1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/spare".
24 let rec cbk (c: C) on c: nat \def (match c with [(CSort m) \Rightarrow m |
25 (CHead c0 _ _) \Rightarrow (cbk c0)]) in cbk.
30 let rec app1 (c: C) on c: (T \to T) \def (\lambda (t: T).(match c with
31 [(CSort _) \Rightarrow t | (CHead c0 k u) \Rightarrow (app1 c0 (THead k u
35 \forall (xs: TList).(\forall (ts: TList).(\forall (h: nat).(\forall (d:
36 nat).((eq TList (lifts h d xs) (lifts h d ts)) \to (eq TList xs ts)))))
38 \lambda (xs: TList).(TList_ind (\lambda (t: TList).(\forall (ts:
39 TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t) (lifts h
40 d ts)) \to (eq TList t ts)))))) (\lambda (ts: TList).(TList_ind (\lambda (t:
41 TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d TNil) (lifts
42 h d t)) \to (eq TList TNil t))))) (\lambda (_: nat).(\lambda (_:
43 nat).(\lambda (H: (eq TList TNil TNil)).H))) (\lambda (t: T).(\lambda (t0:
44 TList).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq TList TNil
45 (lifts h d t0)) \to (eq TList TNil t0)))))).(\lambda (h: nat).(\lambda (d:
46 nat).(\lambda (H0: (eq TList TNil (TCons (lift h d t) (lifts h d t0)))).(let
47 H1 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList return
48 (\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _)
49 \Rightarrow False])) I (TCons (lift h d t) (lifts h d t0)) H0) in (False_ind
50 (eq TList TNil (TCons t t0)) H1)))))))) ts)) (\lambda (t: T).(\lambda (t0:
51 TList).(\lambda (H: ((\forall (ts: TList).(\forall (h: nat).(\forall (d:
52 nat).((eq TList (lifts h d t0) (lifts h d ts)) \to (eq TList t0
53 ts))))))).(\lambda (ts: TList).(TList_ind (\lambda (t1: TList).(\forall (h:
54 nat).(\forall (d: nat).((eq TList (lifts h d (TCons t t0)) (lifts h d t1))
55 \to (eq TList (TCons t t0) t1))))) (\lambda (h: nat).(\lambda (d:
56 nat).(\lambda (H0: (eq TList (TCons (lift h d t) (lifts h d t0)) TNil)).(let
57 H1 \def (eq_ind TList (TCons (lift h d t) (lifts h d t0)) (\lambda (ee:
58 TList).(match ee in TList return (\lambda (_: TList).Prop) with [TNil
59 \Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H0) in (False_ind
60 (eq TList (TCons t t0) TNil) H1))))) (\lambda (t1: T).(\lambda (t2:
61 TList).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq TList (TCons
62 (lift h d t) (lifts h d t0)) (lifts h d t2)) \to (eq TList (TCons t t0)
63 t2)))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq TList (TCons
64 (lift h d t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d t2)))).(let H2
65 \def (f_equal TList T (\lambda (e: TList).(match e in TList return (\lambda
66 (_: TList).T) with [TNil \Rightarrow ((let rec lref_map (f: ((nat \to nat)))
67 (d0: nat) (t3: T) on t3: T \def (match t3 with [(TSort n) \Rightarrow (TSort
68 n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i
69 | false \Rightarrow (f i)])) | (THead k u t4) \Rightarrow (THead k (lref_map
70 f d0 u) (lref_map f (s k d0) t4))]) in lref_map) (\lambda (x: nat).(plus x
71 h)) d t) | (TCons t3 _) \Rightarrow t3])) (TCons (lift h d t) (lifts h d t0))
72 (TCons (lift h d t1) (lifts h d t2)) H1) in ((let H3 \def (f_equal TList
73 TList (\lambda (e: TList).(match e in TList return (\lambda (_: TList).TList)
74 with [TNil \Rightarrow ((let rec lifts (h0: nat) (d0: nat) (ts0: TList) on
75 ts0: TList \def (match ts0 with [TNil \Rightarrow TNil | (TCons t3 ts1)
76 \Rightarrow (TCons (lift h0 d0 t3) (lifts h0 d0 ts1))]) in lifts) h d t0) |
77 (TCons _ t3) \Rightarrow t3])) (TCons (lift h d t) (lifts h d t0)) (TCons
78 (lift h d t1) (lifts h d t2)) H1) in (\lambda (H4: (eq T (lift h d t) (lift h
79 d t1))).(eq_ind T t (\lambda (t3: T).(eq TList (TCons t t0) (TCons t3 t2)))
80 (f_equal2 T TList TList TCons t t t0 t2 (refl_equal T t) (H t2 h d H3)) t1
81 (lift_inj t t1 h d H4)))) H2)))))))) ts))))) xs).
84 \forall (c: C).(\forall (t: T).(\forall (ts: TList).((nfs2 c (TApp ts t))
85 \to (land (nfs2 c ts) (nf2 c t)))))
87 \lambda (c: C).(\lambda (t: T).(\lambda (ts: TList).(TList_ind (\lambda (t0:
88 TList).((nfs2 c (TApp t0 t)) \to (land (nfs2 c t0) (nf2 c t)))) (\lambda (H:
89 (land (nf2 c t) True)).(let H0 \def H in (and_ind (nf2 c t) True (land True
90 (nf2 c t)) (\lambda (H1: (nf2 c t)).(\lambda (_: True).(conj True (nf2 c t) I
91 H1))) H0))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (((nfs2 c
92 (TApp t1 t)) \to (land (nfs2 c t1) (nf2 c t))))).(\lambda (H0: (land (nf2 c
93 t0) (nfs2 c (TApp t1 t)))).(let H1 \def H0 in (and_ind (nf2 c t0) (nfs2 c
94 (TApp t1 t)) (land (land (nf2 c t0) (nfs2 c t1)) (nf2 c t)) (\lambda (H2:
95 (nf2 c t0)).(\lambda (H3: (nfs2 c (TApp t1 t))).(let H_x \def (H H3) in (let
96 H4 \def H_x in (and_ind (nfs2 c t1) (nf2 c t) (land (land (nf2 c t0) (nfs2 c
97 t1)) (nf2 c t)) (\lambda (H5: (nfs2 c t1)).(\lambda (H6: (nf2 c t)).(conj
98 (land (nf2 c t0) (nfs2 c t1)) (nf2 c t) (conj (nf2 c t0) (nfs2 c t1) H2 H5)
99 H6))) H4))))) H1)))))) ts))).
101 theorem pc3_nf2_unfold:
102 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c
103 t2) \to (pr3 c t1 t2)))))
105 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
106 t2)).(\lambda (H0: (nf2 c t2)).(let H1 \def H in (ex2_ind T (\lambda (t:
107 T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pr3 c t1 t2) (\lambda (x:
108 T).(\lambda (H2: (pr3 c t1 x)).(\lambda (H3: (pr3 c t2 x)).(let H_y \def
109 (nf2_pr3_unfold c t2 x H3 H0) in (let H4 \def (eq_ind_r T x (\lambda (t:
110 T).(pr3 c t1 t)) H2 t2 H_y) in H4))))) H1)))))).
112 theorem pc3_pr3_conf:
113 \forall (c: C).(\forall (t: T).(\forall (t1: T).((pc3 c t t1) \to (\forall
114 (t2: T).((pr3 c t t2) \to (pc3 c t2 t1))))))
116 \lambda (c: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pc3 c t
117 t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t t2)).(pc3_t t c t2 (pc3_pr3_x c
120 axiom pc3_gen_appls_sort_abst:
121 \forall (c: C).(\forall (vs: TList).(\forall (w: T).(\forall (u: T).(\forall
122 (n: nat).((pc3 c (THeads (Flat Appl) vs (TSort n)) (THead (Bind Abst) w u))
126 axiom pc3_gen_appls_lref_abst:
127 \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
128 (CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (w: T).(\forall
129 (u: T).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THead (Bind Abst) w u)) \to
133 axiom pc3_gen_appls_lref_sort:
134 \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
135 (CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (ws:
136 TList).(\forall (n: nat).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THeads
137 (Flat Appl) ws (TSort n))) \to False))))))))
140 inductive tys3 (g: G) (c: C): TList \to (T \to Prop) \def
141 | tys3_nil: \forall (u: T).(\forall (u0: T).((ty3 g c u u0) \to (tys3 g c
143 | tys3_cons: \forall (t: T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts:
144 TList).((tys3 g c ts u) \to (tys3 g c (TCons t ts) u))))).
146 theorem tys3_inv_coq:
147 \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (u: T).(\forall
148 (P: ((G \to (C \to (TList \to (T \to Prop)))))).((((tys3 g c ts u) \to
149 (\forall (u0: T).(\forall (u1: T).((eq TList TNil ts) \to ((eq T u0 u) \to
150 ((ty3 g c u0 u1) \to (P g c ts u)))))))) \to ((((tys3 g c ts u) \to (\forall
151 (t: T).(\forall (u0: T).(\forall (ts0: TList).((eq TList (TCons t ts0) ts)
152 \to ((eq T u0 u) \to ((ty3 g c t u0) \to ((tys3 g c ts0 u0) \to (P g c ts
153 u)))))))))) \to ((tys3 g c ts u) \to (P g c ts u))))))))
155 \lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (u: T).(\lambda
156 (P: ((G \to (C \to (TList \to (T \to Prop)))))).(\lambda (H: (((tys3 g c ts
157 u) \to (\forall (u0: T).(\forall (u1: T).((eq TList TNil ts) \to ((eq T u0 u)
158 \to ((ty3 g c u0 u1) \to (P g c ts u))))))))).(\lambda (H0: (((tys3 g c ts u)
159 \to (\forall (t: T).(\forall (u0: T).(\forall (ts0: TList).((eq TList (TCons
160 t ts0) ts) \to ((eq T u0 u) \to ((ty3 g c t u0) \to ((tys3 g c ts0 u0) \to (P
161 g c ts u))))))))))).(\lambda (H1: (tys3 g c ts u)).(let H2 \def (match H1 in
162 tys3 return (\lambda (t: TList).(\lambda (t0: T).(\lambda (_: (tys3 ? ? t
163 t0)).((eq TList t ts) \to ((eq T t0 u) \to (P g c ts u)))))) with [(tys3_nil
164 u0 u1 H2) \Rightarrow (\lambda (H3: (eq TList TNil ts)).(\lambda (H4: (eq T
165 u0 u)).(H H1 u0 u1 H3 H4 H2))) | (tys3_cons t u0 H2 ts0 H3) \Rightarrow
166 (\lambda (H4: (eq TList (TCons t ts0) ts)).(\lambda (H5: (eq T u0 u)).(H0 H1
167 t u0 ts0 H4 H5 H2 H3)))]) in (H2 (refl_equal TList ts) (refl_equal T
170 theorem tys3_gen_nil:
171 \forall (g: G).(\forall (c: C).(\forall (u: T).((tys3 g c TNil u) \to (ex T
172 (\lambda (u0: T).(ty3 g c u u0))))))
174 \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (tys3 g c TNil
175 u)).(tys3_inv_coq g c TNil u (\lambda (g0: G).(\lambda (c0: C).(\lambda (_:
176 TList).(\lambda (t0: T).(ex T (\lambda (u0: T).(ty3 g0 c0 t0 u0)))))))
177 (\lambda (_: (tys3 g c TNil u)).(\lambda (u0: T).(\lambda (u1: T).(\lambda
178 (_: (eq TList TNil TNil)).(\lambda (H2: (eq T u0 u)).(\lambda (H3: (ty3 g c
179 u0 u1)).(let H4 \def (eq_ind T u0 (\lambda (t: T).(ty3 g c t u1)) H3 u H2) in
180 (ex_intro T (\lambda (u2: T).(ty3 g c u u2)) u1 H4)))))))) (\lambda (_: (tys3
181 g c TNil u)).(\lambda (t: T).(\lambda (u0: T).(\lambda (ts0: TList).(\lambda
182 (H2: (eq TList (TCons t ts0) TNil)).(\lambda (H3: (eq T u0 u)).(\lambda (H1:
183 (ty3 g c t u0)).(\lambda (H4: (tys3 g c ts0 u0)).(let H5 \def (eq_ind T u0
184 (\lambda (t0: T).(tys3 g c ts0 t0)) H4 u H3) in (let H6 \def (eq_ind T u0
185 (\lambda (t0: T).(ty3 g c t t0)) H1 u H3) in (let H7 \def (eq_ind TList
186 (TCons t ts0) (\lambda (ee: TList).(match ee in TList return (\lambda (_:
187 TList).Prop) with [TNil \Rightarrow False | (TCons _ _) \Rightarrow True])) I
188 TNil H2) in (False_ind (ex T (\lambda (u1: T).(ty3 g c u u1))) H7))))))))))))
191 theorem tys3_gen_cons:
192 \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (t: T).(\forall
193 (u: T).((tys3 g c (TCons t ts) u) \to (land (ty3 g c t u) (tys3 g c ts
196 \lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (t: T).(\lambda
197 (u: T).(\lambda (H: (tys3 g c (TCons t ts) u)).(tys3_inv_coq g c (TCons t ts)
198 u (\lambda (g0: G).(\lambda (c0: C).(\lambda (_: TList).(\lambda (t1:
199 T).(land (ty3 g0 c0 t t1) (tys3 g0 c0 ts t1)))))) (\lambda (_: (tys3 g c
200 (TCons t ts) u)).(\lambda (u0: T).(\lambda (u1: T).(\lambda (H1: (eq TList
201 TNil (TCons t ts))).(\lambda (H2: (eq T u0 u)).(\lambda (H3: (ty3 g c u0
202 u1)).(let H4 \def (eq_ind T u0 (\lambda (t0: T).(ty3 g c t0 u1)) H3 u H2) in
203 (let H5 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList
204 return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _)
205 \Rightarrow False])) I (TCons t ts) H1) in (False_ind (land (ty3 g c t u)
206 (tys3 g c ts u)) H5))))))))) (\lambda (_: (tys3 g c (TCons t ts) u)).(\lambda
207 (t0: T).(\lambda (u0: T).(\lambda (ts0: TList).(\lambda (H2: (eq TList (TCons
208 t0 ts0) (TCons t ts))).(\lambda (H3: (eq T u0 u)).(\lambda (H1: (ty3 g c t0
209 u0)).(\lambda (H4: (tys3 g c ts0 u0)).(let H5 \def (eq_ind T u0 (\lambda (t1:
210 T).(tys3 g c ts0 t1)) H4 u H3) in (let H6 \def (eq_ind T u0 (\lambda (t1:
211 T).(ty3 g c t0 t1)) H1 u H3) in (let H7 \def (f_equal TList T (\lambda (e:
212 TList).(match e in TList return (\lambda (_: TList).T) with [TNil \Rightarrow
213 t0 | (TCons t1 _) \Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H2) in ((let
214 H8 \def (f_equal TList TList (\lambda (e: TList).(match e in TList return
215 (\lambda (_: TList).TList) with [TNil \Rightarrow ts0 | (TCons _ t1)
216 \Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H2) in (\lambda (H9: (eq T t0
217 t)).(let H10 \def (eq_ind TList ts0 (\lambda (t1: TList).(tys3 g c t1 u)) H5
218 ts H8) in (let H11 \def (eq_ind T t0 (\lambda (t1: T).(ty3 g c t1 u)) H6 t
219 H9) in (conj (ty3 g c t u) (tys3 g c ts u) H11 H10))))) H7))))))))))))
222 theorem ty3_getl_subst0:
223 \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
224 u) \to (\forall (v0: T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 t
225 t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c (CHead d
226 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))))
228 \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
229 (ty3 g c t u)).(ty3_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_:
230 T).(\forall (v0: T).(\forall (t2: T).(\forall (i: nat).((subst0 i v0 t0 t2)
231 \to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d
232 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda
233 (c0: C).(\lambda (t2: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2
234 t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
235 nat).((subst0 i v0 t2 t1) \to (\forall (b: B).(\forall (d: C).(\forall (v:
236 T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
237 w))))))))))))).(\lambda (u0: T).(\lambda (t1: T).(\lambda (_: (ty3 g c0 u0
238 t1)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i:
239 nat).((subst0 i v0 u0 t3) \to (\forall (b: B).(\forall (d: C).(\forall (v:
240 T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
241 w))))))))))))).(\lambda (_: (pc3 c0 t1 t2)).(\lambda (v0: T).(\lambda (t3:
242 T).(\lambda (i: nat).(\lambda (H5: (subst0 i v0 u0 t3)).(\lambda (b:
243 B).(\lambda (d: C).(\lambda (v: T).(\lambda (H6: (getl i c0 (CHead d (Bind b)
244 v))).(H3 v0 t3 i H5 b d v H6))))))))))))))))))) (\lambda (c0: C).(\lambda (m:
245 nat).(\lambda (v0: T).(\lambda (t0: T).(\lambda (i: nat).(\lambda (H0:
246 (subst0 i v0 (TSort m) t0)).(\lambda (b: B).(\lambda (d: C).(\lambda (v:
247 T).(\lambda (_: (getl i c0 (CHead d (Bind b) v))).(subst0_gen_sort v0 t0 i m
248 H0 (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda (n:
249 nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n
250 c0 (CHead d (Bind Abbr) u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0
251 t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
252 nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d0: C).(\forall (v:
253 T).((getl i d (CHead d0 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v
254 w))))))))))))).(\lambda (v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
255 (H3: (subst0 i v0 (TLRef n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda
256 (v: T).(\lambda (H4: (getl i c0 (CHead d0 (Bind b) v))).(and_ind (eq nat n i)
257 (eq T t1 (lift (S n) O v0)) (ex T (\lambda (w: T).(ty3 g d0 v w))) (\lambda
258 (H5: (eq nat n i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7 \def
259 (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v))) H4 n
260 H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl
261 n c0 c1)) H0 (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n
262 H0 (CHead d0 (Bind b) v) H7)) in (let H9 \def (f_equal C C (\lambda (e:
263 C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
264 (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind b)
265 v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in
266 ((let H10 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_:
267 C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k
268 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
269 \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v)
270 (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in
271 ((let H11 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
272 C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t2) \Rightarrow t2]))
273 (CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
274 Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (\lambda (H12: (eq B Abbr
275 b)).(\lambda (H13: (eq C d d0)).(let H14 \def (eq_ind_r T v (\lambda (t2:
276 T).(getl n c0 (CHead d0 (Bind b) t2))) H8 u0 H11) in (eq_ind T u0 (\lambda
277 (t2: T).(ex T (\lambda (w: T).(ty3 g d0 t2 w)))) (let H15 \def (eq_ind_r C d0
278 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind b) u0))) H14 d H13) in (eq_ind C
279 d (\lambda (c1: C).(ex T (\lambda (w: T).(ty3 g c1 u0 w)))) (let H16 \def
280 (eq_ind_r B b (\lambda (b0: B).(getl n c0 (CHead d (Bind b0) u0))) H15 Abbr
281 H12) in (ex_intro T (\lambda (w: T).(ty3 g d u0 w)) t0 H1)) d0 H13)) v
282 H11))))) H10)) H9)))))) (subst0_gen_lref v0 t1 i n H3))))))))))))))))))
283 (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda
284 (H0: (getl n c0 (CHead d (Bind Abst) u0))).(\lambda (t0: T).(\lambda (H1:
285 (ty3 g d u0 t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
286 nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d0: C).(\forall (v:
287 T).((getl i d (CHead d0 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v
288 w))))))))))))).(\lambda (v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
289 (H3: (subst0 i v0 (TLRef n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda
290 (v: T).(\lambda (H4: (getl i c0 (CHead d0 (Bind b) v))).(and_ind (eq nat n i)
291 (eq T t1 (lift (S n) O v0)) (ex T (\lambda (w: T).(ty3 g d0 v w))) (\lambda
292 (H5: (eq nat n i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7 \def
293 (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v))) H4 n
294 H5) in (let H8 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (c1: C).(getl
295 n c0 c1)) H0 (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n
296 H0 (CHead d0 (Bind b) v) H7)) in (let H9 \def (f_equal C C (\lambda (e:
297 C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
298 (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abst) u0) (CHead d0 (Bind b)
299 v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in
300 ((let H10 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_:
301 C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k
302 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
303 \Rightarrow Abst])])) (CHead d (Bind Abst) u0) (CHead d0 (Bind b) v)
304 (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in
305 ((let H11 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
306 C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t2) \Rightarrow t2]))
307 (CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
308 Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in (\lambda (H12: (eq B Abst
309 b)).(\lambda (H13: (eq C d d0)).(let H14 \def (eq_ind_r T v (\lambda (t2:
310 T).(getl n c0 (CHead d0 (Bind b) t2))) H8 u0 H11) in (eq_ind T u0 (\lambda
311 (t2: T).(ex T (\lambda (w: T).(ty3 g d0 t2 w)))) (let H15 \def (eq_ind_r C d0
312 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind b) u0))) H14 d H13) in (eq_ind C
313 d (\lambda (c1: C).(ex T (\lambda (w: T).(ty3 g c1 u0 w)))) (let H16 \def
314 (eq_ind_r B b (\lambda (b0: B).(getl n c0 (CHead d (Bind b0) u0))) H15 Abst
315 H12) in (ex_intro T (\lambda (w: T).(ty3 g d u0 w)) t0 H1)) d0 H13)) v
316 H11))))) H10)) H9)))))) (subst0_gen_lref v0 t1 i n H3))))))))))))))))))
317 (\lambda (c0: C).(\lambda (u0: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0
318 t0)).(\lambda (H1: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
319 nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d: C).(\forall (v:
320 T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
321 w))))))))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_:
322 (ty3 g (CHead c0 (Bind b) u0) t1 t2)).(\lambda (H3: ((\forall (v0:
323 T).(\forall (t3: T).(\forall (i: nat).((subst0 i v0 t1 t3) \to (\forall (b0:
324 B).(\forall (d: C).(\forall (v: T).((getl i (CHead c0 (Bind b) u0) (CHead d
325 (Bind b0) v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda
326 (v0: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead
327 (Bind b) u0 t1) t3)).(\lambda (b0: B).(\lambda (d: C).(\lambda (v:
328 T).(\lambda (H5: (getl i c0 (CHead d (Bind b0) v))).(or3_ind (ex2 T (\lambda
329 (u2: T).(eq T t3 (THead (Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0
330 u2))) (ex2 T (\lambda (t4: T).(eq T t3 (THead (Bind b) u0 t4))) (\lambda (t4:
331 T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex3_2 T T (\lambda (u2: T).(\lambda
332 (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_:
333 T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Bind b)
334 i) v0 t1 t4)))) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (H6: (ex2 T
335 (\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1))) (\lambda (u2: T).(subst0 i
336 v0 u0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1)))
337 (\lambda (u2: T).(subst0 i v0 u0 u2)) (ex T (\lambda (w: T).(ty3 g d v w)))
338 (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Bind b) x t1))).(\lambda (H8:
339 (subst0 i v0 u0 x)).(H1 v0 x i H8 b0 d v H5)))) H6)) (\lambda (H6: (ex2 T
340 (\lambda (t4: T).(eq T t3 (THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0
341 (s (Bind b) i) v0 t1 t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead (Bind
342 b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)) (ex T (\lambda
343 (w: T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Bind b)
344 u0 x))).(\lambda (H8: (subst0 (s (Bind b) i) v0 t1 x)).(H3 v0 x (S i) H8 b0 d
345 v (getl_head (Bind b) i c0 (CHead d (Bind b0) v) H5 u0))))) H6)) (\lambda
346 (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2
347 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_:
348 T).(\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4))))).(ex3_2_ind T T
349 (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda
350 (u2: T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4:
351 T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex T (\lambda (w: T).(ty3 g d v w)))
352 (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t3 (THead (Bind b) x0
353 x1))).(\lambda (H8: (subst0 i v0 u0 x0)).(\lambda (_: (subst0 (s (Bind b) i)
354 v0 t1 x1)).(H1 v0 x0 i H8 b0 d v H5)))))) H6)) (subst0_gen_head (Bind b) v0
355 u0 t1 t3 i H4)))))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda
356 (u0: T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (H1: ((\forall (v0:
357 T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 w t0) \to (\forall (b:
358 B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) v)) \to (ex
359 T (\lambda (w0: T).(ty3 g d v w0))))))))))))).(\lambda (v: T).(\lambda (t0:
360 T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 t0))).(\lambda (H3:
361 ((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 v t1) \to
362 (\forall (b: B).(\forall (d: C).(\forall (v1: T).((getl i c0 (CHead d (Bind
363 b) v1)) \to (ex T (\lambda (w0: T).(ty3 g d v1 w0))))))))))))).(\lambda (v0:
364 T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat
365 Appl) w v) t1)).(\lambda (b: B).(\lambda (d: C).(\lambda (v1: T).(\lambda
366 (H5: (getl i c0 (CHead d (Bind b) v1))).(or3_ind (ex2 T (\lambda (u2: T).(eq
367 T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w u2))) (ex2 T
368 (\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0
369 (s (Flat Appl) i) v0 v t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq
370 T t1 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i
371 v0 w u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v
372 t2)))) (ex T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (H6: (ex2 T (\lambda
373 (u2: T).(eq T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w
374 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t1 (THead (Flat Appl) u2 v)))
375 (\lambda (u2: T).(subst0 i v0 w u2)) (ex T (\lambda (w0: T).(ty3 g d v1 w0)))
376 (\lambda (x: T).(\lambda (_: (eq T t1 (THead (Flat Appl) x v))).(\lambda (H8:
377 (subst0 i v0 w x)).(H1 v0 x i H8 b d v1 H5)))) H6)) (\lambda (H6: (ex2 T
378 (\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0
379 (s (Flat Appl) i) v0 v t2)))).(ex2_ind T (\lambda (t2: T).(eq T t1 (THead
380 (Flat Appl) w t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2)) (ex
381 T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (x: T).(\lambda (_: (eq T t1
382 (THead (Flat Appl) w x))).(\lambda (H8: (subst0 (s (Flat Appl) i) v0 v
383 x)).(H3 v0 x (s (Flat Appl) i) H8 b d v1 H5)))) H6)) (\lambda (H6: (ex3_2 T T
384 (\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2))))
385 (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_:
386 T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))))).(ex3_2_ind T T
387 (\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2))))
388 (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_:
389 T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))) (ex T (\lambda (w0:
390 T).(ty3 g d v1 w0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t1
391 (THead (Flat Appl) x0 x1))).(\lambda (_: (subst0 i v0 w x0)).(\lambda (H9:
392 (subst0 (s (Flat Appl) i) v0 v x1)).(H3 v0 x1 (s (Flat Appl) i) H9 b d v1
393 H5)))))) H6)) (subst0_gen_head (Flat Appl) v0 w v t1 i H4)))))))))))))))))))
394 (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1
395 t2)).(\lambda (H1: ((\forall (v0: T).(\forall (t0: T).(\forall (i:
396 nat).((subst0 i v0 t1 t0) \to (\forall (b: B).(\forall (d: C).(\forall (v:
397 T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
398 w))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (H3:
399 ((\forall (v0: T).(\forall (t3: T).(\forall (i: nat).((subst0 i v0 t2 t3) \to
400 (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b)
401 v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda (v0:
402 T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat
403 Cast) t2 t1) t3)).(\lambda (b: B).(\lambda (d: C).(\lambda (v: T).(\lambda
404 (H5: (getl i c0 (CHead d (Bind b) v))).(or3_ind (ex2 T (\lambda (u2: T).(eq T
405 t3 (THead (Flat Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2))) (ex2 T
406 (\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4:
407 T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex3_2 T T (\lambda (u2: T).(\lambda
408 (t4: T).(eq T t3 (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_:
409 T).(subst0 i v0 t2 u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat
410 Cast) i) v0 t1 t4)))) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (H6:
411 (ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Cast) u2 t1))) (\lambda (u2:
412 T).(subst0 i v0 t2 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat
413 Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2)) (ex T (\lambda (w:
414 T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Flat Cast) x
415 t1))).(\lambda (H8: (subst0 i v0 t2 x)).(H3 v0 x i H8 b d v H5)))) H6))
416 (\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4)))
417 (\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4)))).(ex2_ind T (\lambda
418 (t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4: T).(subst0 (s
419 (Flat Cast) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x:
420 T).(\lambda (_: (eq T t3 (THead (Flat Cast) t2 x))).(\lambda (H8: (subst0 (s
421 (Flat Cast) i) v0 t1 x)).(H1 v0 x (s (Flat Cast) i) H8 b d v H5)))) H6))
422 (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
423 (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2)))
424 (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1
425 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
426 (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2)))
427 (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex T
428 (\lambda (w: T).(ty3 g d v w))) (\lambda (x0: T).(\lambda (x1: T).(\lambda
429 (_: (eq T t3 (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i v0 t2
430 x0)).(\lambda (_: (subst0 (s (Flat Cast) i) v0 t1 x1)).(H3 v0 x0 i H8 b d v
431 H5)))))) H6)) (subst0_gen_head (Flat Cast) v0 t2 t1 t3 i H4))))))))))))))))))
434 theorem ty3_gen_appl_nf2:
435 \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x:
436 T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex4_2 T T (\lambda (u:
437 T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
438 (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
439 (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
440 T).(nf2 c (THead (Bind Abst) u t))))))))))
442 \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x:
443 T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(ex3_2_ind T T (\lambda
444 (u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t))
445 x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
446 (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (ex4_2 T T (\lambda (u:
447 T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
448 (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
449 (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
450 T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x0: T).(\lambda (x1:
451 T).(\lambda (H0: (pc3 c (THead (Flat Appl) w (THead (Bind Abst) x0 x1))
452 x)).(\lambda (H1: (ty3 g c v (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g
453 c w x0)).(let H_x \def (ty3_correct g c v (THead (Bind Abst) x0 x1) H1) in
454 (let H3 \def H_x in (ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) x0
455 x1) t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl)
456 w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v
457 (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u)))
458 (\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) (\lambda
459 (x2: T).(\lambda (H4: (ty3 g c (THead (Bind Abst) x0 x1) x2)).(let H_x0 \def
460 (ty3_correct g c w x0 H2) in (let H5 \def H_x0 in (ex_ind T (\lambda (t:
461 T).(ty3 g c x0 t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead
462 (Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t:
463 T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3
464 g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t)))))
465 (\lambda (x3: T).(\lambda (H6: (ty3 g c x0 x3)).(let H7 \def (ty3_sn3 g c
466 (THead (Bind Abst) x0 x1) x2 H4) in (let H_x1 \def (nf2_sn3 c (THead (Bind
467 Abst) x0 x1) H7) in (let H8 \def H_x1 in (ex2_ind T (\lambda (u: T).(pr3 c
468 (THead (Bind Abst) x0 x1) u)) (\lambda (u: T).(nf2 c u)) (ex4_2 T T (\lambda
469 (u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t))
470 x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
471 (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
472 T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x4: T).(\lambda (H9: (pr3 c
473 (THead (Bind Abst) x0 x1) x4)).(\lambda (H10: (nf2 c x4)).(let H11 \def
474 (pr3_gen_abst c x0 x1 x4 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2:
475 T).(eq T x4 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_:
476 T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
477 (u: T).(pr3 (CHead c (Bind b) u) x1 t2))))) (ex4_2 T T (\lambda (u:
478 T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
479 (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
480 (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
481 T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x5: T).(\lambda (x6:
482 T).(\lambda (H12: (eq T x4 (THead (Bind Abst) x5 x6))).(\lambda (H13: (pr3 c
483 x0 x5)).(\lambda (H14: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind
484 b) u) x1 x6))))).(let H15 \def (eq_ind T x4 (\lambda (t: T).(nf2 c t)) H10
485 (THead (Bind Abst) x5 x6) H12) in (let H16 \def (pr3_head_12 c x0 x5 H13
486 (Bind Abst) x1 x6 (H14 Abst x5)) in (ex4_2_intro T T (\lambda (u: T).(\lambda
487 (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u:
488 T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u:
489 T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c
490 (THead (Bind Abst) u t)))) x5 x6 (pc3_pr3_conf c (THead (Flat Appl) w (THead
491 (Bind Abst) x0 x1)) x H0 (THead (Flat Appl) w (THead (Bind Abst) x5 x6))
492 (pr3_thin_dx c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 w
493 Appl)) (ty3_conv g c (THead (Bind Abst) x5 x6) x2 (ty3_sred_pr3 c (THead
494 (Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 g x2 H4) v (THead (Bind
495 Abst) x0 x1) H1 (pc3_pr3_r c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5
496 x6) H16)) (ty3_conv g c x5 x3 (ty3_sred_pr3 c x0 x5 H13 g x3 H6) w x0 H2
497 (pc3_pr3_r c x0 x5 H13)) H15)))))))) H11))))) H8)))))) H5))))) H3))))))))
498 (ty3_gen_appl g c w v x H))))))).
500 theorem ty3_inv_lref_nf2_pc3:
501 \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (i: nat).((ty3 g c
502 (TLRef i) u1) \to ((nf2 c (TLRef i)) \to (\forall (u2: T).((nf2 c u2) \to
503 ((pc3 c u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))))
505 \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (i: nat).(\lambda
506 (H: (ty3 g c (TLRef i) u1)).(insert_eq T (TLRef i) (\lambda (t: T).(ty3 g c t
507 u1)) (\lambda (t: T).((nf2 c t) \to (\forall (u2: T).((nf2 c u2) \to ((pc3 c
508 u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))) (\lambda
509 (y: T).(\lambda (H0: (ty3 g c y u1)).(ty3_ind g (\lambda (c0: C).(\lambda (t:
510 T).(\lambda (t0: T).((eq T t (TLRef i)) \to ((nf2 c0 t) \to (\forall (u2:
511 T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift
512 (S i) O u)))))))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t:
513 T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (TLRef i)) \to ((nf2
514 c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to (ex T
515 (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (u: T).(\lambda
516 (t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (TLRef i)) \to
517 ((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t1 u2) \to (ex T
518 (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (pc3 c0
519 t1 t2)).(\lambda (H6: (eq T u (TLRef i))).(\lambda (H7: (nf2 c0 u)).(\lambda
520 (u2: T).(\lambda (H8: (nf2 c0 u2)).(\lambda (H9: (pc3 c0 t2 u2)).(let H10
521 \def (eq_ind T u (\lambda (t0: T).(nf2 c0 t0)) H7 (TLRef i) H6) in (let H11
522 \def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef i)) \to ((nf2 c0 t0) \to
523 (\forall (u3: T).((nf2 c0 u3) \to ((pc3 c0 t1 u3) \to (ex T (\lambda (u0:
524 T).(eq T u3 (lift (S i) O u0)))))))))) H4 (TLRef i) H6) in (let H12 \def
525 (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef i) H6) in (let H_y
526 \def (H11 (refl_equal T (TLRef i)) H10 u2 H8) in (H_y (pc3_t t2 c0 t1 H5 u2
527 H9))))))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq
528 T (TSort m) (TLRef i))).(\lambda (_: (nf2 c0 (TSort m))).(\lambda (u2:
529 T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (TSort (next g m))
530 u2)).(let H5 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return
531 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
532 \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i) H1) in
533 (False_ind (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u)))) H5)))))))))
534 (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
535 (H1: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g
536 d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2:
537 T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S
538 i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5:
539 (nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (H7:
540 (pc3 c0 (lift (S n) O t) u2)).(let H8 \def (f_equal T nat (\lambda (e:
541 T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n |
542 (TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef
543 i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0)
544 O t) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0
545 (TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl
546 n0 c0 (CHead d (Bind Abbr) u))) H1 i H8) in (nf2_gen_lref c0 d u i H11 H10
547 (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))))))))))))))
548 (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
549 (H1: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g
550 d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2:
551 T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S
552 i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5:
553 (nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (H6: (nf2 c0 u2)).(\lambda (H7:
554 (pc3 c0 (lift (S n) O u) u2)).(let H8 \def (f_equal T nat (\lambda (e:
555 T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n |
556 (TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef
557 i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0)
558 O u) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0
559 (TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl
560 n0 c0 (CHead d (Bind Abst) u))) H1 i H8) in (let H_y \def (pc3_nf2_unfold c0
561 (lift (S i) O u) u2 H9 H6) in (let H12 \def (pr3_gen_lift c0 u u2 (S i) O H_y
562 d (getl_drop Abst c0 d u i H11)) in (ex2_ind T (\lambda (t2: T).(eq T u2
563 (lift (S i) O t2))) (\lambda (t2: T).(pr3 d u t2)) (ex T (\lambda (u0: T).(eq
564 T u2 (lift (S i) O u0)))) (\lambda (x: T).(\lambda (H13: (eq T u2 (lift (S i)
565 O x))).(\lambda (_: (pr3 d u x)).(eq_ind_r T (lift (S i) O x) (\lambda (t0:
566 T).(ex T (\lambda (u0: T).(eq T t0 (lift (S i) O u0))))) (ex_intro T (\lambda
567 (u0: T).(eq T (lift (S i) O x) (lift (S i) O u0))) x (refl_equal T (lift (S
568 i) O x))) u2 H13)))) H12)))))))))))))))))))) (\lambda (c0: C).(\lambda (u:
569 T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef
570 i)) \to ((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to
571 (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (b:
572 B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b)
573 u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef i)) \to ((nf2 (CHead c0 (Bind b) u)
574 t1) \to (\forall (u2: T).((nf2 (CHead c0 (Bind b) u) u2) \to ((pc3 (CHead c0
575 (Bind b) u) t2 u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O
576 u0))))))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TLRef i))).(\lambda
577 (_: (nf2 c0 (THead (Bind b) u t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0
578 u2)).(\lambda (_: (pc3 c0 (THead (Bind b) u t2) u2)).(let H9 \def (eq_ind T
579 (THead (Bind b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_:
580 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
581 (THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T
582 (\lambda (u0: T).(eq T u2 (lift (S i) O u0)))) H9))))))))))))))))) (\lambda
583 (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda
584 (_: (((eq T w (TLRef i)) \to ((nf2 c0 w) \to (\forall (u2: T).((nf2 c0 u2)
585 \to ((pc3 c0 u u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O
586 u0))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead
587 (Bind Abst) u t))).(\lambda (_: (((eq T v (TLRef i)) \to ((nf2 c0 v) \to
588 (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 (THead (Bind Abst) u t) u2) \to
589 (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (eq
590 T (THead (Flat Appl) w v) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Appl)
591 w v))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (THead
592 (Flat Appl) w (THead (Bind Abst) u t)) u2)).(let H9 \def (eq_ind T (THead
593 (Flat Appl) w v) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
594 with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
595 _) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u0:
596 T).(eq T u2 (lift (S i) O u0)))) H9)))))))))))))))) (\lambda (c0: C).(\lambda
597 (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T
598 t1 (TLRef i)) \to ((nf2 c0 t1) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0
599 t2 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda
600 (t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef i)) \to
601 ((nf2 c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T
602 (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (H5: (eq T
603 (THead (Flat Cast) t2 t1) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Cast)
604 t2 t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0
605 (THead (Flat Cast) t0 t2) u2)).(let H9 \def (eq_ind T (THead (Flat Cast) t2
606 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
607 _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
608 \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u: T).(eq T
609 u2 (lift (S i) O u)))) H9))))))))))))))) c y u1 H0))) H))))).
611 theorem ty3_inv_lref_nf2:
612 \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (i: nat).((ty3 g c
613 (TLRef i) u) \to ((nf2 c (TLRef i)) \to ((nf2 c u) \to (ex T (\lambda (u0:
614 T).(eq T u (lift (S i) O u0))))))))))
616 \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
617 (H: (ty3 g c (TLRef i) u)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1:
618 (nf2 c u)).(ty3_inv_lref_nf2_pc3 g c u i H H0 u H1 (pc3_refl c u)))))))).
620 theorem ty3_inv_appls_lref_nf2:
621 \forall (g: G).(\forall (c: C).(\forall (vs: TList).(\forall (u1:
622 T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) vs (TLRef i)) u1) \to
623 ((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S
624 i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) vs (lift (S i) O u))
627 \lambda (g: G).(\lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t:
628 TList).(\forall (u1: T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) t
629 (TLRef i)) u1) \to ((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u:
630 T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t
631 (lift (S i) O u)) u1))))))))) (\lambda (u1: T).(\lambda (i: nat).(\lambda (H:
632 (ty3 g c (TLRef i) u1)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (nf2 c
633 u1)).(let H_x \def (ty3_inv_lref_nf2 g c u1 i H H0 H1) in (let H2 \def H_x in
634 (ex_ind T (\lambda (u0: T).(eq T u1 (lift (S i) O u0))) (ex2 T (\lambda (u:
635 T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) u1)))
636 (\lambda (x: T).(\lambda (H3: (eq T u1 (lift (S i) O x))).(let H4 \def
637 (eq_ind T u1 (\lambda (t: T).(nf2 c t)) H1 (lift (S i) O x) H3) in (eq_ind_r
638 T (lift (S i) O x) (\lambda (t: T).(ex2 T (\lambda (u: T).(nf2 c (lift (S i)
639 O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) t)))) (ex_intro2 T (\lambda
640 (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u)
641 (lift (S i) O x))) x H4 (pc3_refl c (lift (S i) O x))) u1 H3)))) H2))))))))
642 (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (u1: T).(\forall
643 (i: nat).((ty3 g c (THeads (Flat Appl) t0 (TLRef i)) u1) \to ((nf2 c (TLRef
644 i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u)))
645 (\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) O u))
646 u1)))))))))).(\lambda (u1: T).(\lambda (i: nat).(\lambda (H0: (ty3 g c (THead
647 (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u1)).(\lambda (H1: (nf2 c
648 (TLRef i))).(\lambda (_: (nf2 c u1)).(let H_x \def (ty3_gen_appl_nf2 g c t
649 (THeads (Flat Appl) t0 (TLRef i)) u1 H0) in (let H3 \def H_x in (ex4_2_ind T
650 T (\lambda (u: T).(\lambda (t1: T).(pc3 c (THead (Flat Appl) t (THead (Bind
651 Abst) u t1)) u1))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c (THeads (Flat
652 Appl) t0 (TLRef i)) (THead (Bind Abst) u t1)))) (\lambda (u: T).(\lambda (_:
653 T).(ty3 g c t u))) (\lambda (u: T).(\lambda (t1: T).(nf2 c (THead (Bind Abst)
654 u t1)))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u:
655 T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u)))
656 u1))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (pc3 c (THead (Flat
657 Appl) t (THead (Bind Abst) x0 x1)) u1)).(\lambda (H5: (ty3 g c (THeads (Flat
658 Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c t
659 x0)).(\lambda (H7: (nf2 c (THead (Bind Abst) x0 x1))).(let H8 \def
660 (nf2_gen_abst c x0 x1 H7) in (and_ind (nf2 c x0) (nf2 (CHead c (Bind Abst)
661 x0) x1) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3
662 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) u1)))
663 (\lambda (H9: (nf2 c x0)).(\lambda (H10: (nf2 (CHead c (Bind Abst) x0)
664 x1)).(let H_y \def (H (THead (Bind Abst) x0 x1) i H5 H1) in (let H11 \def
665 (H_y (nf2_abst_shift c x0 H9 x1 H10)) in (ex2_ind T (\lambda (u: T).(nf2 c
666 (lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i)
667 O u)) (THead (Bind Abst) x0 x1))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O
668 u))) (\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift
669 (S i) O u))) u1))) (\lambda (x: T).(\lambda (H12: (nf2 c (lift (S i) O
670 x))).(\lambda (H13: (pc3 c (THeads (Flat Appl) t0 (lift (S i) O x)) (THead
671 (Bind Abst) x0 x1))).(ex_intro2 T (\lambda (u: T).(nf2 c (lift (S i) O u)))
672 (\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S
673 i) O u))) u1)) x H12 (pc3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) c
674 (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O x))) (pc3_thin_dx c
675 (THeads (Flat Appl) t0 (lift (S i) O x)) (THead (Bind Abst) x0 x1) H13 t
676 Appl) u1 H4))))) H11))))) H8)))))))) H3))))))))))) vs))).
678 theorem ty3_inv_lref_lref_nf2:
679 \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (j: nat).((ty3 g c
680 (TLRef i) (TLRef j)) \to ((nf2 c (TLRef i)) \to ((nf2 c (TLRef j)) \to (lt i
683 \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (j: nat).(\lambda
684 (H: (ty3 g c (TLRef i) (TLRef j))).(\lambda (H0: (nf2 c (TLRef i))).(\lambda
685 (H1: (nf2 c (TLRef j))).(let H_x \def (ty3_inv_lref_nf2 g c (TLRef j) i H H0
686 H1) in (let H2 \def H_x in (ex_ind T (\lambda (u0: T).(eq T (TLRef j) (lift
687 (S i) O u0))) (lt i j) (\lambda (x: T).(\lambda (H3: (eq T (TLRef j) (lift (S
688 i) O x))).(let H_x0 \def (lift_gen_lref x O (S i) j H3) in (let H4 \def H_x0
689 in (or_ind (land (lt j O) (eq T x (TLRef j))) (land (le (plus O (S i)) j) (eq
690 T x (TLRef (minus j (S i))))) (lt i j) (\lambda (H5: (land (lt j O) (eq T x
691 (TLRef j)))).(and_ind (lt j O) (eq T x (TLRef j)) (lt i j) (\lambda (H6: (lt
692 j O)).(\lambda (_: (eq T x (TLRef j))).(lt_x_O j H6 (lt i j)))) H5)) (\lambda
693 (H5: (land (le (plus O (S i)) j) (eq T x (TLRef (minus j (S i)))))).(and_ind
694 (le (plus O (S i)) j) (eq T x (TLRef (minus j (S i)))) (lt i j) (\lambda (H6:
695 (le (plus O (S i)) j)).(\lambda (_: (eq T x (TLRef (minus j (S i))))).H6))
696 H5)) H4))))) H2))))))))).
698 inductive wf3 (g: G): C \to (C \to Prop) \def
699 | wf3_sort: \forall (m: nat).(wf3 g (CSort m) (CSort m))
700 | wf3_bind: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u:
701 T).(\forall (t: T).((ty3 g c1 u t) \to (\forall (b: B).(wf3 g (CHead c1 (Bind
702 b) u) (CHead c2 (Bind b) u))))))))
703 | wf3_void: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u:
704 T).(((\forall (t: T).((ty3 g c1 u t) \to False))) \to (\forall (b: B).(wf3 g
705 (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O))))))))
706 | wf3_flat: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u:
707 T).(\forall (f: F).(wf3 g (CHead c1 (Flat f) u) c2))))).
709 theorem wf3_gen_sort1:
710 \forall (g: G).(\forall (x: C).(\forall (m: nat).((wf3 g (CSort m) x) \to
711 (eq C x (CSort m)))))
713 \lambda (g: G).(\lambda (x: C).(\lambda (m: nat).(\lambda (H: (wf3 g (CSort
714 m) x)).(insert_eq C (CSort m) (\lambda (c: C).(wf3 g c x)) (\lambda (c:
715 C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda
716 (c: C).(\lambda (c0: C).((eq C c (CSort m)) \to (eq C c0 c)))) (\lambda (m0:
717 nat).(\lambda (H1: (eq C (CSort m0) (CSort m))).(let H2 \def (f_equal C nat
718 (\lambda (e: C).(match e in C return (\lambda (_: C).nat) with [(CSort n)
719 \Rightarrow n | (CHead _ _ _) \Rightarrow m0])) (CSort m0) (CSort m) H1) in
720 (eq_ind_r nat m (\lambda (n: nat).(eq C (CSort n) (CSort n))) (refl_equal C
721 (CSort m)) m0 H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1
722 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u:
723 T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4:
724 (eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind C (CHead c1
725 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with
726 [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort m)
727 H4) in (False_ind (eq C (CHead c2 (Bind b) u) (CHead c1 (Bind b) u))
728 H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1
729 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u:
730 T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b:
731 B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind
732 C (CHead c1 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_:
733 C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow
734 True])) I (CSort m) H4) in (False_ind (eq C (CHead c2 (Bind Void) (TSort O))
735 (CHead c1 (Bind b) u)) H5)))))))))) (\lambda (c1: C).(\lambda (c2:
736 C).(\lambda (_: (wf3 g c1 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C
737 c2 c1)))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 (Flat
738 f) u) (CSort m))).(let H4 \def (eq_ind C (CHead c1 (Flat f) u) (\lambda (ee:
739 C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
740 False | (CHead _ _ _) \Rightarrow True])) I (CSort m) H3) in (False_ind (eq C
741 c2 (CHead c1 (Flat f) u)) H4))))))))) y x H0))) H)))).
743 theorem wf3_gen_bind1:
744 \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (b:
745 B).((wf3 g (CHead c1 (Bind b) v) x) \to (or (ex3_2 C T (\lambda (c2:
746 C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda
747 (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3
748 C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2:
749 C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
752 \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (b:
753 B).(\lambda (H: (wf3 g (CHead c1 (Bind b) v) x)).(insert_eq C (CHead c1 (Bind
754 b) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(or (ex3_2 C T (\lambda
755 (c2: C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2:
756 C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
757 v w)))) (ex3 C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O))))
758 (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
759 w) \to False)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g
760 (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead c1 (Bind b) v)) \to (or
761 (ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C c0 (CHead c2 (Bind b) v))))
762 (\lambda (c2: C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w:
763 T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: C).(eq C c0 (CHead c2 (Bind Void)
764 (TSort O)))) (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w:
765 T).((ty3 g c1 v w) \to False)))))))) (\lambda (m: nat).(\lambda (H1: (eq C
766 (CSort m) (CHead c1 (Bind b) v))).(let H2 \def (eq_ind C (CSort m) (\lambda
767 (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
768 \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead c1 (Bind b) v)
769 H1) in (False_ind (or (ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C
770 (CSort m) (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g c1
771 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2:
772 C).(eq C (CSort m) (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2: C).(wf3 g
773 c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))))) H2))))
774 (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2:
775 (((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3:
776 C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3:
777 C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
778 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O))))
779 (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
780 w) \to False)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0
781 u t)).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind b0) u) (CHead c1
782 (Bind b) v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return
783 (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow
784 c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H6 \def
785 (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
786 [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return
787 (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
788 b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H7 \def
789 (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
790 [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 (Bind
791 b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (H8: (eq B b0 b)).(\lambda (H9:
792 (eq C c0 c1)).(eq_ind_r B b (\lambda (b1: B).(or (ex3_2 C T (\lambda (c3:
793 C).(\lambda (_: T).(eq C (CHead c2 (Bind b1) u) (CHead c3 (Bind b) v))))
794 (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w:
795 T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C (CHead c2 (Bind b1) u)
796 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda
797 (_: C).(\forall (w: T).((ty3 g c1 v w) \to False)))))) (let H10 \def (eq_ind
798 T u (\lambda (t0: T).(ty3 g c0 t0 t)) H3 v H7) in (eq_ind_r T v (\lambda (t0:
799 T).(or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b)
800 t0) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3)))
801 (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq
802 C (CHead c2 (Bind b) t0) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3:
803 C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
804 False)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).(ty3 g c v t)) H10 c1
805 H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind b)
806 v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
807 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_:
808 C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead
809 c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_:
810 C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13
811 \def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_introl
812 (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b) v)
813 (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3)))
814 (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq
815 C (CHead c2 (Bind b) v) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3:
816 C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
817 False)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2
818 (Bind b) v) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g
819 c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w))) c2 t (refl_equal C
820 (CHead c2 (Bind b) v)) H13 H11))))) u H7)) b0 H8)))) H6)) H5)))))))))))
821 (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2:
822 (((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3:
823 C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3:
824 C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
825 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O))))
826 (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
827 w) \to False)))))))).(\lambda (u: T).(\lambda (H3: ((\forall (t: T).((ty3 g
828 c0 u t) \to False)))).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind
829 b0) u) (CHead c1 (Bind b) v))).(let H5 \def (f_equal C C (\lambda (e:
830 C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 |
831 (CHead c _ _) \Rightarrow c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v)
832 H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e in C return
833 (\lambda (_: C).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow
834 (match k in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 |
835 (Flat _) \Rightarrow b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4)
836 in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
837 (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
838 (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (_: (eq B b0
839 b)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind T u (\lambda (t:
840 T).(\forall (t0: T).((ty3 g c0 t t0) \to False))) H3 v H7) in (let H11 \def
841 (eq_ind C c0 (\lambda (c: C).(\forall (t: T).((ty3 g c v t) \to False))) H10
842 c1 H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind
843 b) v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
844 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_:
845 C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead
846 c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_:
847 C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13
848 \def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_intror
849 (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind Void)
850 (TSort O)) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g
851 c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda
852 (c3: C).(eq C (CHead c2 (Bind Void) (TSort O)) (CHead c3 (Bind Void) (TSort
853 O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
854 c1 v w) \to False)))) (ex3_intro C (\lambda (c3: C).(eq C (CHead c2 (Bind
855 Void) (TSort O)) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g
856 c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))) c2
857 (refl_equal C (CHead c2 (Bind Void) (TSort O))) H13 H11))))))))) H6))
858 H5)))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0
859 c2)).(\lambda (_: (((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T
860 (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda
861 (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3
862 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
863 O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
864 c1 v w) \to False)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C
865 (CHead c0 (Flat f) u) (CHead c1 (Bind b) v))).(let H4 \def (eq_ind C (CHead
866 c0 (Flat f) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
867 with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K
868 return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
869 \Rightarrow True])])) I (CHead c1 (Bind b) v) H3) in (False_ind (or (ex3_2 C
870 T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda
871 (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3
872 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
873 O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
874 c1 v w) \to False))))) H4))))))))) y x H0))) H)))))).
876 theorem wf3_gen_flat1:
877 \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (f:
878 F).((wf3 g (CHead c1 (Flat f) v) x) \to (wf3 g c1 x))))))
880 \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (f:
881 F).(\lambda (H: (wf3 g (CHead c1 (Flat f) v) x)).(insert_eq C (CHead c1 (Flat
882 f) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(wf3 g c1 x)) (\lambda (y:
883 C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda (c: C).(\lambda (c0:
884 C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c0)))) (\lambda (m:
885 nat).(\lambda (H1: (eq C (CSort m) (CHead c1 (Flat f) v))).(let H2 \def
886 (eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return (\lambda (_:
887 C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
888 False])) I (CHead c1 (Flat f) v) H1) in (False_ind (wf3 g c1 (CSort m))
889 H2)))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 c2)).(\lambda
890 (_: (((eq C c0 (CHead c1 (Flat f) v)) \to (wf3 g c1 c2)))).(\lambda (u:
891 T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (b: B).(\lambda (H4:
892 (eq C (CHead c0 (Bind b) u) (CHead c1 (Flat f) v))).(let H5 \def (eq_ind C
893 (CHead c0 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_:
894 C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
895 k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat
896 _) \Rightarrow False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1
897 (CHead c2 (Bind b) u)) H5))))))))))) (\lambda (c0: C).(\lambda (c2:
898 C).(\lambda (_: (wf3 g c0 c2)).(\lambda (_: (((eq C c0 (CHead c1 (Flat f) v))
899 \to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c0
900 u t) \to False)))).(\lambda (b: B).(\lambda (H4: (eq C (CHead c0 (Bind b) u)
901 (CHead c1 (Flat f) v))).(let H5 \def (eq_ind C (CHead c0 (Bind b) u) (\lambda
902 (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
903 \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
904 (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
905 False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1 (CHead c2
906 (Bind Void) (TSort O))) H5)))))))))) (\lambda (c0: C).(\lambda (c2:
907 C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 (Flat f)
908 v)) \to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (f0: F).(\lambda (H3: (eq C
909 (CHead c0 (Flat f0) u) (CHead c1 (Flat f) v))).(let H4 \def (f_equal C C
910 (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
911 \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Flat f0) u) (CHead
912 c1 (Flat f) v) H3) in ((let H5 \def (f_equal C F (\lambda (e: C).(match e in
913 C return (\lambda (_: C).F) with [(CSort _) \Rightarrow f0 | (CHead _ k _)
914 \Rightarrow (match k in K return (\lambda (_: K).F) with [(Bind _)
915 \Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead c0 (Flat f0) u) (CHead
916 c1 (Flat f) v) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in
917 C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
918 \Rightarrow t])) (CHead c0 (Flat f0) u) (CHead c1 (Flat f) v) H3) in (\lambda
919 (_: (eq F f0 f)).(\lambda (H8: (eq C c0 c1)).(let H9 \def (eq_ind C c0
920 (\lambda (c: C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c2))) H2 c1 H8)
921 in (let H10 \def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H8) in
922 H10))))) H5)) H4))))))))) y x H0))) H)))))).
924 theorem wf3_gen_head2:
925 \forall (g: G).(\forall (x: C).(\forall (c: C).(\forall (v: T).(\forall (k:
926 K).((wf3 g x (CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b)))))))))
928 \lambda (g: G).(\lambda (x: C).(\lambda (c: C).(\lambda (v: T).(\lambda (k:
929 K).(\lambda (H: (wf3 g x (CHead c k v))).(insert_eq C (CHead c k v) (\lambda
930 (c0: C).(wf3 g x c0)) (\lambda (_: C).(ex B (\lambda (b: B).(eq K k (Bind
931 b))))) (\lambda (y: C).(\lambda (H0: (wf3 g x y)).(wf3_ind g (\lambda (_:
932 C).(\lambda (c1: C).((eq C c1 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K
933 k (Bind b))))))) (\lambda (m: nat).(\lambda (H1: (eq C (CSort m) (CHead c k
934 v))).(let H2 \def (eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return
935 (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
936 \Rightarrow False])) I (CHead c k v) H1) in (False_ind (ex B (\lambda (b:
937 B).(eq K k (Bind b)))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1:
938 (wf3 g c1 c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b:
939 B).(eq K k (Bind b))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3
940 g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C (CHead c2 (Bind b) u) (CHead c
941 k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return
942 (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _)
943 \Rightarrow c0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in ((let H6 \def
944 (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
945 [(CSort _) \Rightarrow (Bind b) | (CHead _ k0 _) \Rightarrow k0])) (CHead c2
946 (Bind b) u) (CHead c k v) H4) in ((let H7 \def (f_equal C T (\lambda (e:
947 C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
948 (CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in
949 (\lambda (H8: (eq K (Bind b) k)).(\lambda (H9: (eq C c2 c)).(let H10 \def
950 (eq_ind T u (\lambda (t0: T).(ty3 g c1 t0 t)) H3 v H7) in (let H11 \def
951 (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda
952 (b0: B).(eq K k (Bind b0)))))) H2 c H9) in (let H12 \def (eq_ind C c2
953 (\lambda (c0: C).(wf3 g c1 c0)) H1 c H9) in (let H13 \def (eq_ind_r K k
954 (\lambda (k0: K).((eq C c (CHead c k0 v)) \to (ex B (\lambda (b0: B).(eq K k0
955 (Bind b0)))))) H11 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(ex B
956 (\lambda (b0: B).(eq K k0 (Bind b0))))) (ex_intro B (\lambda (b0: B).(eq K
957 (Bind b) (Bind b0))) b (refl_equal K (Bind b))) k H8)))))))) H6))
958 H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1
959 c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K
960 k (Bind b))))))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u
961 t) \to False)))).(\lambda (_: B).(\lambda (H4: (eq C (CHead c2 (Bind Void)
962 (TSort O)) (CHead c k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e
963 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _
964 _) \Rightarrow c0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in
965 ((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_:
966 C).K) with [(CSort _) \Rightarrow (Bind Void) | (CHead _ k0 _) \Rightarrow
967 k0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in ((let H7 \def
968 (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
969 [(CSort _) \Rightarrow (TSort O) | (CHead _ _ t) \Rightarrow t])) (CHead c2
970 (Bind Void) (TSort O)) (CHead c k v) H4) in (\lambda (H8: (eq K (Bind Void)
971 k)).(\lambda (H9: (eq C c2 c)).(let H10 \def (eq_ind C c2 (\lambda (c0:
972 C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b0: B).(eq K k (Bind b0))))))
973 H2 c H9) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c
974 H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c (CHead c k0 v))
975 \to (ex B (\lambda (b0: B).(eq K k0 (Bind b0)))))) H10 (Bind Void) H8) in
976 (eq_ind K (Bind Void) (\lambda (k0: K).(ex B (\lambda (b0: B).(eq K k0 (Bind
977 b0))))) (let H13 \def (eq_ind_r T v (\lambda (t: T).((eq C c (CHead c (Bind
978 Void) t)) \to (ex B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))))) H12
979 (TSort O) H7) in (ex_intro B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))
980 Void (refl_equal K (Bind Void)))) k H8))))))) H6)) H5)))))))))) (\lambda (c1:
981 C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c2
982 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))).(\lambda (_:
983 T).(\lambda (_: F).(\lambda (H3: (eq C c2 (CHead c k v))).(let H4 \def
984 (f_equal C C (\lambda (e: C).e) c2 (CHead c k v) H3) in (let H5 \def (eq_ind
985 C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b: B).(eq
986 K k (Bind b)))))) H2 (CHead c k v) H4) in (let H6 \def (eq_ind C c2 (\lambda
987 (c0: C).(wf3 g c1 c0)) H1 (CHead c k v) H4) in (H5 (refl_equal C (CHead c k
988 v))))))))))))) x y H0))) H)))))).
991 \forall (g: G).(\forall (c: C).(\forall (c1: C).((wf3 g c c1) \to (\forall
992 (c2: C).((wf3 g c c2) \to (eq C c1 c2))))))
994 \lambda (g: G).(\lambda (c: C).(\lambda (c1: C).(\lambda (H: (wf3 g c
995 c1)).(wf3_ind g (\lambda (c0: C).(\lambda (c2: C).(\forall (c3: C).((wf3 g c0
996 c3) \to (eq C c2 c3))))) (\lambda (m: nat).(\lambda (c2: C).(\lambda (H0:
997 (wf3 g (CSort m) c2)).(let H_y \def (wf3_gen_sort1 g c2 m H0) in (eq_ind_r C
998 (CSort m) (\lambda (c0: C).(eq C (CSort m) c0)) (refl_equal C (CSort m)) c2
999 H_y))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2
1000 c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3
1001 c4))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c2 u
1002 t)).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g (CHead c2 (Bind b)
1003 u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in (let H4 \def H_x in
1004 (or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind
1005 b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_:
1006 C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq C c0 (CHead
1007 c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_:
1008 C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3 (Bind b) u)
1009 c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead
1010 c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda
1011 (_: C).(\lambda (w: T).(ty3 g c2 u w))))).(ex3_2_ind C T (\lambda (c4:
1012 C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4:
1013 C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2
1014 u w))) (eq C (CHead c3 (Bind b) u) c0) (\lambda (x0: C).(\lambda (x1:
1015 T).(\lambda (H6: (eq C c0 (CHead x0 (Bind b) u))).(\lambda (H7: (wf3 g c2
1016 x0)).(\lambda (_: (ty3 g c2 u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda
1017 (c4: C).(eq C (CHead c3 (Bind b) u) c4)) (f_equal3 C K T C CHead c3 x0 (Bind
1018 b) (Bind b) u u (H1 x0 H7) (refl_equal K (Bind b)) (refl_equal T u)) c0
1019 H6)))))) H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind
1020 Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall
1021 (w: T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0
1022 (CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda
1023 (_: C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind b)
1024 u) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void) (TSort
1025 O)))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: ((\forall (w: T).((ty3 g c2 u
1026 w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c4:
1027 C).(eq C (CHead c3 (Bind b) u) c4)) (let H_x0 \def (H8 t H2) in (let H9 \def
1028 H_x0 in (False_ind (eq C (CHead c3 (Bind b) u) (CHead x0 (Bind Void) (TSort
1029 O))) H9))) c0 H6))))) H5)) H4))))))))))))) (\lambda (c2: C).(\lambda (c3:
1030 C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4)
1031 \to (eq C c3 c4))))).(\lambda (u: T).(\lambda (H2: ((\forall (t: T).((ty3 g
1032 c2 u t) \to False)))).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g
1033 (CHead c2 (Bind b) u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in
1034 (let H4 \def H_x in (or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C
1035 c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4)))
1036 (\lambda (_: C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq
1037 C c0 (CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4))
1038 (\lambda (_: C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3
1039 (Bind Void) (TSort O)) c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda
1040 (_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_:
1041 T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2 u
1042 w))))).(ex3_2_ind C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4
1043 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_:
1044 C).(\lambda (w: T).(ty3 g c2 u w))) (eq C (CHead c3 (Bind Void) (TSort O))
1045 c0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c0 (CHead x0 (Bind
1046 b) u))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: (ty3 g c2 u x1)).(eq_ind_r
1047 C (CHead x0 (Bind b) u) (\lambda (c4: C).(eq C (CHead c3 (Bind Void) (TSort
1048 O)) c4)) (let H_x0 \def (H2 x1 H8) in (let H9 \def H_x0 in (False_ind (eq C
1049 (CHead c3 (Bind Void) (TSort O)) (CHead x0 (Bind b) u)) H9))) c0 H6))))))
1050 H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind Void)
1051 (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall (w:
1052 T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0 (CHead
1053 c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_:
1054 C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind Void)
1055 (TSort O)) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void)
1056 (TSort O)))).(\lambda (H7: (wf3 g c2 x0)).(\lambda (_: ((\forall (w: T).((ty3
1057 g c2 u w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda
1058 (c4: C).(eq C (CHead c3 (Bind Void) (TSort O)) c4)) (f_equal3 C K T C CHead
1059 c3 x0 (Bind Void) (Bind Void) (TSort O) (TSort O) (H1 x0 H7) (refl_equal K
1060 (Bind Void)) (refl_equal T (TSort O))) c0 H6))))) H5)) H4))))))))))))
1061 (\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1:
1062 ((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3 c4))))).(\lambda (u:
1063 T).(\lambda (f: F).(\lambda (c0: C).(\lambda (H2: (wf3 g (CHead c2 (Flat f)
1064 u) c0)).(let H_y \def (wf3_gen_flat1 g c2 c0 u f H2) in (H1 c0 H_y))))))))))
1067 theorem wf3_clear_conf:
1068 \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (g: G).(\forall
1069 (c2: C).((wf3 g c1 c2) \to (wf3 g c c2))))))
1071 \lambda (c1: C).(\lambda (c: C).(\lambda (H: (clear c1 c)).(clear_ind
1072 (\lambda (c0: C).(\lambda (c2: C).(\forall (g: G).(\forall (c3: C).((wf3 g c0
1073 c3) \to (wf3 g c2 c3)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u:
1074 T).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u)
1075 c2)).H0)))))) (\lambda (e: C).(\lambda (c0: C).(\lambda (_: (clear e
1076 c0)).(\lambda (H1: ((\forall (g: G).(\forall (c2: C).((wf3 g e c2) \to (wf3 g
1077 c0 c2)))))).(\lambda (f: F).(\lambda (u: T).(\lambda (g: G).(\lambda (c2:
1078 C).(\lambda (H2: (wf3 g (CHead e (Flat f) u) c2)).(let H_y \def
1079 (wf3_gen_flat1 g e c2 u f H2) in (H1 g c2 H_y))))))))))) c1 c H))).
1081 theorem clear_wf3_trans:
1082 \forall (c1: C).(\forall (d1: C).((clear c1 d1) \to (\forall (g: G).(\forall
1083 (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda
1084 (c2: C).(clear c2 d2))))))))
1086 \lambda (c1: C).(\lambda (d1: C).(\lambda (H: (clear c1 d1)).(clear_ind
1087 (\lambda (c: C).(\lambda (c0: C).(\forall (g: G).(\forall (d2: C).((wf3 g c0
1088 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(clear c2
1089 d2)))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (g:
1090 G).(\lambda (d2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u) d2)).(let H_x
1091 \def (wf3_gen_bind1 g e d2 u b H0) in (let H1 \def H_x in (or_ind (ex3_2 C T
1092 (\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda
1093 (c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g
1094 e u w)))) (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O))))
1095 (\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w)
1096 \to False)))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2))
1097 (\lambda (c2: C).(clear c2 d2))) (\lambda (H2: (ex3_2 C T (\lambda (c2:
1098 C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda (c2:
1099 C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g e u
1100 w))))).(ex3_2_ind C T (\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2
1101 (Bind b) u)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_:
1102 C).(\lambda (w: T).(ty3 g e u w))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e
1103 (Bind b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda
1104 (x1: T).(\lambda (H3: (eq C d2 (CHead x0 (Bind b) u))).(\lambda (H4: (wf3 g e
1105 x0)).(\lambda (H5: (ty3 g e u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda
1106 (c: C).(ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2:
1107 C).(clear c2 c)))) (ex_intro2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u)
1108 c2)) (\lambda (c2: C).(clear c2 (CHead x0 (Bind b) u))) (CHead x0 (Bind b) u)
1109 (wf3_bind g e x0 H4 u x1 H5 b) (clear_bind b x0 u)) d2 H3)))))) H2)) (\lambda
1110 (H2: (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O))))
1111 (\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w)
1112 \to False))))).(ex3_ind C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void)
1113 (TSort O)))) (\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w:
1114 T).((ty3 g e u w) \to False))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind
1115 b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda (H3:
1116 (eq C d2 (CHead x0 (Bind Void) (TSort O)))).(\lambda (H4: (wf3 g e
1117 x0)).(\lambda (H5: ((\forall (w: T).((ty3 g e u w) \to False)))).(eq_ind_r C
1118 (CHead x0 (Bind Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (c2: C).(wf3
1119 g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2 c)))) (ex_intro2 C
1120 (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2
1121 (CHead x0 (Bind Void) (TSort O)))) (CHead x0 (Bind Void) (TSort O)) (wf3_void
1122 g e x0 H4 u H5 b) (clear_bind Void x0 (TSort O))) d2 H3))))) H2)) H1)))))))))
1123 (\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1:
1124 ((\forall (g: G).(\forall (d2: C).((wf3 g c d2) \to (ex2 C (\lambda (c2:
1125 C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)))))))).(\lambda (f:
1126 F).(\lambda (u: T).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g c
1127 d2)).(let H_x \def (H1 g d2 H2) in (let H3 \def H_x in (ex2_ind C (\lambda
1128 (c2: C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)) (ex2 C (\lambda (c2:
1129 C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda
1130 (x: C).(\lambda (H4: (wf3 g e x)).(\lambda (H5: (clear x d2)).(ex_intro2 C
1131 (\lambda (c2: C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2
1132 d2)) x (wf3_flat g e x H4 u f) H5)))) H3)))))))))))) c1 d1 H))).
1134 theorem wf3_getl_conf:
1135 \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall
1136 (v: T).((getl i c1 (CHead d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2:
1137 C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda
1138 (d2: C).(getl i c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
1141 \lambda (b: B).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1:
1142 C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1 (Bind b) v)) \to
1143 (\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g
1144 d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind b) v)))
1145 (\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda (c1: C).(\lambda (d1:
1146 C).(\lambda (v: T).(\lambda (H: (getl O c1 (CHead d1 (Bind b) v))).(\lambda
1147 (g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda (w: T).(\lambda
1148 (H1: (ty3 g d1 v w)).(let H_y \def (wf3_clear_conf c1 (CHead d1 (Bind b) v)
1149 (getl_gen_O c1 (CHead d1 (Bind b) v) H) g c2 H0) in (let H_x \def
1150 (wf3_gen_bind1 g d1 c2 v b H_y) in (let H2 \def H_x in (or_ind (ex3_2 C T
1151 (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda
1152 (c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3
1153 g d1 v w0)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
1154 O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3
1155 g d1 v w0) \to False)))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind
1156 b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H3: (ex3_2 C T (\lambda
1157 (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3:
1158 C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g d1
1159 v w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
1160 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_:
1161 C).(\lambda (w0: T).(ty3 g d1 v w0))) (ex2 C (\lambda (d2: C).(getl O c2
1162 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0:
1163 C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind b) v))).(\lambda
1164 (H5: (wf3 g d1 x0)).(\lambda (_: (ty3 g d1 v x1)).(eq_ind_r C (CHead x0 (Bind
1165 b) v) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2 (Bind b)
1166 v))) (\lambda (d2: C).(wf3 g d1 d2)))) (ex_intro2 C (\lambda (d2: C).(getl O
1167 (CHead x0 (Bind b) v) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))
1168 x0 (getl_refl b x0 v) H5) c2 H4)))))) H3)) (\lambda (H3: (ex3 C (\lambda (c3:
1169 C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g d1
1170 c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g d1 v w0) \to
1171 False))))).(ex3_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
1172 O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3
1173 g d1 v w0) \to False))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind b)
1174 v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: C).(\lambda (H4: (eq C c2
1175 (CHead x0 (Bind Void) (TSort O)))).(\lambda (_: (wf3 g d1 x0)).(\lambda (H6:
1176 ((\forall (w0: T).((ty3 g d1 v w0) \to False)))).(eq_ind_r C (CHead x0 (Bind
1177 Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2
1178 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H6 w H1) in
1179 (let H7 \def H_x0 in (False_ind (ex2 C (\lambda (d2: C).(getl O (CHead x0
1180 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
1181 d2))) H7))) c2 H4))))) H3)) H2))))))))))))) (\lambda (n: nat).(\lambda (H:
1182 ((\forall (c1: C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1
1183 (Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall
1184 (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind
1185 b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))))).(\lambda (c1: C).(C_ind
1186 (\lambda (c: C).(\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead d1
1187 (Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to (\forall
1188 (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2
1189 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))) (\lambda (n0:
1190 nat).(\lambda (d1: C).(\lambda (v: T).(\lambda (H0: (getl (S n) (CSort n0)
1191 (CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (wf3 g
1192 (CSort n0) c2)).(\lambda (w: T).(\lambda (_: (ty3 g d1 v w)).(getl_gen_sort
1193 n0 (S n) (CHead d1 (Bind b) v) H0 (ex2 C (\lambda (d2: C).(getl (S n) c2
1194 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda
1195 (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead
1196 d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to
1197 (\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2
1198 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))).(\lambda
1199 (k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (v: T).(\lambda (H1: (getl
1200 (S n) (CHead c k t) (CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2:
1201 C).(\lambda (H2: (wf3 g (CHead c k t) c2)).(\lambda (w: T).(\lambda (H3: (ty3
1202 g d1 v w)).(K_ind (\lambda (k0: K).((wf3 g (CHead c k0 t) c2) \to ((getl (r
1203 k0 n) c (CHead d1 (Bind b) v)) \to (ex2 C (\lambda (d2: C).(getl (S n) c2
1204 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))) (\lambda (b0:
1205 B).(\lambda (H4: (wf3 g (CHead c (Bind b0) t) c2)).(\lambda (H5: (getl (r
1206 (Bind b0) n) c (CHead d1 (Bind b) v))).(let H_x \def (wf3_gen_bind1 g c c2 t
1207 b0 H4) in (let H6 \def H_x in (or_ind (ex3_2 C T (\lambda (c3: C).(\lambda
1208 (_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3: C).(\lambda (_:
1209 T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t w0)))) (ex3 C
1210 (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3:
1211 C).(wf3 g c c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to
1212 False)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind b) v)))
1213 (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H7: (ex3_2 C T (\lambda (c3:
1214 C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3:
1215 C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t
1216 w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
1217 (Bind b0) t)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_:
1218 C).(\lambda (w0: T).(ty3 g c t w0))) (ex2 C (\lambda (d2: C).(getl (S n) c2
1219 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0:
1220 C).(\lambda (x1: T).(\lambda (H8: (eq C c2 (CHead x0 (Bind b0) t))).(\lambda
1221 (H9: (wf3 g c x0)).(\lambda (_: (ty3 g c t x1)).(eq_ind_r C (CHead x0 (Bind
1222 b0) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2
1223 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g
1224 x0 H9 w H3) in (let H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0
1225 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2:
1226 C).(getl (S n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2:
1227 C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind
1228 b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S
1229 n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
1230 d2)) x (getl_head (Bind b0) n x0 (CHead x (Bind b) v) H12 t) H13)))) H11)))
1231 c2 H8)))))) H7)) (\lambda (H7: (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3
1232 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3)) (\lambda (_:
1233 C).(\forall (w0: T).((ty3 g c t w0) \to False))))).(ex3_ind C (\lambda (c3:
1234 C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3))
1235 (\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to False))) (ex2 C (\lambda
1236 (d2: C).(getl (S n) c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1
1237 d2))) (\lambda (x0: C).(\lambda (H8: (eq C c2 (CHead x0 (Bind Void) (TSort
1238 O)))).(\lambda (H9: (wf3 g c x0)).(\lambda (_: ((\forall (w0: T).((ty3 g c t
1239 w0) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c0:
1240 C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2 (Bind b) v))) (\lambda
1241 (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g x0 H9 w H3) in (let
1242 H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0 (CHead d2 (Bind b)
1243 v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n)
1244 (CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2:
1245 C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind
1246 b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S
1247 n) (CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2:
1248 C).(wf3 g d1 d2)) x (getl_head (Bind Void) n x0 (CHead x (Bind b) v) H12
1249 (TSort O)) H13)))) H11))) c2 H8))))) H7)) H6)))))) (\lambda (f: F).(\lambda
1250 (H4: (wf3 g (CHead c (Flat f) t) c2)).(\lambda (H5: (getl (r (Flat f) n) c
1251 (CHead d1 (Bind b) v))).(let H_y \def (wf3_gen_flat1 g c c2 t f H4) in (H0 d1
1252 v H5 g c2 H_y w H3))))) k H2 (getl_gen_S k c (CHead d1 (Bind b) v) t n
1253 H1)))))))))))))) c1)))) i)).
1256 \forall (g: G).(\forall (c1: C).(ex C (\lambda (c2: C).(wf3 g c1 c2))))
1258 \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(ex C (\lambda (c2:
1259 C).(wf3 g c c2)))) (\lambda (n: nat).(ex_intro C (\lambda (c2: C).(wf3 g
1260 (CSort n) c2)) (CSort n) (wf3_sort g n))) (\lambda (c: C).(\lambda (H: (ex C
1261 (\lambda (c2: C).(wf3 g c c2)))).(\lambda (k: K).(\lambda (t: T).(let H0 \def
1262 H in (ex_ind C (\lambda (c2: C).(wf3 g c c2)) (ex C (\lambda (c2: C).(wf3 g
1263 (CHead c k t) c2))) (\lambda (x: C).(\lambda (H1: (wf3 g c x)).(K_ind
1264 (\lambda (k0: K).(ex C (\lambda (c2: C).(wf3 g (CHead c k0 t) c2)))) (\lambda
1265 (b: B).(let H_x \def (ty3_inference g c t) in (let H2 \def H_x in (or_ind (ex
1266 T (\lambda (t2: T).(ty3 g c t t2))) (\forall (t2: T).((ty3 g c t t2) \to
1267 False)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda
1268 (H3: (ex T (\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3
1269 g c t t2)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda
1270 (x0: T).(\lambda (H4: (ty3 g c t x0)).(ex_intro C (\lambda (c2: C).(wf3 g
1271 (CHead c (Bind b) t) c2)) (CHead x (Bind b) t) (wf3_bind g c x H1 t x0 H4
1272 b)))) H3)) (\lambda (H3: ((\forall (t2: T).((ty3 g c t t2) \to
1273 False)))).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))
1274 (CHead x (Bind Void) (TSort O)) (wf3_void g c x H1 t H3 b))) H2)))) (\lambda
1275 (f: F).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) x
1276 (wf3_flat g c x H1 t f))) k))) H0)))))) c1)).
1278 theorem getl_wf3_trans:
1279 \forall (i: nat).(\forall (c1: C).(\forall (d1: C).((getl i c1 d1) \to
1280 (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2:
1281 C).(wf3 g c1 c2)) (\lambda (c2: C).(getl i c2 d2)))))))))
1283 \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1:
1284 C).((getl n c1 d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to
1285 (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2
1286 d2)))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (H: (getl O c1
1287 d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H0: (wf3 g d1 d2)).(let H_x
1288 \def (clear_wf3_trans c1 d1 (getl_gen_O c1 d1 H) g d2 H0) in (let H1 \def H_x
1289 in (ex2_ind C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(clear c2 d2))
1290 (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2 d2)))
1291 (\lambda (x: C).(\lambda (H2: (wf3 g c1 x)).(\lambda (H3: (clear x
1292 d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2
1293 d2)) x H2 (getl_intro O x d2 x (drop_refl x) H3))))) H1))))))))) (\lambda (n:
1294 nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).((getl n c1 d1) \to
1295 (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2:
1296 C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2 d2))))))))))).(\lambda (c1:
1297 C).(C_ind (\lambda (c: C).(\forall (d1: C).((getl (S n) c d1) \to (\forall
1298 (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c
1299 c2)) (\lambda (c2: C).(getl (S n) c2 d2))))))))) (\lambda (n0: nat).(\lambda
1300 (d1: C).(\lambda (H0: (getl (S n) (CSort n0) d1)).(\lambda (g: G).(\lambda
1301 (d2: C).(\lambda (_: (wf3 g d1 d2)).(getl_gen_sort n0 (S n) d1 H0 (ex2 C
1302 (\lambda (c2: C).(wf3 g (CSort n0) c2)) (\lambda (c2: C).(getl (S n) c2
1303 d2)))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).((getl (S n) c
1304 d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda
1305 (c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)))))))))).(\lambda
1306 (k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (H1: (getl (S n) (CHead c k
1307 t) d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g d1 d2)).(K_ind
1308 (\lambda (k0: K).((getl (r k0 n) c d1) \to (ex2 C (\lambda (c2: C).(wf3 g
1309 (CHead c k0 t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))))) (\lambda (b:
1310 B).(\lambda (H3: (getl (r (Bind b) n) c d1)).(let H_x \def (H c d1 H3 g d2
1311 H2) in (let H4 \def H_x in (ex2_ind C (\lambda (c2: C).(wf3 g c c2)) (\lambda
1312 (c2: C).(getl n c2 d2)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t)
1313 c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3
1314 g c x)).(\lambda (H6: (getl n x d2)).(let H_x0 \def (ty3_inference g c t) in
1315 (let H7 \def H_x0 in (or_ind (ex T (\lambda (t2: T).(ty3 g c t t2))) (\forall
1316 (t2: T).((ty3 g c t t2) \to False)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c
1317 (Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (H8: (ex T
1318 (\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3 g c t t2))
1319 (ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda (c2:
1320 C).(getl (S n) c2 d2))) (\lambda (x0: T).(\lambda (H9: (ty3 g c t
1321 x0)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda
1322 (c2: C).(getl (S n) c2 d2)) (CHead x (Bind b) t) (wf3_bind g c x H5 t x0 H9
1323 b) (getl_head (Bind b) n x d2 H6 t)))) H8)) (\lambda (H8: ((\forall (t2:
1324 T).((ty3 g c t t2) \to False)))).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead
1325 c (Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (CHead x (Bind Void)
1326 (TSort O)) (wf3_void g c x H5 t H8 b) (getl_head (Bind Void) n x d2 H6 (TSort
1327 O)))) H7)))))) H4))))) (\lambda (f: F).(\lambda (H3: (getl (r (Flat f) n) c
1328 d1)).(let H_x \def (H0 d1 H3 g d2 H2) in (let H4 \def H_x in (ex2_ind C
1329 (\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (ex2 C
1330 (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) (\lambda (c2: C).(getl (S
1331 n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3 g c x)).(\lambda (H6: (getl (S
1332 n) x d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2))
1333 (\lambda (c2: C).(getl (S n) c2 d2)) x (wf3_flat g c x H5 t f) H6)))) H4)))))
1334 k (getl_gen_S k c d1 t n H1))))))))))) c1)))) i).
1337 \forall (g: G).(\forall (c: C).((wf3 g c c) \to (\forall (t1: T).(\forall
1338 (t2: T).((ty3 g c t1 t2) \to (ty3 g (CSort (cbk c)) (app1 c t1) (app1 c
1341 \lambda (g: G).(\lambda (c: C).(\lambda (H: (wf3 g c c)).(insert_eq C c
1342 (\lambda (c0: C).(wf3 g c0 c)) (\lambda (c0: C).(\forall (t1: T).(\forall
1343 (t2: T).((ty3 g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0
1344 t2)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y c)).(wf3_ind g (\lambda (c0:
1345 C).(\lambda (c1: C).((eq C c0 c1) \to (\forall (t1: T).(\forall (t2: T).((ty3
1346 g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0 t2))))))))
1347 (\lambda (m: nat).(\lambda (_: (eq C (CSort m) (CSort m))).(\lambda (t1:
1348 T).(\lambda (t2: T).(\lambda (H2: (ty3 g (CSort m) t1 t2)).H2))))) (\lambda
1349 (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C
1350 c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to (ty3 g
1351 (CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u: T).(\lambda
1352 (t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C
1353 (CHead c1 (Bind b) u) (CHead c2 (Bind b) u))).(\lambda (t1: T).(\lambda (t2:
1354 T).(\lambda (H5: (ty3 g (CHead c1 (Bind b) u) t1 t2)).(let H6 \def (f_equal C
1355 C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
1356 \Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind b) u)
1357 (CHead c2 (Bind b) u) H4) in (let H7 \def (eq_ind_r C c2 (\lambda (c0:
1358 C).((eq C c1 c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to
1359 (ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 c1 H6) in (let H8
1360 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c1 H6) in (ex_ind T
1361 (\lambda (t0: T).(ty3 g (CHead c1 (Bind b) u) t2 t0)) (ty3 g (CSort (cbk c1))
1362 (app1 c1 (THead (Bind b) u t1)) (app1 c1 (THead (Bind b) u t2))) (\lambda (x:
1363 T).(\lambda (_: (ty3 g (CHead c1 (Bind b) u) t2 x)).(H7 (refl_equal C c1)
1364 (THead (Bind b) u t1) (THead (Bind b) u t2) (ty3_bind g c1 u t H3 b t1 t2
1365 H5)))) (ty3_correct g (CHead c1 (Bind b) u) t1 t2 H5)))))))))))))))))
1366 (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2:
1367 (((eq C c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to
1368 (ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u:
1369 T).(\lambda (H3: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b:
1370 B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort
1371 O)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H5: (ty3 g (CHead c1 (Bind
1372 b) u) t1 t2)).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return
1373 (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _)
1374 \Rightarrow c0])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4)
1375 in ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda
1376 (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k
1377 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
1378 \Rightarrow b])])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4)
1379 in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
1380 (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
1381 (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) in (\lambda (H9:
1382 (eq B b Void)).(\lambda (H10: (eq C c1 c2)).(let H11 \def (eq_ind B b
1383 (\lambda (b0: B).(ty3 g (CHead c1 (Bind b0) u) t1 t2)) H5 Void H9) in
1384 (eq_ind_r B Void (\lambda (b0: B).(ty3 g (CSort (cbk (CHead c1 (Bind b0) u)))
1385 (app1 (CHead c1 (Bind b0) u) t1) (app1 (CHead c1 (Bind b0) u) t2))) (let H12
1386 \def (eq_ind T u (\lambda (t: T).(ty3 g (CHead c1 (Bind Void) t) t1 t2)) H11
1387 (TSort O) H8) in (let H13 \def (eq_ind T u (\lambda (t: T).(\forall (t0:
1388 T).((ty3 g c1 t t0) \to False))) H3 (TSort O) H8) in (eq_ind_r T (TSort O)
1389 (\lambda (t: T).(ty3 g (CSort (cbk (CHead c1 (Bind Void) t))) (app1 (CHead c1
1390 (Bind Void) t) t1) (app1 (CHead c1 (Bind Void) t) t2))) (let H14 \def
1391 (eq_ind_r C c2 (\lambda (c0: C).((eq C c1 c0) \to (\forall (t3: T).(\forall
1392 (t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1
1393 t4))))))) H2 c1 H10) in (let H15 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g
1394 c1 c0)) H1 c1 H10) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c1 (Bind Void)
1395 (TSort O)) t2 t)) (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Bind Void) (TSort
1396 O) t1)) (app1 c1 (THead (Bind Void) (TSort O) t2))) (\lambda (x: T).(\lambda
1397 (_: (ty3 g (CHead c1 (Bind Void) (TSort O)) t2 x)).(H14 (refl_equal C c1)
1398 (THead (Bind Void) (TSort O) t1) (THead (Bind Void) (TSort O) t2) (ty3_bind g
1399 c1 (TSort O) (TSort (next g O)) (ty3_sort g c1 O) Void t1 t2 H12))))
1400 (ty3_correct g (CHead c1 (Bind Void) (TSort O)) t1 t2 H12)))) u H8))) b
1401 H9))))) H7)) H6))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1:
1402 (wf3 g c1 c2)).(\lambda (H2: (((eq C c1 c2) \to (\forall (t1: T).(\forall
1403 (t2: T).((ty3 g c1 t1 t2) \to (ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1
1404 t2)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1
1405 (Flat f) u) c2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead
1406 c1 (Flat f) u) t1 t2)).(let H5 \def (f_equal C C (\lambda (e: C).e) (CHead c1
1407 (Flat f) u) c2 H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c0: C).((eq C c1
1408 c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort
1409 (cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 (CHead c1 (Flat f) u) H5) in
1410 (let H7 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 (CHead c1
1411 (Flat f) u) H5) in (let H_x \def (wf3_gen_head2 g c1 c1 u (Flat f) H7) in
1412 (let H8 \def H_x in (ex_ind B (\lambda (b: B).(eq K (Flat f) (Bind b))) (ty3
1413 g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u t1)) (app1 c1 (THead (Flat f) u
1414 t2))) (\lambda (x: B).(\lambda (H9: (eq K (Flat f) (Bind x))).(let H10 \def
1415 (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return (\lambda (_:
1416 K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) I
1417 (Bind x) H9) in (False_ind (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u
1418 t1)) (app1 c1 (THead (Flat f) u t2))) H10)))) H8)))))))))))))))) y c H0)))
1421 theorem wf3_pr2_conf:
1422 \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1
1423 t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1
1424 u) \to (pr2 c2 t1 t2)))))))))
1426 \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
1427 (H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0:
1428 T).(\forall (c2: C).((wf3 g c c2) \to (\forall (u: T).((ty3 g c t u) \to (pr2
1429 c2 t t0)))))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda
1430 (H0: (pr0 t3 t4)).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(\lambda (u:
1431 T).(\lambda (_: (ty3 g c t3 u)).(pr2_free c2 t3 t4 H0))))))))) (\lambda (c:
1432 C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
1433 (CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
1434 (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2:
1435 C).(\lambda (H3: (wf3 g c c2)).(\lambda (u0: T).(\lambda (H4: (ty3 g c t3
1436 u0)).(let H_y \def (ty3_sred_pr0 t3 t4 H1 g c u0 H4) in (let H_x \def
1437 (ty3_getl_subst0 g c t4 u0 H_y u t i H2 Abbr d u H0) in (let H5 \def H_x in
1438 (ex_ind T (\lambda (w: T).(ty3 g d u w)) (pr2 c2 t3 t) (\lambda (x:
1439 T).(\lambda (H6: (ty3 g d u x)).(let H_x0 \def (wf3_getl_conf Abbr i c d u H0
1440 g c2 H3 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl i c2
1441 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(wf3 g d d2)) (pr2 c2 t3 t)
1442 (\lambda (x0: C).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(\lambda
1443 (_: (wf3 g d x0)).(pr2_delta c2 x0 u i H8 t3 t4 H1 t H2)))) H7)))))
1444 H5)))))))))))))))))) c1 t1 t2 H))))).
1446 theorem wf3_pr3_conf:
1447 \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr3 c1
1448 t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1
1449 u) \to (pr3 c2 t1 t2)))))))))
1451 \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
1452 (H: (pr3 c1 t1 t2)).(pr3_ind c1 (\lambda (t: T).(\lambda (t0: T).(\forall
1453 (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t u) \to (pr3 c2 t
1454 t0))))))) (\lambda (t: T).(\lambda (c2: C).(\lambda (_: (wf3 g c1
1455 c2)).(\lambda (u: T).(\lambda (_: (ty3 g c1 t u)).(pr3_refl c2 t))))))
1456 (\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr2 c1 t4 t3)).(\lambda (t5:
1457 T).(\lambda (_: (pr3 c1 t3 t5)).(\lambda (H2: ((\forall (c2: C).((wf3 g c1
1458 c2) \to (\forall (u: T).((ty3 g c1 t3 u) \to (pr3 c2 t3 t5))))))).(\lambda
1459 (c2: C).(\lambda (H3: (wf3 g c1 c2)).(\lambda (u: T).(\lambda (H4: (ty3 g c1
1460 t4 u)).(pr3_sing c2 t3 t4 (wf3_pr2_conf g c1 t4 t3 H0 c2 H3 u H4) t5 (H2 c2
1461 H3 u (ty3_sred_pr2 c1 t4 t3 H0 g u H4))))))))))))) t1 t2 H))))).
1463 theorem wf3_pc3_conf:
1464 \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1
1465 t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u1: T).((ty3 g c1 t1
1466 u1) \to (\forall (u2: T).((ty3 g c1 t2 u2) \to (pc3 c2 t1 t2)))))))))))
1468 \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
1469 (H: (pc3 c1 t1 t2)).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda
1470 (u1: T).(\lambda (H1: (ty3 g c1 t1 u1)).(\lambda (u2: T).(\lambda (H2: (ty3 g
1471 c1 t2 u2)).(let H3 \def H in (ex2_ind T (\lambda (t: T).(pr3 c1 t1 t))
1472 (\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1 t2) (\lambda (x: T).(\lambda (H4:
1473 (pr3 c1 t1 x)).(\lambda (H5: (pr3 c1 t2 x)).(pc3_pr3_t c2 t1 x (wf3_pr3_conf
1474 g c1 t1 x H4 c2 H0 u1 H1) t2 (wf3_pr3_conf g c1 t2 x H5 c2 H0 u2 H2)))))
1477 theorem wf3_ty3_conf:
1478 \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1
1479 t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (ty3 g c2 t1 t2)))))))
1481 \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
1482 (H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
1483 (t0: T).(\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t t0)))))) (\lambda (c:
1484 C).(\lambda (t3: T).(\lambda (t: T).(\lambda (H0: (ty3 g c t3 t)).(\lambda
1485 (H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t))))).(\lambda (u:
1486 T).(\lambda (t4: T).(\lambda (H2: (ty3 g c u t4)).(\lambda (H3: ((\forall
1487 (c2: C).((wf3 g c c2) \to (ty3 g c2 u t4))))).(\lambda (H4: (pc3 c t4
1488 t3)).(\lambda (c2: C).(\lambda (H5: (wf3 g c c2)).(ex_ind T (\lambda (t0:
1489 T).(ty3 g c t4 t0)) (ty3 g c2 u t3) (\lambda (x: T).(\lambda (H6: (ty3 g c t4
1490 x)).(ty3_conv g c2 t3 t (H1 c2 H5) u t4 (H3 c2 H5) (wf3_pc3_conf g c t4 t3 H4
1491 c2 H5 x H6 t H0)))) (ty3_correct g c u t4 H2)))))))))))))) (\lambda (c:
1492 C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(ty3_sort g
1493 c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u:
1494 T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda
1495 (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g
1496 c2 u t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def
1497 (wf3_getl_conf Abbr n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind
1498 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
1499 C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x:
1500 C).(\lambda (H5: (getl n c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (wf3 g d
1501 x)).(ty3_abbr g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (n:
1502 nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c
1503 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u
1504 t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g c2 u
1505 t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def
1506 (wf3_getl_conf Abst n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind
1507 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
1508 C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x:
1509 C).(\lambda (H5: (getl n c2 (CHead x (Bind Abst) u))).(\lambda (H6: (wf3 g d
1510 x)).(ty3_abst g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (c:
1511 C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c u t)).(\lambda (H1:
1512 ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 u t))))).(\lambda (b:
1513 B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u)
1514 t3 t4)).(\lambda (H3: ((\forall (c2: C).((wf3 g (CHead c (Bind b) u) c2) \to
1515 (ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c c2)).(ty3_bind g
1516 c2 u t (H1 c2 H4) b t3 t4 (H3 (CHead c2 (Bind b) u) (wf3_bind g c c2 H4 u t
1517 H0 b))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda
1518 (_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g
1519 c2 w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead
1520 (Bind Abst) u t))).(\lambda (H3: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g
1521 c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c
1522 c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c:
1523 C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda
1524 (H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t4))))).(\lambda (t0:
1525 T).(\lambda (_: (ty3 g c t4 t0)).(\lambda (H3: ((\forall (c2: C).((wf3 g c
1526 c2) \to (ty3 g c2 t4 t0))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c
1527 c2)).(ty3_cast g c2 t3 t4 (H1 c2 H4) t0 (H3 c2 H4)))))))))))) c1 t1 t2 H))))).
1530 \forall (g: G).(\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (wf3 g
1533 \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wf3 g c1
1534 c2)).(wf3_ind g (\lambda (_: C).(\lambda (c0: C).(wf3 g c0 c0))) (\lambda (m:
1535 nat).(wf3_sort g m)) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wf3 g
1536 c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (t: T).(\lambda
1537 (H2: (ty3 g c3 u t)).(\lambda (b: B).(wf3_bind g c4 c4 H1 u t (wf3_ty3_conf g
1538 c3 u t H2 c4 H0) b))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_:
1539 (wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (_:
1540 ((\forall (t: T).((ty3 g c3 u t) \to False)))).(\lambda (_: B).(wf3_bind g c4
1541 c4 H1 (TSort O) (TSort (next g O)) (ty3_sort g c4 O) Void)))))))) (\lambda
1542 (c3: C).(\lambda (c4: C).(\lambda (_: (wf3 g c3 c4)).(\lambda (H1: (wf3 g c4
1543 c4)).(\lambda (_: T).(\lambda (_: F).H1)))))) c1 c2 H)))).
1546 \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (u: T).((ty3 g c1 t
1547 u) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t
1550 \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
1551 (ty3 g c1 t u)).(let H_x \def (wf3_total g c1) in (let H0 \def H_x in (ex_ind
1552 C (\lambda (c2: C).(wf3 g c1 c2)) (ex2 C (\lambda (c2: C).(wf3 g c1 c2))
1553 (\lambda (c2: C).(ty3 g c2 t u))) (\lambda (x: C).(\lambda (H1: (wf3 g c1
1554 x)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t
1555 u)) x H1 (wf3_ty3_conf g c1 t u H x H1)))) H0))))))).
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