pr3 c t2 x)).(let H_y \def 
+(nf2_pr3_unfold c t2 x H3 H0) in (let H4 \def (eq_ind_r T x (\lambda (t: 
+T).(pr3 c t1 t)) H2 t2 H_y) in H4))))) H1)))))).
+
+theorem pc3_pr3_conf:
+ \forall (c: C).(\forall (t: T).(\forall (t1: T).((pc3 c t t1) \to (\forall 
+(t2: T).((pr3 c t t2) \to (pc3 c t2 t1))))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pc3 c t 
+t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t t2)).(pc3_t t c t2 (pc3_pr3_x c 
+t2 t H0) t1 H)))))).
+
+axiom pc3_gen_appls_sort_abst:
+ \forall (c: C).(\forall (vs: TList).(\forall (w: T).(\forall (u: T).(\forall 
+(n: nat).((pc3 c (THeads (Flat Appl) vs (TSort n)) (THead (Bind Abst) w u)) 
+\to False)))))
+.
+
+axiom pc3_gen_appls_lref_abst:
+ \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (w: T).(\forall 
+(u: T).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THead (Bind Abst) w u)) \to 
+False))))))))
+.
 
-inductive sort: T \to Prop \def
-| sort_sort: \forall (n: nat).(sort (TSort n))
-| sort_abst: \forall (u: T).((sort u) \to (\forall (t: T).((sort t) \to (sort 
-(THead (Bind Abst) u t))))).
+axiom pc3_gen_appls_lref_sort:
+ \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (ws: 
+TList).(\forall (n: nat).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THeads 
+(Flat Appl) ws (TSort n))) \to False))))))))
+.
 
-theorem sort_nf2:
- \forall (t: T).((sort t) \to (\forall (c: C).(nf2 c t)))
+inductive tys3 (g: G) (c: C): TList \to (T \to Prop) \def
+| tys3_nil: \forall (u: T).(\forall (u0: T).((ty3 g c u u0) \to (tys3 g c 
+TNil u)))
+| tys3_cons: \forall (t: T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts: 
+TList).((tys3 g c ts u) \to (tys3 g c (TCons t ts) u))))).
+
+theorem tys3_gen_nil:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).((tys3 g c TNil u) \to (ex T 
+(\lambda (u0: T).(ty3 g c u u0))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (tys3 g c TNil 
+u)).(insert_eq TList TNil (\lambda (t: TList).(tys3 g c t u)) (\lambda (_: 
+TList).(ex T (\lambda (u0: T).(ty3 g c u u0)))) (\lambda (y: TList).(\lambda 
+(H0: (tys3 g c y u)).(tys3_ind g c (\lambda (t: TList).(\lambda (t0: T).((eq 
+TList t TNil) \to (ex T (\lambda (u0: T).(ty3 g c t0 u0)))))) (\lambda (u0: 
+T).(\lambda (u1: T).(\lambda (H1: (ty3 g c u0 u1)).(\lambda (_: (eq TList 
+TNil TNil)).(ex_intro T (\lambda (u2: T).(ty3 g c u0 u2)) u1 H1))))) (\lambda 
+(t: T).(\lambda (u0: T).(\lambda (_: (ty3 g c t u0)).(\lambda (ts: 
+TList).(\lambda (_: (tys3 g c ts u0)).(\lambda (_: (((eq TList ts TNil) \to 
+(ex T (\lambda (u1: T).(ty3 g c u0 u1)))))).(\lambda (H4: (eq TList (TCons t 
+ts) TNil)).(let H5 \def (eq_ind TList (TCons t ts) (\lambda (ee: 
+TList).(match ee in TList return (\lambda (_: TList).Prop) with [TNil 
+\Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H4) in (False_ind 
+(ex T (\lambda (u1: T).(ty3 g c u0 u1))) H5))))))))) y u H0))) H)))).
+
+theorem tys3_gen_cons:
+ \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (t: T).(\forall 
+(u: T).((tys3 g c (TCons t ts) u) \to (land (ty3 g c t u) (tys3 g c ts 
+u)))))))
 \def
- \lambda (t: T).(\lambda (H: (sort t)).(sort_ind (\lambda (t0: T).(\forall 
-(c: C).(nf2 c t0))) (\lambda (n: nat).(\lambda (c: C).(nf2_sort c n))) 
-(\lambda (u: T).(\lambda (_: (sort u)).(\lambda (H1: ((\forall (c: C).(nf2 c 
-u)))).(\lambda (t0: T).(\lambda (_: (sort t0)).(\lambda (H3: ((\forall (c: 
-C).(nf2 c t0)))).(\lambda (c: C).(let H_y \def (H3 (CHead c (Bind Abst) u)) 
-in (nf2_abst_shift c u (H1 c) t0 H_y))))))))) t H)).
-
-theorem sort_pc3:
- \forall (t1: T).((sort t1) \to (\forall (t2: T).((sort t2) \to (\forall (c: 
-C).((pc3 c t1 t2) \to (eq T t1 t2))))))
+ \lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (t: T).(\lambda 
+(u: T).(\lambda (H: (tys3 g c (TCons t ts) u)).(insert_eq TList (TCons t ts) 
+(\lambda (t0: TList).(tys3 g c t0 u)) (\lambda (_: TList).(land (ty3 g c t u) 
+(tys3 g c ts u))) (\lambda (y: TList).(\lambda (H0: (tys3 g c y u)).(tys3_ind 
+g c (\lambda (t0: TList).(\lambda (t1: T).((eq TList t0 (TCons t ts)) \to 
+(land (ty3 g c t t1) (tys3 g c ts t1))))) (\lambda (u0: T).(\lambda (u1: 
+T).(\lambda (_: (ty3 g c u0 u1)).(\lambda (H2: (eq TList TNil (TCons t 
+ts))).(let H3 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList 
+return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _) 
+\Rightarrow False])) I (TCons t ts) H2) in (False_ind (land (ty3 g c t u0) 
+(tys3 g c ts u0)) H3)))))) (\lambda (t0: T).(\lambda (u0: T).(\lambda (H1: 
+(ty3 g c t0 u0)).(\lambda (ts0: TList).(\lambda (H2: (tys3 g c ts0 
+u0)).(\lambda (H3: (((eq TList ts0 (TCons t ts)) \to (land (ty3 g c t u0) 
+(tys3 g c ts u0))))).(\lambda (H4: (eq TList (TCons t0 ts0) (TCons t 
+ts))).(let H5 \def (f_equal TList T (\lambda (e: TList).(match e in TList 
+return (\lambda (_: TList).T) with [TNil \Rightarrow t0 | (TCons t1 _) 
+\Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H4) in ((let H6 \def (f_equal 
+TList TList (\lambda (e: TList).(match e in TList return (\lambda (_: 
+TList).TList) with [TNil \Rightarrow ts0 | (TCons _ t1) \Rightarrow t1])) 
+(TCons t0 ts0) (TCons t ts) H4) in (\lambda (H7: (eq T t0 t)).(let H8 \def 
+(eq_ind TList ts0 (\lambda (t1: TList).((eq TList t1 (TCons t ts)) \to (land 
+(ty3 g c t u0) (tys3 g c ts u0)))) H3 ts H6) in (let H9 \def (eq_ind TList 
+ts0 (\lambda (t1: TList).(tys3 g c t1 u0)) H2 ts H6) in (let H10 \def (eq_ind 
+T t0 (\lambda (t1: T).(ty3 g c t1 u0)) H1 t H7) in (conj (ty3 g c t u0) (tys3 
+g c ts u0) H10 H9)))))) H5))))))))) y u H0))) H)))))).
+
+theorem ty3_gen_appl_nf2:
+ \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x: 
+T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex4_2 T T (\lambda (u: 
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) 
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) 
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: 
+T).(nf2 c (THead (Bind Abst) u t))))))))))
 \def
- \lambda (t1: T).(\lambda (H: (sort t1)).(sort_ind (\lambda (t: T).(\forall 
-(t2: T).((sort t2) \to (\forall (c: C).((pc3 c t t2) \to (eq T t t2)))))) 
-(\lambda (n: nat).(\lambda (t2: T).(\lambda (H0: (sort t2)).(sort_ind 
-(\lambda (t: T).(\forall (c: C).((pc3 c (TSort n) t) \to (eq T (TSort n) 
-t)))) (\lambda (n0: nat).(\lambda (c: C).(\lambda (H1: (pc3 c (TSort n) 
-(TSort n0))).(eq_ind nat n (\lambda (n1: nat).(eq T (TSort n) (TSort n1))) 
-(refl_equal T (TSort n)) n0 (pc3_gen_sort c n n0 H1))))) (\lambda (u: 
-T).(\lambda (_: (sort u)).(\lambda (_: ((\forall (c: C).((pc3 c (TSort n) u) 
-\to (eq T (TSort n) u))))).(\lambda (t: T).(\lambda (_: (sort t)).(\lambda 
-(_: ((\forall (c: C).((pc3 c (TSort n) t) \to (eq T (TSort n) t))))).(\lambda 
-(c: C).(\lambda (H5: (pc3 c (TSort n) (THead (Bind Abst) u 
-t))).(pc3_gen_sort_abst c u t n H5 (eq T (TSort n) (THead (Bind Abst) u 
-t))))))))))) t2 H0)))) (\lambda (u: T).(\lambda (_: (sort u)).(\lambda (H1: 
-((\forall (t2: T).((sort t2) \to (\forall (c: C).((pc3 c u t2) \to (eq T u 
-t2))))))).(\lambda (t: T).(\lambda (_: (sort t)).(\lambda (H3: ((\forall (t2: 
-T).((sort t2) \to (\forall (c: C).((pc3 c t t2) \to (eq T t 
-t2))))))).(\lambda (t2: T).(\lambda (H4: (sort t2)).(sort_ind (\lambda (t0: 
-T).(\forall (c: C).((pc3 c (THead (Bind Abst) u t) t0) \to (eq T (THead (Bind 
-Abst) u t) t0)))) (\lambda (n: nat).(\lambda (c: C).(\lambda (H5: (pc3 c 
-(THead (Bind Abst) u t) (TSort n))).(pc3_gen_sort_abst c u t n (pc3_s c 
-(TSort n) (THead (Bind Abst) u t) H5) (eq T (THead (Bind Abst) u t) (TSort 
-n)))))) (\lambda (u0: T).(\lambda (H5: (sort u0)).(\lambda (_: ((\forall (c: 
-C).((pc3 c (THead (Bind Abst) u t) u0) \to (eq T (THead (Bind Abst) u t) 
-u0))))).(\lambda (t0: T).(\lambda (H7: (sort t0)).(\lambda (_: ((\forall (c: 
-C).((pc3 c (THead (Bind Abst) u t) t0) \to (eq T (THead (Bind Abst) u t) 
-t0))))).(\lambda (c: C).(\lambda (H9: (pc3 c (THead (Bind Abst) u t) (THead 
-(Bind Abst) u0 t0))).(and_ind (pc3 c u u0) (\forall (b: B).(\forall (u1: 
-T).(pc3 (CHead c (Bind b) u1) t t0))) (eq T (THead (Bind Abst) u t) (THead 
-(Bind Abst) u0 t0)) (\lambda (H10: (pc3 c u u0)).(\lambda (H11: ((\forall (b: 
-B).(\forall (u1: T).(pc3 (CHead c (Bind b) u1) t t0))))).(let H_y \def (H11 
-Abbr u) in (let H_y0 \def (H1 u0 H5 c H10) in (let H_y1 \def (H3 t0 H7 (CHead 
-c (Bind Abbr) u) H_y) in (let H12 \def (eq_ind_r T t0 (\lambda (t3: T).(pc3 
-(CHead c (Bind Abbr) u) t t3)) H_y t H_y1) in (let H13 \def (eq_ind_r T t0 
-(\lambda (t3: T).(sort t3)) H7 t H_y1) in (eq_ind T t (\lambda (t3: T).(eq T 
-(THead (Bind Abst) u t) (THead (Bind Abst) u0 t3))) (let H14 \def (eq_ind_r T 
-u0 (\lambda (t3: T).(pc3 c u t3)) H10 u H_y0) in (let H15 \def (eq_ind_r T u0 
-(\lambda (t3: T).(sort t3)) H5 u H_y0) in (eq_ind T u (\lambda (t3: T).(eq T 
-(THead (Bind Abst) u t) (THead (Bind Abst) t3 t))) (refl_equal T (THead (Bind 
-Abst) u t)) u0 H_y0))) t0 H_y1)))))))) (pc3_gen_abst c u u0 t t0 H9)))))))))) 
-t2 H4))))))))) t1 H)).
-
-theorem sort_correct:
- \forall (g: G).(\forall (t1: T).((sort t1) \to (\forall (c: C).(ex3 T 
-(\lambda (t2: T).(tau0 g c t1 t2)) (\lambda (t2: T).(ty3 g c t1 t2)) (\lambda 
-(t2: T).(sort t2))))))
+ \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x: 
+T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(ex3_2_ind T T (\lambda 
+(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) 
+x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) 
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (ex4_2 T T (\lambda (u: 
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) 
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) 
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: 
+T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H0: (pc3 c (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) 
+x)).(\lambda (H1: (ty3 g c v (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g 
+c w x0)).(let H_x \def (ty3_correct g c v (THead (Bind Abst) x0 x1) H1) in 
+(let H3 \def H_x in (ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) x0 
+x1) t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) 
+w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v 
+(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) 
+(\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) (\lambda 
+(x2: T).(\lambda (H4: (ty3 g c (THead (Bind Abst) x0 x1) x2)).(let H_x0 \def 
+(ty3_correct g c w x0 H2) in (let H5 \def H_x0 in (ex_ind T (\lambda (t: 
+T).(ty3 g c x0 t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead 
+(Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: 
+T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 
+g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) 
+(\lambda (x3: T).(\lambda (H6: (ty3 g c x0 x3)).(let H7 \def (ty3_sn3 g c 
+(THead (Bind Abst) x0 x1) x2 H4) in (let H_x1 \def (nf2_sn3 c (THead (Bind 
+Abst) x0 x1) H7) in (let H8 \def H_x1 in (ex2_ind T (\lambda (u: T).(pr3 c 
+(THead (Bind Abst) x0 x1) u)) (\lambda (u: T).(nf2 c u)) (ex4_2 T T (\lambda 
+(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) 
+x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) 
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: 
+T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x4: T).(\lambda (H9: (pr3 c 
+(THead (Bind Abst) x0 x1) x4)).(\lambda (H10: (nf2 c x4)).(let H11 \def 
+(pr3_gen_abst c x0 x1 x4 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x4 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))) (ex4_2 T T (\lambda (u: 
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) 
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) 
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: 
+T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x5: T).(\lambda (x6: 
+T).(\lambda (H12: (eq T x4 (THead (Bind Abst) x5 x6))).(\lambda (H13: (pr3 c 
+x0 x5)).(\lambda (H14: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind 
+b) u) x1 x6))))).(let H15 \def (eq_ind T x4 (\lambda (t: T).(nf2 c t)) H10 
+(THead (Bind Abst) x5 x6) H12) in (let H16 \def (pr3_head_12 c x0 x5 H13 
+(Bind Abst) x1 x6 (H14 Abst x5)) in (ex4_2_intro T T (\lambda (u: T).(\lambda 
+(t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: 
+T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: 
+T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c 
+(THead (Bind Abst) u t)))) x5 x6 (pc3_pr3_conf c (THead (Flat Appl) w (THead 
+(Bind Abst) x0 x1)) x H0 (THead (Flat Appl) w (THead (Bind Abst) x5 x6)) 
+(pr3_thin_dx c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 w 
+Appl)) (ty3_conv g c (THead (Bind Abst) x5 x6) x2 (ty3_sred_pr3 c (THead 
+(Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 g x2 H4) v (THead (Bind 
+Abst) x0 x1) H1 (pc3_pr3_r c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 
+x6) H16)) (ty3_conv g c x5 x3 (ty3_sred_pr3 c x0 x5 H13 g x3 H6) w x0 H2 
+(pc3_pr3_r c x0 x5 H13)) H15)))))))) H11))))) H8)))))) H5))))) H3)))))))) 
+(ty3_gen_appl g c w v x H))))))).
+
+theorem ty3_inv_lref_nf2_pc3:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (i: nat).((ty3 g c 
+(TLRef i) u1) \to ((nf2 c (TLRef i)) \to (\forall (u2: T).((nf2 c u2) \to 
+((pc3 c u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (i: nat).(\lambda 
+(H: (ty3 g c (TLRef i) u1)).(insert_eq T (TLRef i) (\lambda (t: T).(ty3 g c t 
+u1)) (\lambda (t: T).((nf2 c t) \to (\forall (u2: T).((nf2 c u2) \to ((pc3 c 
+u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))) (\lambda 
+(y: T).(\lambda (H0: (ty3 g c y u1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: 
+T).(\lambda (t0: T).((eq T t (TLRef i)) \to ((nf2 c0 t) \to (\forall (u2: 
+T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift 
+(S i) O u)))))))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t: 
+T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (TLRef i)) \to ((nf2 
+c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to (ex T 
+(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (u: T).(\lambda 
+(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (TLRef i)) \to 
+((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t1 u2) \to (ex T 
+(\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (pc3 c0 
+t1 t2)).(\lambda (H6: (eq T u (TLRef i))).(\lambda (H7: (nf2 c0 u)).(\lambda 
+(u2: T).(\lambda (H8: (nf2 c0 u2)).(\lambda (H9: (pc3 c0 t2 u2)).(let H10 
+\def (eq_ind T u (\lambda (t0: T).(nf2 c0 t0)) H7 (TLRef i) H6) in (let H11 
+\def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef i)) \to ((nf2 c0 t0) \to 
+(\forall (u3: T).((nf2 c0 u3) \to ((pc3 c0 t1 u3) \to (ex T (\lambda (u0: 
+T).(eq T u3 (lift (S i) O u0)))))))))) H4 (TLRef i) H6) in (let H12 \def 
+(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef i) H6) in (let H_y 
+\def (H11 (refl_equal T (TLRef i)) H10 u2 H8) in (H_y (pc3_t t2 c0 t1 H5 u2 
+H9))))))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq 
+T (TSort m) (TLRef i))).(\lambda (_: (nf2 c0 (TSort m))).(\lambda (u2: 
+T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (TSort (next g m)) 
+u2)).(let H5 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i) H1) in 
+(False_ind (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u)))) H5))))))))) 
+(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
+(H1: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g 
+d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2: 
+T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S 
+i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5: 
+(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (H7: 
+(pc3 c0 (lift (S n) O t) u2)).(let H8 \def (f_equal T nat (\lambda (e: 
+T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | 
+(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef 
+i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) 
+O t) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0 
+(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl 
+n0 c0 (CHead d (Bind Abbr) u))) H1 i H8) in (nf2_gen_lref c0 d u i H11 H10 
+(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0)))))))))))))))))))))) 
+(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
+(H1: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g 
+d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2: 
+T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S 
+i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5: 
+(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (H6: (nf2 c0 u2)).(\lambda (H7: 
+(pc3 c0 (lift (S n) O u) u2)).(let H8 \def (f_equal T nat (\lambda (e: 
+T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | 
+(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef 
+i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) 
+O u) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0 
+(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl 
+n0 c0 (CHead d (Bind Abst) u))) H1 i H8) in (let H_y \def (pc3_nf2_unfold c0 
+(lift (S i) O u) u2 H9 H6) in (let H12 \def (pr3_gen_lift c0 u u2 (S i) O H_y 
+d (getl_drop Abst c0 d u i H11)) in (ex2_ind T (\lambda (t2: T).(eq T u2 
+(lift (S i) O t2))) (\lambda (t2: T).(pr3 d u t2)) (ex T (\lambda (u0: T).(eq 
+T u2 (lift (S i) O u0)))) (\lambda (x: T).(\lambda (H13: (eq T u2 (lift (S i) 
+O x))).(\lambda (_: (pr3 d u x)).(eq_ind_r T (lift (S i) O x) (\lambda (t0: 
+T).(ex T (\lambda (u0: T).(eq T t0 (lift (S i) O u0))))) (ex_intro T (\lambda 
+(u0: T).(eq T (lift (S i) O x) (lift (S i) O u0))) x (refl_equal T (lift (S 
+i) O x))) u2 H13)))) H12)))))))))))))))))))) (\lambda (c0: C).(\lambda (u: 
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef 
+i)) \to ((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to 
+(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (b: 
+B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) 
+u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef i)) \to ((nf2 (CHead c0 (Bind b) u) 
+t1) \to (\forall (u2: T).((nf2 (CHead c0 (Bind b) u) u2) \to ((pc3 (CHead c0 
+(Bind b) u) t2 u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O 
+u0))))))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TLRef i))).(\lambda 
+(_: (nf2 c0 (THead (Bind b) u t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 
+u2)).(\lambda (_: (pc3 c0 (THead (Bind b) u t2) u2)).(let H9 \def (eq_ind T 
+(THead (Bind b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T 
+(\lambda (u0: T).(eq T u2 (lift (S i) O u0)))) H9))))))))))))))))) (\lambda 
+(c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda 
+(_: (((eq T w (TLRef i)) \to ((nf2 c0 w) \to (\forall (u2: T).((nf2 c0 u2) 
+\to ((pc3 c0 u u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O 
+u0))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead 
+(Bind Abst) u t))).(\lambda (_: (((eq T v (TLRef i)) \to ((nf2 c0 v) \to 
+(\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 (THead (Bind Abst) u t) u2) \to 
+(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (eq 
+T (THead (Flat Appl) w v) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Appl) 
+w v))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (THead 
+(Flat Appl) w (THead (Bind Abst) u t)) u2)).(let H9 \def (eq_ind T (THead 
+(Flat Appl) w v) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) 
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ 
+_) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u0: 
+T).(eq T u2 (lift (S i) O u0)))) H9)))))))))))))))) (\lambda (c0: C).(\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T 
+t1 (TLRef i)) \to ((nf2 c0 t1) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 
+t2 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda 
+(t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef i)) \to 
+((nf2 c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T 
+(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (H5: (eq T 
+(THead (Flat Cast) t2 t1) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Cast) 
+t2 t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 
+(THead (Flat Cast) t0 t2) u2)).(let H9 \def (eq_ind T (THead (Flat Cast) t2 
+t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort 
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u: T).(eq T 
+u2 (lift (S i) O u)))) H9))))))))))))))) c y u1 H0))) H))))).
+
+theorem ty3_inv_lref_nf2:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (i: nat).((ty3 g c 
+(TLRef i) u) \to ((nf2 c (TLRef i)) \to ((nf2 c u) \to (ex T (\lambda (u0: 
+T).(eq T u (lift (S i) O u0))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H: (ty3 g c (TLRef i) u)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: 
+(nf2 c u)).(ty3_inv_lref_nf2_pc3 g c u i H H0 u H1 (pc3_refl c u)))))))).
+
+theorem ty3_inv_appls_lref_nf2:
+ \forall (g: G).(\forall (c: C).(\forall (vs: TList).(\forall (u1: 
+T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) vs (TLRef i)) u1) \to 
+((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S 
+i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) vs (lift (S i) O u)) 
+u1))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: 
+TList).(\forall (u1: T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) t 
+(TLRef i)) u1) \to ((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: 
+T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t 
+(lift (S i) O u)) u1))))))))) (\lambda (u1: T).(\lambda (i: nat).(\lambda (H: 
+(ty3 g c (TLRef i) u1)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (nf2 c 
+u1)).(let H_x \def (ty3_inv_lref_nf2 g c u1 i H H0 H1) in (let H2 \def H_x in 
+(ex_ind T (\lambda (u0: T).(eq T u1 (lift (S i) O u0))) (ex2 T (\lambda (u: 
+T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) u1))) 
+(\lambda (x: T).(\lambda (H3: (eq T u1 (lift (S i) O x))).(let H4 \def 
+(eq_ind T u1 (\lambda (t: T).(nf2 c t)) H1 (lift (S i) O x) H3) in (eq_ind_r 
+T (lift (S i) O x) (\lambda (t: T).(ex2 T (\lambda (u: T).(nf2 c (lift (S i) 
+O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) t)))) (ex_intro2 T (\lambda 
+(u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) 
+(lift (S i) O x))) x H4 (pc3_refl c (lift (S i) O x))) u1 H3)))) H2)))))))) 
+(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (u1: T).(\forall 
+(i: nat).((ty3 g c (THeads (Flat Appl) t0 (TLRef i)) u1) \to ((nf2 c (TLRef 
+i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) 
+(\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) O u)) 
+u1)))))))))).(\lambda (u1: T).(\lambda (i: nat).(\lambda (H0: (ty3 g c (THead 
+(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u1)).(\lambda (H1: (nf2 c 
+(TLRef i))).(\lambda (_: (nf2 c u1)).(let H_x \def (ty3_gen_appl_nf2 g c t 
+(THeads (Flat Appl) t0 (TLRef i)) u1 H0) in (let H3 \def H_x in (ex4_2_ind T 
+T (\lambda (u: T).(\lambda (t1: T).(pc3 c (THead (Flat Appl) t (THead (Bind 
+Abst) u t1)) u1))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c (THeads (Flat 
+Appl) t0 (TLRef i)) (THead (Bind Abst) u t1)))) (\lambda (u: T).(\lambda (_: 
+T).(ty3 g c t u))) (\lambda (u: T).(\lambda (t1: T).(nf2 c (THead (Bind Abst) 
+u t1)))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: 
+T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) 
+u1))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (pc3 c (THead (Flat 
+Appl) t (THead (Bind Abst) x0 x1)) u1)).(\lambda (H5: (ty3 g c (THeads (Flat 
+Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c t 
+x0)).(\lambda (H7: (nf2 c (THead (Bind Abst) x0 x1))).(let H8 \def 
+(nf2_gen_abst c x0 x1 H7) in (and_ind (nf2 c x0) (nf2 (CHead c (Bind Abst) 
+x0) x1) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 
+c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) u1))) 
+(\lambda (H9: (nf2 c x0)).(\lambda (H10: (nf2 (CHead c (Bind Abst) x0) 
+x1)).(let H_y \def (H (THead (Bind Abst) x0 x1) i H5 H1) in (let H11 \def 
+(H_y (nf2_abst_shift c x0 H9 x1 H10)) in (ex2_ind T (\lambda (u: T).(nf2 c 
+(lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) 
+O u)) (THead (Bind Abst) x0 x1))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O 
+u))) (\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift 
+(S i) O u))) u1))) (\lambda (x: T).(\lambda (H12: (nf2 c (lift (S i) O 
+x))).(\lambda (H13: (pc3 c (THeads (Flat Appl) t0 (lift (S i) O x)) (THead 
+(Bind Abst) x0 x1))).(ex_intro2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) 
+(\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S 
+i) O u))) u1)) x H12 (pc3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) c 
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O x))) (pc3_thin_dx c 
+(THeads (Flat Appl) t0 (lift (S i) O x)) (THead (Bind Abst) x0 x1) H13 t 
+Appl) u1 H4))))) H11))))) H8)))))))) H3))))))))))) vs))).
+
+theorem ty3_inv_lref_lref_nf2:
+ \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (j: nat).((ty3 g c 
+(TLRef i) (TLRef j)) \to ((nf2 c (TLRef i)) \to ((nf2 c (TLRef j)) \to (lt i 
+j)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (j: nat).(\lambda 
+(H: (ty3 g c (TLRef i) (TLRef j))).(\lambda (H0: (nf2 c (TLRef i))).(\lambda 
+(H1: (nf2 c (TLRef j))).(let H_x \def (ty3_inv_lref_nf2 g c (TLRef j) i H H0 
+H1) in (let H2 \def H_x in (ex_ind T (\lambda (u0: T).(eq T (TLRef j) (lift 
+(S i) O u0))) (lt i j) (\lambda (x: T).(\lambda (H3: (eq T (TLRef j) (lift (S 
+i) O x))).(let H_x0 \def (lift_gen_lref x O (S i) j H3) in (let H4 \def H_x0 
+in (or_ind (land (lt j O) (eq T x (TLRef j))) (land (le (plus O (S i)) j) (eq 
+T x (TLRef (minus j (S i))))) (lt i j) (\lambda (H5: (land (lt j O) (eq T x 
+(TLRef j)))).(and_ind (lt j O) (eq T x (TLRef j)) (lt i j) (\lambda (H6: (lt 
+j O)).(\lambda (_: (eq T x (TLRef j))).(lt_x_O j H6 (lt i j)))) H5)) (\lambda 
+(H5: (land (le (plus O (S i)) j) (eq T x (TLRef (minus j (S i)))))).(and_ind 
+(le (plus O (S i)) j) (eq T x (TLRef (minus j (S i)))) (lt i j) (\lambda (H6: 
+(le (plus O (S i)) j)).(\lambda (_: (eq T x (TLRef (minus j (S i))))).H6)) 
+H5)) H4))))) H2))))))))).
+
+theorem ty3_shift1:
+ \forall (g: G).(\forall (c: C).((wf3 g c c) \to (\forall (t1: T).(\forall 
+(t2: T).((ty3 g c t1 t2) \to (ty3 g (CSort (cbk c)) (app1 c t1) (app1 c 
+t2)))))))
 \def
- \lambda (g: G).(\lambda (t1: T).(\lambda (H: (sort t1)).(sort_ind (\lambda 
-(t: T).(\forall (c: C).(ex3 T (\lambda (t2: T).(tau0 g c t t2)) (\lambda (t2: 
-T).(ty3 g c t t2)) (\lambda (t2: T).(sort t2))))) (\lambda (n: nat).(\lambda 
-(c: C).(ex3_intro T (\lambda (t2: T).(tau0 g c (TSort n) t2)) (\lambda (t2: 
-T).(ty3 g c (TSort n) t2)) (\lambda (t2: T).(sort t2)) (TSort (next g n)) 
-(tau0_sort g c n) (ty3_sort g c n) (sort_sort (next g n))))) (\lambda (u: 
-T).(\lambda (H0: (sort u)).(\lambda (H1: ((\forall (c: C).(ex3 T (\lambda 
-(t2: T).(tau0 g c u t2)) (\lambda (t2: T).(ty3 g c u t2)) (\lambda (t2: 
-T).(sort t2)))))).(\lambda (t: T).(\lambda (_: (sort t)).(\lambda (H3: 
-((\forall (c: C).(ex3 T (\lambda (t2: T).(tau0 g c t t2)) (\lambda (t2: 
-T).(ty3 g c t t2)) (\lambda (t2: T).(sort t2)))))).(\lambda (c: C).(let H_x 
-\def (H1 c) in (let H4 \def H_x in (ex3_ind T (\lambda (t2: T).(tau0 g c u 
-t2)) (\lambda (t2: T).(ty3 g c u t2)) (\lambda (t2: T).(sort t2)) (ex3 T 
-(\lambda (t2: T).(tau0 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(ty3 
-g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(sort t2))) (\lambda (x0: 
-T).(\lambda (_: (tau0 g c u x0)).(\lambda (H6: (ty3 g c u x0)).(\lambda (_: 
-(sort x0)).(let H_x0 \def (H3 (CHead c (Bind Abst) u)) in (let H8 \def H_x0 
-in (ex3_ind T (\lambda (t2: T).(tau0 g (CHead c (Bind Abst) u) t t2)) 
-(\lambda (t2: T).(ty3 g (CHead c (Bind Abst) u) t t2)) (\lambda (t2: T).(sort 
-t2)) (ex3 T (\lambda (t2: T).(tau0 g c (THead (Bind Abst) u t) t2)) (\lambda 
-(t2: T).(ty3 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(sort t2))) 
-(\lambda (x1: T).(\lambda (H9: (tau0 g (CHead c (Bind Abst) u) t 
-x1)).(\lambda (H10: (ty3 g (CHead c (Bind Abst) u) t x1)).(\lambda (H11: 
-(sort x1)).(ex_ind T (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) u) x1 t0)) 
-(ex3 T (\lambda (t2: T).(tau0 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: 
-T).(ty3 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(sort t2))) 
-(\lambda (x: T).(\lambda (H12: (ty3 g (CHead c (Bind Abst) u) x1 
-x)).(ex3_intro T (\lambda (t2: T).(tau0 g c (THead (Bind Abst) u t) t2)) 
-(\lambda (t2: T).(ty3 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(sort 
-t2)) (THead (Bind Abst) u x1) (tau0_bind g Abst c u t x1 H9) (ty3_bind g c u 
-x0 H6 Abst t x1 H10 x H12) (sort_abst u H0 x1 H11)))) (ty3_correct g (CHead c 
-(Bind Abst) u) t x1 H10)))))) H8))))))) H4)))))))))) t1 H))).
+ \lambda (g: G).(\lambda (c: C).(\lambda (H: (wf3 g c c)).(insert_eq C c 
+(\lambda (c0: C).(wf3 g c0 c)) (\lambda (c0: C).(\forall (t1: T).(\forall 
+(t2: T).((ty3 g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0 
+t2)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y c)).(wf3_ind g (\lambda (c0: 
+C).(\lambda (c1: C).((eq C c0 c1) \to (\forall (t1: T).(\forall (t2: T).((ty3 
+g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0 t2)))))))) 
+(\lambda (m: nat).(\lambda (_: (eq C (CSort m) (CSort m))).(\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (H2: (ty3 g (CSort m) t1 t2)).H2))))) (\lambda 
+(c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C 
+c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to (ty3 g 
+(CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u: T).(\lambda 
+(t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C 
+(CHead c1 (Bind b) u) (CHead c2 (Bind b) u))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (H5: (ty3 g (CHead c1 (Bind b) u) t1 t2)).(let H6 \def (f_equal C 
+C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
+\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind b) u) 
+(CHead c2 (Bind b) u) H4) in (let H7 \def (eq_ind_r C c2 (\lambda (c0: 
+C).((eq C c1 c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to 
+(ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 c1 H6) in (let H8 
+\def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c1 H6) in (ex_ind T 
+(\lambda (t0: T).(ty3 g (CHead c1 (Bind b) u) t2 t0)) (ty3 g (CSort (cbk c1)) 
+(app1 c1 (THead (Bind b) u t1)) (app1 c1 (THead (Bind b) u t2))) (\lambda (x: 
+T).(\lambda (_: (ty3 g (CHead c1 (Bind b) u) t2 x)).(H7 (refl_equal C c1) 
+(THead (Bind b) u t1) (THead (Bind b) u t2) (ty3_bind g c1 u t H3 b t1 t2 
+H5)))) (ty3_correct g (CHead c1 (Bind b) u) t1 t2 H5))))))))))))))))) 
+(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: 
+(((eq C c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to 
+(ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u: 
+T).(\lambda (H3: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b: 
+B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort 
+O)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H5: (ty3 g (CHead c1 (Bind 
+b) u) t1 t2)).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return 
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) 
+\Rightarrow c0])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) 
+in ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda 
+(_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k 
+in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
+\Rightarrow b])])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) 
+in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda 
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) 
+(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) in (\lambda (H9: 
+(eq B b Void)).(\lambda (H10: (eq C c1 c2)).(let H11 \def (eq_ind B b 
+(\lambda (b0: B).(ty3 g (CHead c1 (Bind b0) u) t1 t2)) H5 Void H9) in 
+(eq_ind_r B Void (\lambda (b0: B).(ty3 g (CSort (cbk (CHead c1 (Bind b0) u))) 
+(app1 (CHead c1 (Bind b0) u) t1) (app1 (CHead c1 (Bind b0) u) t2))) (let H12 
+\def (eq_ind T u (\lambda (t: T).(ty3 g (CHead c1 (Bind Void) t) t1 t2)) H11 
+(TSort O) H8) in (let H13 \def (eq_ind T u (\lambda (t: T).(\forall (t0: 
+T).((ty3 g c1 t t0) \to False))) H3 (TSort O) H8) in (eq_ind_r T (TSort O) 
+(\lambda (t: T).(ty3 g (CSort (cbk (CHead c1 (Bind Void) t))) (app1 (CHead c1 
+(Bind Void) t) t1) (app1 (CHead c1 (Bind Void) t) t2))) (let H14 \def 
+(eq_ind_r C c2 (\lambda (c0: C).((eq C c1 c0) \to (\forall (t3: T).(\forall 
+(t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1 
+t4))))))) H2 c1 H10) in (let H15 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g 
+c1 c0)) H1 c1 H10) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c1 (Bind Void) 
+(TSort O)) t2 t)) (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Bind Void) (TSort 
+O) t1)) (app1 c1 (THead (Bind Void) (TSort O) t2))) (\lambda (x: T).(\lambda 
+(_: (ty3 g (CHead c1 (Bind Void) (TSort O)) t2 x)).(H14 (refl_equal C c1) 
+(THead (Bind Void) (TSort O) t1) (THead (Bind Void) (TSort O) t2) (ty3_bind g 
+c1 (TSort O) (TSort (next g O)) (ty3_sort g c1 O) Void t1 t2 H12)))) 
+(ty3_correct g (CHead c1 (Bind Void) (TSort O)) t1 t2 H12)))) u H8))) b 
+H9))))) H7)) H6))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: 
+(wf3 g c1 c2)).(\lambda (H2: (((eq C c1 c2) \to (\forall (t1: T).(\forall 
+(t2: T).((ty3 g c1 t1 t2) \to (ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1 
+t2)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 
+(Flat f) u) c2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead 
+c1 (Flat f) u) t1 t2)).(let H5 \def (f_equal C C (\lambda (e: C).e) (CHead c1 
+(Flat f) u) c2 H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c0: C).((eq C c1 
+c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort 
+(cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 (CHead c1 (Flat f) u) H5) in 
+(let H7 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 (CHead c1 
+(Flat f) u) H5) in (let H_x \def (wf3_gen_head2 g c1 c1 u (Flat f) H7) in 
+(let H8 \def H_x in (ex_ind B (\lambda (b: B).(eq K (Flat f) (Bind b))) (ty3 
+g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u t1)) (app1 c1 (THead (Flat f) u 
+t2))) (\lambda (x: B).(\lambda (H9: (eq K (Flat f) (Bind x))).(let H10 \def 
+(eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return (\lambda (_: 
+K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) I 
+(Bind x) H9) in (False_ind (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u 
+t1)) (app1 c1 (THead (Flat f) u t2))) H10)))) H8)))))))))))))))) y c H0))) 
+H))).