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 include "LambdaDelta-1/sn3/nf2.ma".
19 include "LambdaDelta-1/sn3/fwd.ma".
21 include "LambdaDelta-1/nf2/iso.ma".
23 include "LambdaDelta-1/pr3/iso.ma".
25 theorem sn3_pr3_trans:
26 \forall (c: C).(\forall (t1: T).((sn3 c t1) \to (\forall (t2: T).((pr3 c t1
27 t2) \to (sn3 c t2)))))
29 \lambda (c: C).(\lambda (t1: T).(\lambda (H: (sn3 c t1)).(sn3_ind c (\lambda
30 (t: T).(\forall (t2: T).((pr3 c t t2) \to (sn3 c t2)))) (\lambda (t2:
31 T).(\lambda (H0: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
32 Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H1: ((\forall
33 (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to
34 (\forall (t4: T).((pr3 c t3 t4) \to (sn3 c t4)))))))).(\lambda (t3:
35 T).(\lambda (H2: (pr3 c t2 t3)).(sn3_sing c t3 (\lambda (t0: T).(\lambda (H3:
36 (((eq T t3 t0) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 c t3 t0)).(let
37 H_x \def (term_dec t2 t3) in (let H5 \def H_x in (or_ind (eq T t2 t3) ((eq T
38 t2 t3) \to (\forall (P: Prop).P)) (sn3 c t0) (\lambda (H6: (eq T t2 t3)).(let
39 H7 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t t0)) H4 t2 H6) in (let H8
40 \def (eq_ind_r T t3 (\lambda (t: T).((eq T t t0) \to (\forall (P: Prop).P)))
41 H3 t2 H6) in (let H9 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t2 t)) H2 t2
42 H6) in (H0 t0 H8 H7))))) (\lambda (H6: (((eq T t2 t3) \to (\forall (P:
43 Prop).P)))).(H1 t3 H6 H2 t0 H4)) H5)))))))))))) t1 H))).
45 theorem sn3_pr2_intro:
46 \forall (c: C).(\forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to
47 (\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)))
49 \lambda (c: C).(\lambda (t1: T).(\lambda (H: ((\forall (t2: T).((((eq T t1
50 t2) \to (\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c
51 t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H0: (((eq T t1 t2) \to
52 (\forall (P: Prop).P)))).(\lambda (H1: (pr3 c t1 t2)).(let H2 \def H0 in
53 ((let H3 \def H in (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(((\forall
54 (t3: T).((((eq T t t3) \to (\forall (P: Prop).P))) \to ((pr2 c t t3) \to (sn3
55 c t3))))) \to ((((eq T t t0) \to (\forall (P: Prop).P))) \to (sn3 c t0)))))
56 (\lambda (t: T).(\lambda (H4: ((\forall (t3: T).((((eq T t t3) \to (\forall
57 (P: Prop).P))) \to ((pr2 c t t3) \to (sn3 c t3)))))).(\lambda (H5: (((eq T t
58 t) \to (\forall (P: Prop).P)))).(H4 t H5 (pr2_free c t t (pr0_refl t))))))
59 (\lambda (t3: T).(\lambda (t4: T).(\lambda (H4: (pr2 c t4 t3)).(\lambda (t5:
60 T).(\lambda (H5: (pr3 c t3 t5)).(\lambda (H6: ((((\forall (t6: T).((((eq T t3
61 t6) \to (\forall (P: Prop).P))) \to ((pr2 c t3 t6) \to (sn3 c t6))))) \to
62 ((((eq T t3 t5) \to (\forall (P: Prop).P))) \to (sn3 c t5))))).(\lambda (H7:
63 ((\forall (t6: T).((((eq T t4 t6) \to (\forall (P: Prop).P))) \to ((pr2 c t4
64 t6) \to (sn3 c t6)))))).(\lambda (H8: (((eq T t4 t5) \to (\forall (P:
65 Prop).P)))).(let H_x \def (term_dec t4 t3) in (let H9 \def H_x in (or_ind (eq
66 T t4 t3) ((eq T t4 t3) \to (\forall (P: Prop).P)) (sn3 c t5) (\lambda (H10:
67 (eq T t4 t3)).(let H11 \def (eq_ind T t4 (\lambda (t: T).((eq T t t5) \to
68 (\forall (P: Prop).P))) H8 t3 H10) in (let H12 \def (eq_ind T t4 (\lambda (t:
69 T).(\forall (t6: T).((((eq T t t6) \to (\forall (P: Prop).P))) \to ((pr2 c t
70 t6) \to (sn3 c t6))))) H7 t3 H10) in (let H13 \def (eq_ind T t4 (\lambda (t:
71 T).(pr2 c t t3)) H4 t3 H10) in (H6 H12 H11))))) (\lambda (H10: (((eq T t4 t3)
72 \to (\forall (P: Prop).P)))).(sn3_pr3_trans c t3 (H7 t3 H10 H4) t5 H5))
73 H9))))))))))) t1 t2 H1 H3)) H2)))))))).
76 \forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: T).((sn3 c t) \to
77 (sn3 c (THead (Flat Cast) u t))))))
79 \lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c u)).(sn3_ind c (\lambda
80 (t: T).(\forall (t0: T).((sn3 c t0) \to (sn3 c (THead (Flat Cast) t t0)))))
81 (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
82 (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall
83 (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
84 (\forall (t: T).((sn3 c t) \to (sn3 c (THead (Flat Cast) t2
85 t))))))))).(\lambda (t: T).(\lambda (H2: (sn3 c t)).(sn3_ind c (\lambda (t0:
86 T).(sn3 c (THead (Flat Cast) t1 t0))) (\lambda (t0: T).(\lambda (H3:
87 ((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0
88 t2) \to (sn3 c t2)))))).(\lambda (H4: ((\forall (t2: T).((((eq T t0 t2) \to
89 (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c (THead (Flat Cast) t1
90 t2))))))).(sn3_pr2_intro c (THead (Flat Cast) t1 t0) (\lambda (t2:
91 T).(\lambda (H5: (((eq T (THead (Flat Cast) t1 t0) t2) \to (\forall (P:
92 Prop).P)))).(\lambda (H6: (pr2 c (THead (Flat Cast) t1 t0) t2)).(let H7 \def
93 (pr2_gen_cast c t1 t0 t2 H6) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda
94 (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
95 T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3)))) (pr2 c
96 t0 t2) (sn3 c t2) (\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
97 T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
98 T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0
99 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
100 (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2)))
101 (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3))) (sn3 c t2) (\lambda (x0:
102 T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead (Flat Cast) x0
103 x1))).(\lambda (H10: (pr2 c t1 x0)).(\lambda (H11: (pr2 c t0 x1)).(let H12
104 \def (eq_ind T t2 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) t3) \to
105 (\forall (P: Prop).P))) H5 (THead (Flat Cast) x0 x1) H9) in (eq_ind_r T
106 (THead (Flat Cast) x0 x1) (\lambda (t3: T).(sn3 c t3)) (let H_x \def
107 (term_dec x0 t1) in (let H13 \def H_x in (or_ind (eq T x0 t1) ((eq T x0 t1)
108 \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) x0 x1)) (\lambda (H14:
109 (eq T x0 t1)).(let H15 \def (eq_ind T x0 (\lambda (t3: T).((eq T (THead (Flat
110 Cast) t1 t0) (THead (Flat Cast) t3 x1)) \to (\forall (P: Prop).P))) H12 t1
111 H14) in (let H16 \def (eq_ind T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1
112 H14) in (eq_ind_r T t1 (\lambda (t3: T).(sn3 c (THead (Flat Cast) t3 x1)))
113 (let H_x0 \def (term_dec t0 x1) in (let H17 \def H_x0 in (or_ind (eq T t0 x1)
114 ((eq T t0 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) t1 x1))
115 (\lambda (H18: (eq T t0 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t3:
116 T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat Cast) t1 t3)) \to (\forall
117 (P: Prop).P))) H15 t0 H18) in (let H20 \def (eq_ind_r T x1 (\lambda (t3:
118 T).(pr2 c t0 t3)) H11 t0 H18) in (eq_ind T t0 (\lambda (t3: T).(sn3 c (THead
119 (Flat Cast) t1 t3))) (H19 (refl_equal T (THead (Flat Cast) t1 t0)) (sn3 c
120 (THead (Flat Cast) t1 t0))) x1 H18)))) (\lambda (H18: (((eq T t0 x1) \to
121 (\forall (P: Prop).P)))).(H4 x1 H18 (pr3_pr2 c t0 x1 H11))) H17))) x0 H14))))
122 (\lambda (H14: (((eq T x0 t1) \to (\forall (P: Prop).P)))).(H1 x0 (\lambda
123 (H15: (eq T t1 x0)).(\lambda (P: Prop).(let H16 \def (eq_ind_r T x0 (\lambda
124 (t3: T).((eq T t3 t1) \to (\forall (P0: Prop).P0))) H14 t1 H15) in (let H17
125 \def (eq_ind_r T x0 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead
126 (Flat Cast) t3 x1)) \to (\forall (P0: Prop).P0))) H12 t1 H15) in (let H18
127 \def (eq_ind_r T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1 H15) in (H16
128 (refl_equal T t1) P)))))) (pr3_pr2 c t1 x0 H10) x1 (let H_x0 \def (term_dec
129 t0 x1) in (let H15 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to
130 (\forall (P: Prop).P)) (sn3 c x1) (\lambda (H16: (eq T t0 x1)).(let H17 \def
131 (eq_ind_r T x1 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat
132 Cast) x0 t3)) \to (\forall (P: Prop).P))) H12 t0 H16) in (let H18 \def
133 (eq_ind_r T x1 (\lambda (t3: T).(pr2 c t0 t3)) H11 t0 H16) in (eq_ind T t0
134 (\lambda (t3: T).(sn3 c t3)) (sn3_sing c t0 H3) x1 H16)))) (\lambda (H16:
135 (((eq T t0 x1) \to (\forall (P: Prop).P)))).(H3 x1 H16 (pr3_pr2 c t0 x1
136 H11))) H15))))) H13))) t2 H9))))))) H8)) (\lambda (H8: (pr2 c t0
137 t2)).(sn3_pr3_trans c t0 (sn3_sing c t0 H3) t2 (pr3_pr2 c t0 t2 H8)))
138 H7))))))))) t H2)))))) u H))).
141 \forall (c: C).(\forall (t: T).((sn3 c t) \to (\forall (f: F).(\forall (u:
142 T).(sn3 (CHead c (Flat f) u) t)))))
144 \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(\lambda (f:
145 F).(\lambda (u: T).(sn3_ind c (\lambda (t0: T).(sn3 (CHead c (Flat f) u) t0))
146 (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
147 (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall
148 (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
149 (sn3 (CHead c (Flat f) u) t2)))))).(sn3_pr2_intro (CHead c (Flat f) u) t1
150 (\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) \to (\forall (P:
151 Prop).P)))).(\lambda (H3: (pr2 (CHead c (Flat f) u) t1 t2)).(H1 t2 H2
152 (pr3_pr2 c t1 t2 (pr2_gen_cflat f c u t1 t2 H3)))))))))) t H))))).
155 \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c
156 (THead (Bind b) v t)) \to (sn3 (CHead c (Bind b) v) t)))))
158 \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
159 (sn3 c (THead (Bind b) v t))).(let H_x \def (sn3_gen_bind b c v t H) in (let
160 H0 \def H_x in (land_ind (sn3 c v) (sn3 (CHead c (Bind b) v) t) (sn3 (CHead c
161 (Bind b) v) t) (\lambda (_: (sn3 c v)).(\lambda (H2: (sn3 (CHead c (Bind b)
162 v) t)).H2)) H0))))))).
165 \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1:
166 T).(\forall (t: T).((sn3 (CHead c (Bind b) v1) t) \to (\forall (v2: T).(sn3
167 (CHead c (Bind b) v2) t)))))))
169 \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda
170 (v1: T).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c (Bind b) v1) t)).(\lambda
171 (v2: T).(sn3_ind (CHead c (Bind b) v1) (\lambda (t0: T).(sn3 (CHead c (Bind
172 b) v2) t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2)
173 \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to (sn3
174 (CHead c (Bind b) v1) t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1
175 t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to
176 (sn3 (CHead c (Bind b) v2) t2)))))).(sn3_pr2_intro (CHead c (Bind b) v2) t1
177 (\lambda (t2: T).(\lambda (H3: (((eq T t1 t2) \to (\forall (P:
178 Prop).P)))).(\lambda (H4: (pr2 (CHead c (Bind b) v2) t1 t2)).(H2 t2 H3
179 (pr3_pr2 (CHead c (Bind b) v1) t1 t2 (pr2_change b H c v2 t1 t2 H4
180 v1)))))))))) t H0))))))).
183 \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
184 (CHead d (Bind Abbr) v)) \to ((sn3 c (TLRef i)) \to (sn3 d v))))))
186 \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
187 (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 c (TLRef
188 i))).(sn3_gen_lift c v (S i) O (sn3_pr3_trans c (TLRef i) H0 (lift (S i) O v)
189 (pr3_pr2 c (TLRef i) (lift (S i) O v) (pr2_delta c d v i H (TLRef i) (TLRef
190 i) (pr0_refl (TLRef i)) (lift (S i) O v) (subst0_lref v i)))) d (getl_drop
191 Abbr c d v i H))))))).
194 \forall (v: T).(\forall (t: T).(\forall (i: nat).(((\forall (w: T).(ex T
195 (\lambda (u: T).(subst0 i w t u))))) \to (\forall (c: C).(\forall (d:
196 C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to (sn3 d v))))))))
198 \lambda (v: T).(\lambda (t: T).(\lambda (i: nat).(\lambda (H: ((\forall (w:
199 T).(ex T (\lambda (u: T).(subst0 i w t u)))))).(let H_x \def (H v) in (let H0
200 \def H_x in (ex_ind T (\lambda (u: T).(subst0 i v t u)) (\forall (c:
201 C).(\forall (d: C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to
202 (sn3 d v))))) (\lambda (x: T).(\lambda (H1: (subst0 i v t x)).(subst0_ind
203 (\lambda (n: nat).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_: T).(\forall
204 (c: C).(\forall (d: C).((getl n c (CHead d (Bind Abbr) t0)) \to ((sn3 c t1)
205 \to (sn3 d t0))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (c:
206 C).(\lambda (d: C).(\lambda (H2: (getl i0 c (CHead d (Bind Abbr)
207 v0))).(\lambda (H3: (sn3 c (TLRef i0))).(sn3_gen_def c d v0 i0 H2 H3)))))))
208 (\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0:
209 nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c:
210 C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to
211 (sn3 d v0))))))).(\lambda (t0: T).(\lambda (k: K).(\lambda (c: C).(\lambda
212 (d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) v0))).(\lambda (H5: (sn3
213 c (THead k u1 t0))).(let H_y \def (sn3_gen_head k c u1 t0 H5) in (H3 c d H4
214 H_y)))))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: T).(\lambda
215 (t1: T).(\lambda (i0: nat).(\lambda (H2: (subst0 (s k i0) v0 t1 t2)).(\lambda
216 (H3: ((\forall (c: C).(\forall (d: C).((getl (s k i0) c (CHead d (Bind Abbr)
217 v0)) \to ((sn3 c t1) \to (sn3 d v0))))))).(\lambda (u: T).(\lambda (c:
218 C).(\lambda (d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr)
219 v0))).(\lambda (H5: (sn3 c (THead k u t1))).(K_ind (\lambda (k0: K).((subst0
220 (s k0 i0) v0 t1 t2) \to (((\forall (c0: C).(\forall (d0: C).((getl (s k0 i0)
221 c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0)))))) \to ((sn3
222 c (THead k0 u t1)) \to (sn3 d v0))))) (\lambda (b: B).(\lambda (_: (subst0 (s
223 (Bind b) i0) v0 t1 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0:
224 C).((getl (s (Bind b) i0) c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to
225 (sn3 d0 v0))))))).(\lambda (H8: (sn3 c (THead (Bind b) u t1))).(let H_x0 \def
226 (sn3_gen_bind b c u t1 H8) in (let H9 \def H_x0 in (land_ind (sn3 c u) (sn3
227 (CHead c (Bind b) u) t1) (sn3 d v0) (\lambda (_: (sn3 c u)).(\lambda (H11:
228 (sn3 (CHead c (Bind b) u) t1)).(H7 (CHead c (Bind b) u) d (getl_clear_bind b
229 (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d (Bind Abbr) v0) i0 H4)
230 H11))) H9))))))) (\lambda (f: F).(\lambda (_: (subst0 (s (Flat f) i0) v0 t1
231 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: C).((getl (s (Flat f) i0)
232 c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0))))))).(\lambda
233 (H8: (sn3 c (THead (Flat f) u t1))).(let H_x0 \def (sn3_gen_flat f c u t1 H8)
234 in (let H9 \def H_x0 in (land_ind (sn3 c u) (sn3 c t1) (sn3 d v0) (\lambda
235 (_: (sn3 c u)).(\lambda (H11: (sn3 c t1)).(H7 c d H4 H11))) H9))))))) k H2 H3
236 H5))))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda
237 (i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c:
238 C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to
239 (sn3 d v0))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda
240 (_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: ((\forall (c: C).(\forall (d:
241 C).((getl (s k i0) c (CHead d (Bind Abbr) v0)) \to ((sn3 c t1) \to (sn3 d
242 v0))))))).(\lambda (c: C).(\lambda (d: C).(\lambda (H6: (getl i0 c (CHead d
243 (Bind Abbr) v0))).(\lambda (H7: (sn3 c (THead k u1 t1))).(let H_y \def
244 (sn3_gen_head k c u1 t1 H7) in (H3 c d H6 H_y))))))))))))))))) i v t x H1)))
247 theorem sn3_cpr3_trans:
248 \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
249 (k: K).(\forall (t: T).((sn3 (CHead c k u1) t) \to (sn3 (CHead c k u2)
252 \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
253 u2)).(\lambda (k: K).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c k u1)
254 t)).(sn3_ind (CHead c k u1) (\lambda (t0: T).(sn3 (CHead c k u2) t0))
255 (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
256 (P: Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u1)
257 t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
258 Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u2)
259 t2)))))).(sn3_sing (CHead c k u2) t1 (\lambda (t2: T).(\lambda (H3: (((eq T
260 t1 t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 (CHead c k u2) t1
261 t2)).(H2 t2 H3 (pr3_pr3_pr3_t c u1 u2 H t1 t2 k H4))))))))) t H0))))))).
264 \forall (b: B).(\forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t:
265 T).((sn3 (CHead c (Bind b) u) t) \to (sn3 c (THead (Bind b) u t)))))))
267 \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c
268 u)).(sn3_ind c (\lambda (t: T).(\forall (t0: T).((sn3 (CHead c (Bind b) t)
269 t0) \to (sn3 c (THead (Bind b) t t0))))) (\lambda (t1: T).(\lambda (_:
270 ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1
271 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to
272 (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (t: T).((sn3 (CHead c
273 (Bind b) t2) t) \to (sn3 c (THead (Bind b) t2 t))))))))).(\lambda (t:
274 T).(\lambda (H2: (sn3 (CHead c (Bind b) t1) t)).(sn3_ind (CHead c (Bind b)
275 t1) (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (\lambda (t2:
276 T).(\lambda (H3: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
277 Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind b)
278 t1) t3)))))).(\lambda (H4: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
279 Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 c (THead (Bind b)
280 t1 t3))))))).(sn3_sing c (THead (Bind b) t1 t2) (\lambda (t3: T).(\lambda
281 (H5: (((eq T (THead (Bind b) t1 t2) t3) \to (\forall (P: Prop).P)))).(\lambda
282 (H6: (pr3 c (THead (Bind b) t1 t2) t3)).(let H_x \def (bind_dec_not b Abst)
283 in (let H7 \def H_x in (or_ind (eq B b Abst) (not (eq B b Abst)) (sn3 c t3)
284 (\lambda (H8: (eq B b Abst)).(let H9 \def (eq_ind B b (\lambda (b0: B).(pr3 c
285 (THead (Bind b0) t1 t2) t3)) H6 Abst H8) in (let H10 \def (eq_ind B b
286 (\lambda (b0: B).((eq T (THead (Bind b0) t1 t2) t3) \to (\forall (P:
287 Prop).P))) H5 Abst H8) in (let H11 \def (eq_ind B b (\lambda (b0: B).(\forall
288 (t4: T).((((eq T t2 t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind
289 b0) t1) t2 t4) \to (sn3 c (THead (Bind b0) t1 t4)))))) H4 Abst H8) in (let
290 H12 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T t2 t4) \to
291 (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) t2 t4) \to (sn3
292 (CHead c (Bind b0) t1) t4))))) H3 Abst H8) in (let H13 \def (eq_ind B b
293 (\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P)))
294 \to ((pr3 c t1 t4) \to (\forall (t0: T).((sn3 (CHead c (Bind b0) t4) t0) \to
295 (sn3 c (THead (Bind b0) t4 t0)))))))) H1 Abst H8) in (let H14 \def
296 (pr3_gen_abst c t1 t2 t3 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4:
297 T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_:
298 T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall
299 (u0: T).(pr3 (CHead c (Bind b0) u0) t2 t4))))) (sn3 c t3) (\lambda (x0:
300 T).(\lambda (x1: T).(\lambda (H15: (eq T t3 (THead (Bind Abst) x0
301 x1))).(\lambda (H16: (pr3 c t1 x0)).(\lambda (H17: ((\forall (b0: B).(\forall
302 (u0: T).(pr3 (CHead c (Bind b0) u0) t2 x1))))).(let H18 \def (eq_ind T t3
303 (\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) t0) \to (\forall (P:
304 Prop).P))) H10 (THead (Bind Abst) x0 x1) H15) in (eq_ind_r T (THead (Bind
305 Abst) x0 x1) (\lambda (t0: T).(sn3 c t0)) (let H_x0 \def (term_dec t1 x0) in
306 (let H19 \def H_x0 in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P:
307 Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H20: (eq T t1 x0)).(let
308 H21 \def (eq_ind_r T x0 (\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2)
309 (THead (Bind Abst) t0 x1)) \to (\forall (P: Prop).P))) H18 t1 H20) in (let
310 H22 \def (eq_ind_r T x0 (\lambda (t0: T).(pr3 c t1 t0)) H16 t1 H20) in
311 (eq_ind T t1 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t0 x1))) (let H_x1
312 \def (term_dec t2 x1) in (let H23 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2
313 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abst) t1 x1)) (\lambda
314 (H24: (eq T t2 x1)).(let H25 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T
315 (THead (Bind Abst) t1 t2) (THead (Bind Abst) t1 t0)) \to (\forall (P:
316 Prop).P))) H21 t2 H24) in (let H26 \def (eq_ind_r T x1 (\lambda (t0:
317 T).(\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) t2 t0))))
318 H17 t2 H24) in (eq_ind T t2 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t1
319 t0))) (H25 (refl_equal T (THead (Bind Abst) t1 t2)) (sn3 c (THead (Bind Abst)
320 t1 t2))) x1 H24)))) (\lambda (H24: (((eq T t2 x1) \to (\forall (P:
321 Prop).P)))).(H11 x1 H24 (H17 Abst t1))) H23))) x0 H20)))) (\lambda (H20:
322 (((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x1 \def (term_dec t2 x1)
323 in (let H21 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P:
324 Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H22: (eq T t2 x1)).(let
325 H23 \def (eq_ind_r T x1 (\lambda (t0: T).(\forall (b0: B).(\forall (u0:
326 T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) H17 t2 H22) in (eq_ind T t2 (\lambda
327 (t0: T).(sn3 c (THead (Bind Abst) x0 t0))) (H13 x0 H20 H16 t2 (sn3_cpr3_trans
328 c t1 x0 H16 (Bind Abst) t2 (sn3_sing (CHead c (Bind Abst) t1) t2 H12))) x1
329 H22))) (\lambda (H22: (((eq T t2 x1) \to (\forall (P: Prop).P)))).(H13 x0 H20
330 H16 x1 (sn3_cpr3_trans c t1 x0 H16 (Bind Abst) x1 (H12 x1 H22 (H17 Abst
331 t1))))) H21)))) H19))) t3 H15))))))) H14)))))))) (\lambda (H8: (not (eq B b
332 Abst))).(let H_x0 \def (pr3_gen_bind b H8 c t1 t2 t3 H6) in (let H9 \def H_x0
333 in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
334 b) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
335 T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) t2 t4)))) (pr3 (CHead c (Bind
336 b) t1) t2 (lift (S O) O t3)) (sn3 c t3) (\lambda (H10: (ex3_2 T T (\lambda
337 (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2:
338 T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3
339 (CHead c (Bind b) t1) t2 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
340 (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_:
341 T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 (CHead c (Bind b)
342 t1) t2 t4))) (sn3 c t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq
343 T t3 (THead (Bind b) x0 x1))).(\lambda (H12: (pr3 c t1 x0)).(\lambda (H13:
344 (pr3 (CHead c (Bind b) t1) t2 x1)).(let H14 \def (eq_ind T t3 (\lambda (t0:
345 T).((eq T (THead (Bind b) t1 t2) t0) \to (\forall (P: Prop).P))) H5 (THead
346 (Bind b) x0 x1) H11) in (eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0:
347 T).(sn3 c t0)) (let H_x1 \def (term_dec t1 x0) in (let H15 \def H_x1 in
348 (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: Prop).P)) (sn3 c (THead
349 (Bind b) x0 x1)) (\lambda (H16: (eq T t1 x0)).(let H17 \def (eq_ind_r T x0
350 (\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t0 x1)) \to
351 (\forall (P: Prop).P))) H14 t1 H16) in (let H18 \def (eq_ind_r T x0 (\lambda
352 (t0: T).(pr3 c t1 t0)) H12 t1 H16) in (eq_ind T t1 (\lambda (t0: T).(sn3 c
353 (THead (Bind b) t0 x1))) (let H_x2 \def (term_dec t2 x1) in (let H19 \def
354 H_x2 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: Prop).P)) (sn3 c
355 (THead (Bind b) t1 x1)) (\lambda (H20: (eq T t2 x1)).(let H21 \def (eq_ind_r
356 T x1 (\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t1 t0))
357 \to (\forall (P: Prop).P))) H17 t2 H20) in (let H22 \def (eq_ind_r T x1
358 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) t2 t0)) H13 t2 H20) in (eq_ind T
359 t2 (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (H21 (refl_equal T (THead
360 (Bind b) t1 t2)) (sn3 c (THead (Bind b) t1 t2))) x1 H20)))) (\lambda (H20:
361 (((eq T t2 x1) \to (\forall (P: Prop).P)))).(H4 x1 H20 H13)) H19))) x0
362 H16)))) (\lambda (H16: (((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x2
363 \def (term_dec t2 x1) in (let H17 \def H_x2 in (or_ind (eq T t2 x1) ((eq T t2
364 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind b) x0 x1)) (\lambda (H18:
365 (eq T t2 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t0: T).(pr3 (CHead c
366 (Bind b) t1) t2 t0)) H13 t2 H18) in (eq_ind T t2 (\lambda (t0: T).(sn3 c
367 (THead (Bind b) x0 t0))) (H1 x0 H16 H12 t2 (sn3_cpr3_trans c t1 x0 H12 (Bind
368 b) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3))) x1 H18))) (\lambda (H18: (((eq
369 T t2 x1) \to (\forall (P: Prop).P)))).(H1 x0 H16 H12 x1 (sn3_cpr3_trans c t1
370 x0 H12 (Bind b) x1 (H3 x1 H18 H13)))) H17)))) H15))) t3 H11))))))) H10))
371 (\lambda (H10: (pr3 (CHead c (Bind b) t1) t2 (lift (S O) O
372 t3))).(sn3_gen_lift (CHead c (Bind b) t1) t3 (S O) O (sn3_pr3_trans (CHead c
373 (Bind b) t1) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3) (lift (S O) O t3) H10)
374 c (drop_drop (Bind b) O c c (drop_refl c) t1))) H9)))) H7)))))))))) t
378 \forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c (THead (Bind Abbr) v
379 t)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead
380 (Bind Abst) w t))))))))
382 \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
383 (Bind Abbr) v t))).(insert_eq T (THead (Bind Abbr) v t) (\lambda (t0: T).(sn3
384 c t0)) (\lambda (_: T).(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat
385 Appl) v (THead (Bind Abst) w t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c
386 y)).(unintro T t (\lambda (t0: T).((eq T y (THead (Bind Abbr) v t0)) \to
387 (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead (Bind Abst)
388 w t0))))))) (unintro T v (\lambda (t0: T).(\forall (x: T).((eq T y (THead
389 (Bind Abbr) t0 x)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat
390 Appl) t0 (THead (Bind Abst) w x)))))))) (sn3_ind c (\lambda (t0: T).(\forall
391 (x: T).(\forall (x0: T).((eq T t0 (THead (Bind Abbr) x x0)) \to (\forall (w:
392 T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) w
393 x0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2)
394 \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda
395 (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3
396 c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Bind Abbr) x
397 x0)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead
398 (Bind Abst) w x0))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3:
399 (eq T t1 (THead (Bind Abbr) x x0))).(\lambda (w: T).(\lambda (H4: (sn3 c
400 w)).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0
401 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (\forall (x1:
402 T).(\forall (x2: T).((eq T t2 (THead (Bind Abbr) x1 x2)) \to (\forall (w0:
403 T).((sn3 c w0) \to (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) w0
404 x2)))))))))))) H2 (THead (Bind Abbr) x x0) H3) in (let H6 \def (eq_ind T t1
405 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P)))
406 \to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead (Bind Abbr) x x0) H3) in
407 (sn3_ind c (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t0
408 x0)))) (\lambda (t2: T).(\lambda (H7: ((\forall (t3: T).((((eq T t2 t3) \to
409 (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H8:
410 ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2
411 t3) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t3
412 x0)))))))).(sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind Abst) t2 x0))
413 (\lambda (t3: T).(\lambda (H9: (((eq T (THead (Flat Appl) x (THead (Bind
414 Abst) t2 x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H10: (pr2 c (THead
415 (Flat Appl) x (THead (Bind Abst) t2 x0)) t3)).(let H11 \def (pr2_gen_appl c x
416 (THead (Bind Abst) t2 x0) t3 H10) in (or3_ind (ex3_2 T T (\lambda (u2:
417 T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
418 T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
419 (THead (Bind Abst) t2 x0) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
420 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0)
421 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
422 T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
423 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
424 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
425 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T
426 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
427 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
428 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
429 (THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
430 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
431 (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
432 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
433 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_:
434 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
435 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
436 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
437 y2) z1 z2)))))))) (sn3 c t3) (\lambda (H12: (ex3_2 T T (\lambda (u2:
438 T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
439 T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
440 (THead (Bind Abst) t2 x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
441 (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_:
442 T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst)
443 t2 x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H13: (eq
444 T t3 (THead (Flat Appl) x1 x2))).(\lambda (H14: (pr2 c x x1)).(\lambda (H15:
445 (pr2 c (THead (Bind Abst) t2 x0) x2)).(let H16 \def (eq_ind T t3 (\lambda
446 (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to
447 (\forall (P: Prop).P))) H9 (THead (Flat Appl) x1 x2) H13) in (eq_ind_r T
448 (THead (Flat Appl) x1 x2) (\lambda (t0: T).(sn3 c t0)) (let H17 \def
449 (pr2_gen_abst c t2 x0 x2 H15) in (ex3_2_ind T T (\lambda (u2: T).(\lambda
450 (t4: T).(eq T x2 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_:
451 T).(pr2 c t2 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
452 (u: T).(pr2 (CHead c (Bind b) u) x0 t4))))) (sn3 c (THead (Flat Appl) x1 x2))
453 (\lambda (x3: T).(\lambda (x4: T).(\lambda (H18: (eq T x2 (THead (Bind Abst)
454 x3 x4))).(\lambda (H19: (pr2 c t2 x3)).(\lambda (H20: ((\forall (b:
455 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 x4))))).(let H21 \def (eq_ind
456 T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0))
457 (THead (Flat Appl) x1 t0)) \to (\forall (P: Prop).P))) H16 (THead (Bind Abst)
458 x3 x4) H18) in (eq_ind_r T (THead (Bind Abst) x3 x4) (\lambda (t0: T).(sn3 c
459 (THead (Flat Appl) x1 t0))) (let H_x \def (term_dec t2 x3) in (let H22 \def
460 H_x in (or_ind (eq T t2 x3) ((eq T t2 x3) \to (\forall (P: Prop).P)) (sn3 c
461 (THead (Flat Appl) x1 (THead (Bind Abst) x3 x4))) (\lambda (H23: (eq T t2
462 x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: T).((eq T (THead (Flat Appl)
463 x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) x1 (THead (Bind Abst) t0
464 x4))) \to (\forall (P: Prop).P))) H21 t2 H23) in (let H25 \def (eq_ind_r T x3
465 (\lambda (t0: T).(pr2 c t2 t0)) H19 t2 H23) in (eq_ind T t2 (\lambda (t0:
466 T).(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t0 x4)))) (let H_x0 \def
467 (term_dec x x1) in (let H26 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to
468 (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t2
469 x4))) (\lambda (H27: (eq T x x1)).(let H28 \def (eq_ind_r T x1 (\lambda (t0:
470 T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (THead (Flat Appl)
471 t0 (THead (Bind Abst) t2 x4))) \to (\forall (P: Prop).P))) H24 x H27) in (let
472 H29 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H27) in (eq_ind
473 T x (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) t2
474 x4)))) (let H_x1 \def (term_dec x0 x4) in (let H30 \def H_x1 in (or_ind (eq T
475 x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x
476 (THead (Bind Abst) t2 x4))) (\lambda (H31: (eq T x0 x4)).(let H32 \def
477 (eq_ind_r T x4 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind
478 Abst) t2 x0)) (THead (Flat Appl) x (THead (Bind Abst) t2 t0))) \to (\forall
479 (P: Prop).P))) H28 x0 H31) in (let H33 \def (eq_ind_r T x4 (\lambda (t0:
480 T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0
481 H31) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead
482 (Bind Abst) t2 t0)))) (H32 (refl_equal T (THead (Flat Appl) x (THead (Bind
483 Abst) t2 x0))) (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)))) x4
484 H31)))) (\lambda (H31: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead
485 (Bind Abbr) x x4) (\lambda (H32: (eq T (THead (Bind Abbr) x x0) (THead (Bind
486 Abbr) x x4))).(\lambda (P: Prop).(let H33 \def (f_equal T T (\lambda (e:
487 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
488 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
489 Abbr) x x0) (THead (Bind Abbr) x x4) H32) in (let H34 \def (eq_ind_r T x4
490 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H31 x0 H33) in
491 (let H35 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u:
492 T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H33) in (H34 (refl_equal T x0)
493 P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4)
494 (pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead
495 (Bind Abbr) x x4)) t2 (sn3_sing c t2 H7))) H30))) x1 H27)))) (\lambda (H27:
496 (((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind Abbr) x1 x4)
497 (\lambda (H28: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x1
498 x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e
499 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _)
500 \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0)
501 (THead (Bind Abbr) x1 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e:
502 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
503 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
504 Abbr) x x0) (THead (Bind Abbr) x1 x4) H28) in (\lambda (H31: (eq T x
505 x1)).(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall
506 (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (let H33 \def
507 (eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
508 H27 x H31) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14
509 x H31) in (H33 (refl_equal T x) P)))))) H29)))) (pr3_head_12 c x x1 (pr3_pr2
510 c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20
511 Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) t2 (sn3_sing c t2
512 H7))) H26))) x3 H23)))) (\lambda (H23: (((eq T t2 x3) \to (\forall (P:
513 Prop).P)))).(let H_x0 \def (term_dec x x1) in (let H24 \def H_x0 in (or_ind
514 (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl)
515 x1 (THead (Bind Abst) x3 x4))) (\lambda (H25: (eq T x x1)).(let H26 \def
516 (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H25) in (eq_ind T x
517 (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4))))
518 (let H_x1 \def (term_dec x0 x4) in (let H27 \def H_x1 in (or_ind (eq T x0 x4)
519 ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x (THead
520 (Bind Abst) x3 x4))) (\lambda (H28: (eq T x0 x4)).(let H29 \def (eq_ind_r T
521 x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
522 x0 t0)))) H20 x0 H28) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat
523 Appl) x (THead (Bind Abst) x3 t0)))) (H8 x3 H23 (pr3_pr2 c t2 x3 H19)) x4
524 H28))) (\lambda (H28: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead
525 (Bind Abbr) x x4) (\lambda (H29: (eq T (THead (Bind Abbr) x x0) (THead (Bind
526 Abbr) x x4))).(\lambda (P: Prop).(let H30 \def (f_equal T T (\lambda (e:
527 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
528 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
529 Abbr) x x0) (THead (Bind Abbr) x x4) H29) in (let H31 \def (eq_ind_r T x4
530 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H28 x0 H30) in
531 (let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u:
532 T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (H31 (refl_equal T x0)
533 P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4)
534 (pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead
535 (Bind Abbr) x x4)) x3 (H7 x3 H23 (pr3_pr2 c t2 x3 H19)))) H27))) x1 H25)))
536 (\lambda (H25: (((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind
537 Abbr) x1 x4) (\lambda (H26: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr)
538 x1 x4))).(\lambda (P: Prop).(let H27 \def (f_equal T T (\lambda (e: T).(match
539 e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _)
540 \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0)
541 (THead (Bind Abbr) x1 x4) H26) in ((let H28 \def (f_equal T T (\lambda (e:
542 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
543 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
544 Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in (\lambda (H29: (eq T x
545 x1)).(let H30 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall
546 (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (let H31 \def
547 (eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
548 H25 x H29) in (let H32 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14
549 x H29) in (H31 (refl_equal T x) P)))))) H27)))) (pr3_head_12 c x x1 (pr3_pr2
550 c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20
551 Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) x3 (H7 x3 H23
552 (pr3_pr2 c t2 x3 H19)))) H24)))) H22))) x2 H18))))))) H17)) t3 H13)))))))
553 H12)) (\lambda (H12: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
554 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead
555 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
556 T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
557 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
558 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
559 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T
560 T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
561 (THead (Bind Abst) t2 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
562 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
563 Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
564 (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
565 T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
566 z1 t4))))))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
567 T).(\lambda (x4: T).(\lambda (H13: (eq T (THead (Bind Abst) t2 x0) (THead
568 (Bind Abst) x1 x2))).(\lambda (H14: (eq T t3 (THead (Bind Abbr) x3
569 x4))).(\lambda (H15: (pr2 c x x3)).(\lambda (H16: ((\forall (b: B).(\forall
570 (u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H17 \def (eq_ind T t3
571 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0)
572 \to (\forall (P: Prop).P))) H9 (THead (Bind Abbr) x3 x4) H14) in (eq_ind_r T
573 (THead (Bind Abbr) x3 x4) (\lambda (t0: T).(sn3 c t0)) (let H18 \def (f_equal
574 T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
575 \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) \Rightarrow t0]))
576 (THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in ((let H19 \def
577 (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
578 [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0)
579 \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in
580 (\lambda (_: (eq T t2 x1)).(let H21 \def (eq_ind_r T x2 (\lambda (t0:
581 T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t0 x4)))) H16 x0
582 H19) in (let H_x \def (term_dec x x3) in (let H22 \def H_x in (or_ind (eq T x
583 x3) ((eq T x x3) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abbr) x3 x4))
584 (\lambda (H23: (eq T x x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0:
585 T).(pr2 c x t0)) H15 x H23) in (eq_ind T x (\lambda (t0: T).(sn3 c (THead
586 (Bind Abbr) t0 x4))) (let H_x0 \def (term_dec x0 x4) in (let H25 \def H_x0 in
587 (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead
588 (Bind Abbr) x x4)) (\lambda (H26: (eq T x0 x4)).(let H27 \def (eq_ind_r T x4
589 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0
590 t0)))) H21 x0 H26) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Bind Abbr)
591 x t0))) (sn3_sing c (THead (Bind Abbr) x x0) H6) x4 H26))) (\lambda (H26:
592 (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x x4)
593 (\lambda (H27: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x
594 x4))).(\lambda (P: Prop).(let H28 \def (f_equal T T (\lambda (e: T).(match e
595 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
596 \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0)
597 (THead (Bind Abbr) x x4) H27) in (let H29 \def (eq_ind_r T x4 (\lambda (t0:
598 T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H26 x0 H28) in (let H30 \def
599 (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c
600 (Bind b) u) x0 t0)))) H21 x0 H28) in (H29 (refl_equal T x0) P)))))) (pr3_pr2
601 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4
602 (Bind Abbr) (H21 Abbr x))))) H25))) x3 H23))) (\lambda (H23: (((eq T x x3)
603 \to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x3 x4) (\lambda (H24: (eq
604 T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x3 x4))).(\lambda (P:
605 Prop).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T return
606 (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x |
607 (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr)
608 x3 x4) H24) in ((let H26 \def (f_equal T T (\lambda (e: T).(match e in T
609 return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
610 \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0)
611 (THead (Bind Abbr) x3 x4) H24) in (\lambda (H27: (eq T x x3)).(let H28 \def
612 (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c
613 (Bind b) u) x0 t0)))) H21 x0 H26) in (let H29 \def (eq_ind_r T x3 (\lambda
614 (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H23 x H27) in (let H30
615 \def (eq_ind_r T x3 (\lambda (t0: T).(pr2 c x t0)) H15 x H27) in (H29
616 (refl_equal T x) P)))))) H25)))) (pr3_head_12 c x x3 (pr3_pr2 c x x3 H15)
617 (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x3) x0 x4 (H21 Abbr x3)))))
618 H22)))))) H18)) t3 H14)))))))))) H12)) (\lambda (H12: (ex6_6 B T T T T T
619 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
620 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
621 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
622 (THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
623 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
624 (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
625 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
626 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_:
627 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
628 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
629 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
630 y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_:
631 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
632 b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
633 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead
634 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
635 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
636 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
637 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
638 (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
639 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
640 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
641 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3)
642 (\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda
643 (x5: T).(\lambda (x6: T).(\lambda (H13: (not (eq B x1 Abst))).(\lambda (H14:
644 (eq T (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3))).(\lambda (H15: (eq
645 T t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
646 (_: (pr2 c x x5)).(\lambda (H17: (pr2 c x2 x6)).(\lambda (H18: (pr2 (CHead c
647 (Bind x1) x6) x3 x4)).(let H19 \def (eq_ind T t3 (\lambda (t0: T).((eq T
648 (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P:
649 Prop).P))) H9 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))
650 H15) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5)
651 x4)) (\lambda (t0: T).(sn3 c t0)) (let H20 \def (f_equal T B (\lambda (e:
652 T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst |
653 (TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return
654 (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
655 Abst])])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in ((let H21
656 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
657 with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _)
658 \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in
659 ((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
660 T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
661 t0) \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14)
662 in (\lambda (H23: (eq T t2 x2)).(\lambda (H24: (eq B Abst x1)).(let H25 \def
663 (eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H18 x0
664 H22) in (let H26 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H17 t2
665 H23) in (let H27 \def (eq_ind_r B x1 (\lambda (b: B).(pr2 (CHead c (Bind b)
666 x6) x0 x4)) H25 Abst H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b:
667 B).(not (eq B b Abst))) H13 Abst H24) in (eq_ind B Abst (\lambda (b: B).(sn3
668 c (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (let H29
669 \def (match (H28 (refl_equal B Abst)) in False return (\lambda (_:
670 False).(sn3 c (THead (Bind Abst) x6 (THead (Flat Appl) (lift (S O) O x5)
671 x4)))) with []) in H29) x1 H24)))))))) H21)) H20)) t3 H15)))))))))))))) H12))
672 H11))))))))) w H4))))))))))) y H0))))) H)))).
674 theorem sn3_appl_lref:
675 \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (v:
676 T).((sn3 c v) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))
678 \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda
679 (v: T).(\lambda (H0: (sn3 c v)).(sn3_ind c (\lambda (t: T).(sn3 c (THead
680 (Flat Appl) t (TLRef i)))) (\lambda (t1: T).(\lambda (_: ((\forall (t2:
681 T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
682 t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
683 Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c (THead (Flat Appl) t2 (TLRef
684 i)))))))).(sn3_pr2_intro c (THead (Flat Appl) t1 (TLRef i)) (\lambda (t2:
685 T).(\lambda (H3: (((eq T (THead (Flat Appl) t1 (TLRef i)) t2) \to (\forall
686 (P: Prop).P)))).(\lambda (H4: (pr2 c (THead (Flat Appl) t1 (TLRef i))
687 t2)).(let H5 \def (pr2_gen_appl c t1 (TLRef i) t2 H4) in (or3_ind (ex3_2 T T
688 (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
689 (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda
690 (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
691 T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1
692 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
693 T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
694 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))) (\lambda (_:
695 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
696 (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
697 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
698 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
699 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
700 (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
701 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
702 t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
703 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
704 T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1:
705 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
706 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
707 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))
708 (sn3 c t2) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
709 t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1
710 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T
711 T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
712 (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda
713 (t3: T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1:
714 T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr2 c t1
715 x0)).(\lambda (H9: (pr2 c (TLRef i) x1)).(let H10 \def (eq_ind T t2 (\lambda
716 (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to (\forall (P: Prop).P)))
717 H3 (THead (Flat Appl) x0 x1) H7) in (eq_ind_r T (THead (Flat Appl) x0 x1)
718 (\lambda (t: T).(sn3 c t)) (let H11 \def (eq_ind_r T x1 (\lambda (t: T).((eq
719 T (THead (Flat Appl) t1 (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P:
720 Prop).P))) H10 (TLRef i) (H x1 H9)) in (let H12 \def (eq_ind_r T x1 (\lambda
721 (t: T).(pr2 c (TLRef i) t)) H9 (TLRef i) (H x1 H9)) in (eq_ind T (TLRef i)
722 (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H_x \def (term_dec t1
723 x0) in (let H13 \def H_x in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall
724 (P: Prop).P)) (sn3 c (THead (Flat Appl) x0 (TLRef i))) (\lambda (H14: (eq T
725 t1 x0)).(let H15 \def (eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat
726 Appl) t1 (TLRef i)) (THead (Flat Appl) t (TLRef i))) \to (\forall (P:
727 Prop).P))) H11 t1 H14) in (let H16 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c
728 t1 t)) H8 t1 H14) in (eq_ind T t1 (\lambda (t: T).(sn3 c (THead (Flat Appl) t
729 (TLRef i)))) (H15 (refl_equal T (THead (Flat Appl) t1 (TLRef i))) (sn3 c
730 (THead (Flat Appl) t1 (TLRef i)))) x0 H14)))) (\lambda (H14: (((eq T t1 x0)
731 \to (\forall (P: Prop).P)))).(H2 x0 H14 (pr3_pr2 c t1 x0 H8))) H13))) x1 (H
732 x1 H9)))) t2 H7))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1:
733 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead
734 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
735 T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
736 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2)))))
737 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
738 (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T
739 T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
740 (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
741 T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))))
742 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1
743 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
744 T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))
745 (sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
746 T).(\lambda (H7: (eq T (TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H8:
747 (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c t1 x2)).(\lambda (_:
748 ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let
749 H11 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i))
750 t) \to (\forall (P: Prop).P))) H3 (THead (Bind Abbr) x2 x3) H8) in (eq_ind_r
751 T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) (let H12 \def (eq_ind
752 T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
753 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
754 \Rightarrow False])) I (THead (Bind Abst) x0 x1) H7) in (False_ind (sn3 c
755 (THead (Bind Abbr) x2 x3)) H12)) t2 H8)))))))))) H6)) (\lambda (H6: (ex6_6 B
756 T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
757 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
758 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
759 (_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
760 (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
761 T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
762 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
763 T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1:
764 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
765 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
766 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
767 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
768 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
769 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
770 T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind b) y1
771 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
772 T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat
773 Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
774 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2)))))))
775 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
776 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
777 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
778 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2) (\lambda (x0: B).(\lambda (x1:
779 T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
780 T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H8: (eq T (TLRef i) (THead
781 (Bind x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x5 (THead (Flat
782 Appl) (lift (S O) O x4) x3)))).(\lambda (_: (pr2 c t1 x4)).(\lambda (_: (pr2
783 c x1 x5)).(\lambda (_: (pr2 (CHead c (Bind x0) x5) x2 x3)).(let H13 \def
784 (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to
785 (\forall (P: Prop).P))) H3 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O)
786 O x4) x3)) H9) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S
787 O) O x4) x3)) (\lambda (t: T).(sn3 c t)) (let H14 \def (eq_ind T (TLRef i)
788 (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
789 \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
790 False])) I (THead (Bind x0) x1 x2) H8) in (False_ind (sn3 c (THead (Bind x0)
791 x5 (THead (Flat Appl) (lift (S O) O x4) x3))) H14)) t2 H9)))))))))))))) H6))
792 H5))))))))) v H0))))).
794 theorem sn3_appl_abbr:
795 \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c
796 (CHead d (Bind Abbr) w)) \to (\forall (v: T).((sn3 c (THead (Flat Appl) v
797 (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))))
799 \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda
800 (H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (v: T).(\lambda (H0: (sn3 c
801 (THead (Flat Appl) v (lift (S i) O w)))).(insert_eq T (THead (Flat Appl) v
802 (lift (S i) O w)) (\lambda (t: T).(sn3 c t)) (\lambda (_: T).(sn3 c (THead
803 (Flat Appl) v (TLRef i)))) (\lambda (y: T).(\lambda (H1: (sn3 c y)).(unintro
804 T v (\lambda (t: T).((eq T y (THead (Flat Appl) t (lift (S i) O w))) \to (sn3
805 c (THead (Flat Appl) t (TLRef i))))) (sn3_ind c (\lambda (t: T).(\forall (x:
806 T).((eq T t (THead (Flat Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat
807 Appl) x (TLRef i)))))) (\lambda (t1: T).(\lambda (H2: ((\forall (t2:
808 T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
809 t2)))))).(\lambda (H3: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
810 Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).((eq T t2 (THead (Flat
811 Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x (TLRef
812 i)))))))))).(\lambda (x: T).(\lambda (H4: (eq T t1 (THead (Flat Appl) x (lift
813 (S i) O w)))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2:
814 T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall
815 (x0: T).((eq T t2 (THead (Flat Appl) x0 (lift (S i) O w))) \to (sn3 c (THead
816 (Flat Appl) x0 (TLRef i))))))))) H3 (THead (Flat Appl) x (lift (S i) O w))
817 H4) in (let H6 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: T).((((eq T t
818 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (sn3 c t2))))) H2
819 (THead (Flat Appl) x (lift (S i) O w)) H4) in (sn3_pr2_intro c (THead (Flat
820 Appl) x (TLRef i)) (\lambda (t2: T).(\lambda (H7: (((eq T (THead (Flat Appl)
821 x (TLRef i)) t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr2 c (THead
822 (Flat Appl) x (TLRef i)) t2)).(let H9 \def (pr2_gen_appl c x (TLRef i) t2 H8)
823 in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
824 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
825 (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T
826 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
827 (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
828 T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))))
829 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x
830 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
831 T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))
832 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
833 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
834 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
835 (_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
836 (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
837 T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
838 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
839 T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
840 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
841 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
842 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))
843 (sn3 c t2) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
844 t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x
845 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T
846 T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
847 (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3:
848 T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1:
849 T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H12: (pr2 c
850 x x0)).(\lambda (H13: (pr2 c (TLRef i) x1)).(let H14 \def (eq_ind T t2
851 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P:
852 Prop).P))) H7 (THead (Flat Appl) x0 x1) H11) in (eq_ind_r T (THead (Flat
853 Appl) x0 x1) (\lambda (t: T).(sn3 c t)) (let H15 \def (pr2_gen_lref c x1 i
854 H13) in (or_ind (eq T x1 (TLRef i)) (ex2_2 C T (\lambda (d0: C).(\lambda (u:
855 T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq
856 T x1 (lift (S i) O u))))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda (H16:
857 (eq T x1 (TLRef i))).(let H17 \def (eq_ind T x1 (\lambda (t: T).((eq T (THead
858 (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P:
859 Prop).P))) H14 (TLRef i) H16) in (eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c
860 (THead (Flat Appl) x0 t))) (let H_x \def (term_dec x x0) in (let H18 \def H_x
861 in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c (THead
862 (Flat Appl) x0 (TLRef i))) (\lambda (H19: (eq T x x0)).(let H20 \def
863 (eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead
864 (Flat Appl) t (TLRef i))) \to (\forall (P: Prop).P))) H17 x H19) in (let H21
865 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H19) in (eq_ind T x
866 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (TLRef i)))) (H20 (refl_equal T
867 (THead (Flat Appl) x (TLRef i))) (sn3 c (THead (Flat Appl) x (TLRef i)))) x0
868 H19)))) (\lambda (H19: (((eq T x x0) \to (\forall (P: Prop).P)))).(H5 (THead
869 (Flat Appl) x0 (lift (S i) O w)) (\lambda (H20: (eq T (THead (Flat Appl) x
870 (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)))).(\lambda (P:
871 Prop).(let H21 \def (f_equal T T (\lambda (e: T).(match e in T return
872 (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x |
873 (THead _ t _) \Rightarrow t])) (THead (Flat Appl) x (lift (S i) O w)) (THead
874 (Flat Appl) x0 (lift (S i) O w)) H20) in (let H22 \def (eq_ind_r T x0
875 (\lambda (t: T).((eq T x t) \to (\forall (P0: Prop).P0))) H19 x H21) in (let
876 H23 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H21) in (H22
877 (refl_equal T x) P)))))) (pr3_pr2 c (THead (Flat Appl) x (lift (S i) O w))
878 (THead (Flat Appl) x0 (lift (S i) O w)) (pr2_head_1 c x x0 H12 (Flat Appl)
879 (lift (S i) O w))) x0 (refl_equal T (THead (Flat Appl) x0 (lift (S i) O
880 w))))) H18))) x1 H16))) (\lambda (H16: (ex2_2 C T (\lambda (d0: C).(\lambda
881 (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
882 T).(eq T x1 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda
883 (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
884 T).(eq T x1 (lift (S i) O u)))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda
885 (x2: C).(\lambda (x3: T).(\lambda (H17: (getl i c (CHead x2 (Bind Abbr)
886 x3))).(\lambda (H18: (eq T x1 (lift (S i) O x3))).(let H19 \def (eq_ind T x1
887 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0
888 t)) \to (\forall (P: Prop).P))) H14 (lift (S i) O x3) H18) in (eq_ind_r T
889 (lift (S i) O x3) (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H20
890 \def (eq_ind C (CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H
891 (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2
892 (Bind Abbr) x3) H17)) in (let H21 \def (f_equal C C (\lambda (e: C).(match e
893 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
894 \Rightarrow c0])) (CHead d (Bind Abbr) w) (CHead x2 (Bind Abbr) x3)
895 (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 (Bind Abbr) x3) H17)) in
896 ((let H22 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
897 C).T) with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d
898 (Bind Abbr) w) (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w)
899 i H (CHead x2 (Bind Abbr) x3) H17)) in (\lambda (H23: (eq C d x2)).(let H24
900 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c (CHead x2 (Bind Abbr) t))) H20
901 w H22) in (eq_ind T w (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 (lift (S
902 i) O t)))) (let H25 \def (eq_ind_r C x2 (\lambda (c0: C).(getl i c (CHead c0
903 (Bind Abbr) w))) H24 d H23) in (let H_x \def (term_dec x x0) in (let H26 \def
904 H_x in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c
905 (THead (Flat Appl) x0 (lift (S i) O w))) (\lambda (H27: (eq T x x0)).(let H28
906 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H27) in (eq_ind T x
907 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (lift (S i) O w)))) (sn3_sing c
908 (THead (Flat Appl) x (lift (S i) O w)) H6) x0 H27))) (\lambda (H27: (((eq T x
909 x0) \to (\forall (P: Prop).P)))).(H6 (THead (Flat Appl) x0 (lift (S i) O w))
910 (\lambda (H28: (eq T (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat
911 Appl) x0 (lift (S i) O w)))).(\lambda (P: Prop).(let H29 \def (f_equal T T
912 (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
913 \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t]))
914 (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O
915 w)) H28) in (let H30 \def (eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to
916 (\forall (P0: Prop).P0))) H27 x H29) in (let H31 \def (eq_ind_r T x0 (\lambda
917 (t: T).(pr2 c x t)) H12 x H29) in (H30 (refl_equal T x) P)))))) (pr3_pr2 c
918 (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O
919 w)) (pr2_head_1 c x x0 H12 (Flat Appl) (lift (S i) O w))))) H26)))) x3
920 H22)))) H21))) x1 H18)))))) H16)) H15)) t2 H11))))))) H10)) (\lambda (H10:
921 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
922 T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
923 (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
924 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
925 T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
926 (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1
927 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
928 T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda
929 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
930 (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
931 T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
932 (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
933 b) u) z1 t3))))))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2:
934 T).(\lambda (x3: T).(\lambda (H11: (eq T (TLRef i) (THead (Bind Abst) x0
935 x1))).(\lambda (H12: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c
936 x x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b)
937 u) x1 x3))))).(let H15 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat
938 Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 (THead (Bind Abbr) x2
939 x3) H12) in (eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t))
940 (let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return
941 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
942 \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0
943 x1) H11) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H16)) t2
944 H12)))))))))) H10)) (\lambda (H10: (ex6_6 B T T T T T (\lambda (b:
945 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
946 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
947 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i)
948 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
949 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
950 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
951 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
952 (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
953 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
954 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
955 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind
956 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
957 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
958 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
959 (_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
960 (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
961 T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
962 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
963 T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
964 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
965 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
966 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
967 (sn3 c t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
968 T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0
969 Abst))).(\lambda (H12: (eq T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda
970 (H13: (eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4)
971 x3)))).(\lambda (_: (pr2 c x x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_:
972 (pr2 (CHead c (Bind x0) x5) x2 x3)).(let H17 \def (eq_ind T t2 (\lambda (t:
973 T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7
974 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) H13) in
975 (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))
976 (\lambda (t: T).(sn3 c t)) (let H18 \def (eq_ind T (TLRef i) (\lambda (ee:
977 T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
978 False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
979 (THead (Bind x0) x1 x2) H12) in (False_ind (sn3 c (THead (Bind x0) x5 (THead
980 (Flat Appl) (lift (S O) O x4) x3))) H18)) t2 H13)))))))))))))) H10))
981 H9))))))))))))) y H1)))) H0))))))).
983 theorem sn3_appl_cast:
984 \forall (c: C).(\forall (v: T).(\forall (u: T).((sn3 c (THead (Flat Appl) v
985 u)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) v t)) \to (sn3 c (THead
986 (Flat Appl) v (THead (Flat Cast) u t))))))))
988 \lambda (c: C).(\lambda (v: T).(\lambda (u: T).(\lambda (H: (sn3 c (THead
989 (Flat Appl) v u))).(insert_eq T (THead (Flat Appl) v u) (\lambda (t: T).(sn3
990 c t)) (\lambda (_: T).(\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to
991 (sn3 c (THead (Flat Appl) v (THead (Flat Cast) u t0)))))) (\lambda (y:
992 T).(\lambda (H0: (sn3 c y)).(unintro T u (\lambda (t: T).((eq T y (THead
993 (Flat Appl) v t)) \to (\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to
994 (sn3 c (THead (Flat Appl) v (THead (Flat Cast) t t0))))))) (unintro T v
995 (\lambda (t: T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to
996 (\forall (t0: T).((sn3 c (THead (Flat Appl) t t0)) \to (sn3 c (THead (Flat
997 Appl) t (THead (Flat Cast) x t0)))))))) (sn3_ind c (\lambda (t: T).(\forall
998 (x: T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (t0:
999 T).((sn3 c (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead
1000 (Flat Cast) x0 t0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2:
1001 T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
1002 t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
1003 Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2
1004 (THead (Flat Appl) x x0)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) x
1005 t)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0
1006 t))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1 (THead
1007 (Flat Appl) x x0))).(\lambda (t: T).(\lambda (H4: (sn3 c (THead (Flat Appl) x
1008 t))).(insert_eq T (THead (Flat Appl) x t) (\lambda (t0: T).(sn3 c t0))
1009 (\lambda (_: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 t))))
1010 (\lambda (y0: T).(\lambda (H5: (sn3 c y0)).(unintro T t (\lambda (t0: T).((eq
1011 T y0 (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead (Flat
1012 Cast) x0 t0))))) (sn3_ind c (\lambda (t0: T).(\forall (x1: T).((eq T t0
1013 (THead (Flat Appl) x x1)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast)
1014 x0 x1)))))) (\lambda (t0: T).(\lambda (H6: ((\forall (t2: T).((((eq T t0 t2)
1015 \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2)))))).(\lambda
1016 (H7: ((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3
1017 c t0 t2) \to (\forall (x1: T).((eq T t2 (THead (Flat Appl) x x1)) \to (sn3 c
1018 (THead (Flat Appl) x (THead (Flat Cast) x0 x1)))))))))).(\lambda (x1:
1019 T).(\lambda (H8: (eq T t0 (THead (Flat Appl) x x1))).(let H9 \def (eq_ind T
1020 t0 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
1021 Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).((eq T t3 (THead (Flat
1022 Appl) x x2)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0
1023 x2))))))))) H7 (THead (Flat Appl) x x1) H8) in (let H10 \def (eq_ind T t0
1024 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P)))
1025 \to ((pr3 c t2 t3) \to (sn3 c t3))))) H6 (THead (Flat Appl) x x1) H8) in (let
1026 H11 \def (eq_ind T t1 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to
1027 (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).(\forall (x3:
1028 T).((eq T t3 (THead (Flat Appl) x2 x3)) \to (\forall (t4: T).((sn3 c (THead
1029 (Flat Appl) x2 t4)) \to (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x3
1030 t4)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H12 \def (eq_ind T t1
1031 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P)))
1032 \to ((pr3 c t2 t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in
1033 (sn3_pr2_intro c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (\lambda
1034 (t2: T).(\lambda (H13: (((eq T (THead (Flat Appl) x (THead (Flat Cast) x0
1035 x1)) t2) \to (\forall (P: Prop).P)))).(\lambda (H14: (pr2 c (THead (Flat
1036 Appl) x (THead (Flat Cast) x0 x1)) t2)).(let H15 \def (pr2_gen_appl c x
1037 (THead (Flat Cast) x0 x1) t2 H14) in (or3_ind (ex3_2 T T (\lambda (u2:
1038 T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1039 T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
1040 (THead (Flat Cast) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
1041 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1)
1042 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1043 T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
1044 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
1045 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b:
1046 B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T
1047 T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1048 (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
1049 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
1050 T (THead (Flat Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b:
1051 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
1052 (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1053 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1054 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_:
1055 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1056 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1057 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
1058 y2) z1 z2)))))))) (sn3 c t2) (\lambda (H16: (ex3_2 T T (\lambda (u2:
1059 T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1060 T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
1061 (THead (Flat Cast) x0 x1) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
1062 (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
1063 T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Flat Cast)
1064 x0 x1) t3))) (sn3 c t2) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H17: (eq
1065 T t2 (THead (Flat Appl) x2 x3))).(\lambda (H18: (pr2 c x x2)).(\lambda (H19:
1066 (pr2 c (THead (Flat Cast) x0 x1) x3)).(let H20 \def (eq_ind T t2 (\lambda
1067 (t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to
1068 (\forall (P: Prop).P))) H13 (THead (Flat Appl) x2 x3) H17) in (eq_ind_r T
1069 (THead (Flat Appl) x2 x3) (\lambda (t3: T).(sn3 c t3)) (let H21 \def
1070 (pr2_gen_cast c x0 x1 x3 H19) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda
1071 (t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
1072 T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3)))) (pr2 c
1073 x1 x3) (sn3 c (THead (Flat Appl) x2 x3)) (\lambda (H22: (ex3_2 T T (\lambda
1074 (u2: T).(\lambda (t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
1075 T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1
1076 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x3 (THead
1077 (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
1078 (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3))) (sn3 c (THead (Flat Appl) x2
1079 x3)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H23: (eq T x3 (THead (Flat
1080 Cast) x4 x5))).(\lambda (H24: (pr2 c x0 x4)).(\lambda (H25: (pr2 c x1
1081 x5)).(let H26 \def (eq_ind T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x
1082 (THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P:
1083 Prop).P))) H20 (THead (Flat Cast) x4 x5) H23) in (eq_ind_r T (THead (Flat
1084 Cast) x4 x5) (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 t3))) (let H_x
1085 \def (term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) in (let
1086 H27 \def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2
1087 x4)) ((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) \to (\forall
1088 (P: Prop).P)) (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x4 x5)))
1089 (\lambda (H28: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2
1090 x4))).(let H29 \def (f_equal T T (\lambda (e: T).(match e in T return
1091 (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x |
1092 (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x0) (THead (Flat Appl)
1093 x2 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T
1094 return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
1095 \Rightarrow x0 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x0)
1096 (THead (Flat Appl) x2 x4) H28) in (\lambda (H31: (eq T x x2)).(let H32 \def
1097 (eq_ind_r T x4 (\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat
1098 Cast) x0 x1)) (THead (Flat Appl) x2 (THead (Flat Cast) t3 x5))) \to (\forall
1099 (P: Prop).P))) H26 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t3:
1100 T).(pr2 c x0 t3)) H24 x0 H30) in (eq_ind T x0 (\lambda (t3: T).(sn3 c (THead
1101 (Flat Appl) x2 (THead (Flat Cast) t3 x5)))) (let H34 \def (eq_ind_r T x2
1102 (\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1))
1103 (THead (Flat Appl) t3 (THead (Flat Cast) x0 x5))) \to (\forall (P: Prop).P)))
1104 H32 x H31) in (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18
1105 x H31) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead
1106 (Flat Cast) x0 x5)))) (let H_x0 \def (term_dec (THead (Flat Appl) x x1)
1107 (THead (Flat Appl) x x5)) in (let H36 \def H_x0 in (or_ind (eq T (THead (Flat
1108 Appl) x x1) (THead (Flat Appl) x x5)) ((eq T (THead (Flat Appl) x x1) (THead
1109 (Flat Appl) x x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x
1110 (THead (Flat Cast) x0 x5))) (\lambda (H37: (eq T (THead (Flat Appl) x x1)
1111 (THead (Flat Appl) x x5))).(let H38 \def (f_equal T T (\lambda (e: T).(match
1112 e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _)
1113 \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x1)
1114 (THead (Flat Appl) x x5) H37) in (let H39 \def (eq_ind_r T x5 (\lambda (t3:
1115 T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl)
1116 x (THead (Flat Cast) x0 t3))) \to (\forall (P: Prop).P))) H34 x1 H38) in (let
1117 H40 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H38) in
1118 (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast)
1119 x0 t3)))) (H39 (refl_equal T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)))
1120 (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)))) x5 H38))))) (\lambda
1121 (H37: (((eq T (THead (Flat Appl) x x1) (THead (Flat Appl) x x5)) \to (\forall
1122 (P: Prop).P)))).(H9 (THead (Flat Appl) x x5) H37 (pr3_pr2 c (THead (Flat
1123 Appl) x x1) (THead (Flat Appl) x x5) (pr2_thin_dx c x1 x5 H25 x Appl)) x5
1124 (refl_equal T (THead (Flat Appl) x x5)))) H36))) x2 H31))) x4 H30))))) H29)))
1125 (\lambda (H28: (((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4))
1126 \to (\forall (P: Prop).P)))).(let H_x0 \def (term_dec (THead (Flat Appl) x
1127 x1) (THead (Flat Appl) x2 x5)) in (let H29 \def H_x0 in (or_ind (eq T (THead
1128 (Flat Appl) x x1) (THead (Flat Appl) x2 x5)) ((eq T (THead (Flat Appl) x x1)
1129 (THead (Flat Appl) x2 x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat
1130 Appl) x2 (THead (Flat Cast) x4 x5))) (\lambda (H30: (eq T (THead (Flat Appl)
1131 x x1) (THead (Flat Appl) x2 x5))).(let H31 \def (f_equal T T (\lambda (e:
1132 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x |
1133 (TLRef _) \Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl)
1134 x x1) (THead (Flat Appl) x2 x5) H30) in ((let H32 \def (f_equal T T (\lambda
1135 (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1
1136 | (TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat
1137 Appl) x x1) (THead (Flat Appl) x2 x5) H30) in (\lambda (H33: (eq T x
1138 x2)).(let H34 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H32)
1139 in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 (THead (Flat
1140 Cast) x4 t3)))) (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead
1141 (Flat Appl) x x0) (THead (Flat Appl) t3 x4)) \to (\forall (P: Prop).P))) H28
1142 x H33) in (let H36 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18 x
1143 H33) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead
1144 (Flat Cast) x4 x1)))) (H11 (THead (Flat Appl) x x4) H35 (pr3_pr2 c (THead
1145 (Flat Appl) x x0) (THead (Flat Appl) x x4) (pr2_thin_dx c x0 x4 H24 x Appl))
1146 x x4 (refl_equal T (THead (Flat Appl) x x4)) x1 (sn3_sing c (THead (Flat
1147 Appl) x x1) H10)) x2 H33))) x5 H32)))) H31))) (\lambda (H30: (((eq T (THead
1148 (Flat Appl) x x1) (THead (Flat Appl) x2 x5)) \to (\forall (P:
1149 Prop).P)))).(H11 (THead (Flat Appl) x2 x4) H28 (pr3_flat c x x2 (pr3_pr2 c x
1150 x2 H18) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x2 x4 (refl_equal T (THead (Flat
1151 Appl) x2 x4)) x5 (H10 (THead (Flat Appl) x2 x5) H30 (pr3_flat c x x2 (pr3_pr2
1152 c x x2 H18) x1 x5 (pr3_pr2 c x1 x5 H25) Appl)))) H29)))) H27))) x3 H23)))))))
1153 H22)) (\lambda (H22: (pr2 c x1 x3)).(let H_x \def (term_dec (THead (Flat
1154 Appl) x x1) (THead (Flat Appl) x2 x3)) in (let H23 \def H_x in (or_ind (eq T
1155 (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3)) ((eq T (THead (Flat Appl)
1156 x x1) (THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)) (sn3 c (THead
1157 (Flat Appl) x2 x3)) (\lambda (H24: (eq T (THead (Flat Appl) x x1) (THead
1158 (Flat Appl) x2 x3))).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T
1159 return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _)
1160 \Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x1)
1161 (THead (Flat Appl) x2 x3) H24) in ((let H26 \def (f_equal T T (\lambda (e:
1162 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 |
1163 (TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat
1164 Appl) x x1) (THead (Flat Appl) x2 x3) H24) in (\lambda (H27: (eq T x
1165 x2)).(let H28 \def (eq_ind_r T x3 (\lambda (t3: T).(pr2 c x1 t3)) H22 x1 H26)
1166 in (let H29 \def (eq_ind_r T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x
1167 (THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P:
1168 Prop).P))) H20 x1 H26) in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat
1169 Appl) x2 t3))) (let H30 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead
1170 (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) t3 x1)) \to
1171 (\forall (P: Prop).P))) H29 x H27) in (let H31 \def (eq_ind_r T x2 (\lambda
1172 (t3: T).(pr2 c x t3)) H18 x H27) in (eq_ind T x (\lambda (t3: T).(sn3 c
1173 (THead (Flat Appl) t3 x1))) (sn3_sing c (THead (Flat Appl) x x1) H10) x2
1174 H27))) x3 H26))))) H25))) (\lambda (H24: (((eq T (THead (Flat Appl) x x1)
1175 (THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat
1176 Appl) x2 x3) H24 (pr3_flat c x x2 (pr3_pr2 c x x2 H18) x1 x3 (pr3_pr2 c x1 x3
1177 H22) Appl))) H23)))) H21)) t2 H17))))))) H16)) (\lambda (H16: (ex4_4 T T T T
1178 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
1179 (THead (Flat Cast) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1180 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
1181 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1182 (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1183 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1184 u0) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1:
1185 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead
1186 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1187 T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
1188 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
1189 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b:
1190 B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))) (sn3 c t2)
1191 (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda
1192 (H17: (eq T (THead (Flat Cast) x0 x1) (THead (Bind Abst) x2 x3))).(\lambda
1193 (H18: (eq T t2 (THead (Bind Abbr) x4 x5))).(\lambda (_: (pr2 c x
1194 x4)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1195 u0) x3 x5))))).(let H21 \def (eq_ind T t2 (\lambda (t3: T).((eq T (THead
1196 (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P: Prop).P))) H13
1197 (THead (Bind Abbr) x4 x5) H18) in (eq_ind_r T (THead (Bind Abbr) x4 x5)
1198 (\lambda (t3: T).(sn3 c t3)) (let H22 \def (eq_ind T (THead (Flat Cast) x0
1199 x1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
1200 _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
1201 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
1202 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) x2
1203 x3) H17) in (False_ind (sn3 c (THead (Bind Abbr) x4 x5)) H22)) t2
1204 H18)))))))))) H16)) (\lambda (H16: (ex6_6 B T T T T T (\lambda (b:
1205 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1206 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
1207 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat
1208 Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
1209 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
1210 t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
1211 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1212 T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
1213 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
1214 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
1215 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
1216 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1217 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1218 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1219 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead
1220 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1221 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
1222 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
1223 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1224 (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
1225 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
1226 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
1227 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2)
1228 (\lambda (x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda
1229 (x6: T).(\lambda (x7: T).(\lambda (_: (not (eq B x2 Abst))).(\lambda (H18:
1230 (eq T (THead (Flat Cast) x0 x1) (THead (Bind x2) x3 x4))).(\lambda (H19: (eq
1231 T t2 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda
1232 (_: (pr2 c x x6)).(\lambda (_: (pr2 c x3 x7)).(\lambda (_: (pr2 (CHead c
1233 (Bind x2) x7) x4 x5)).(let H23 \def (eq_ind T t2 (\lambda (t3: T).((eq T
1234 (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P:
1235 Prop).P))) H13 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5))
1236 H19) in (eq_ind_r T (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6)
1237 x5)) (\lambda (t3: T).(sn3 c t3)) (let H24 \def (eq_ind T (THead (Flat Cast)
1238 x0 x1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
1239 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
1240 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
1241 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x2) x3 x4)
1242 H18) in (False_ind (sn3 c (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O)
1243 O x6) x5))) H24)) t2 H19)))))))))))))) H16)) H15))))))))))))))) y0 H5))))
1244 H4))))))))) y H0))))) H)))).
1246 theorem sn3_appl_bind:
1247 \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u:
1248 T).((sn3 c u) \to (\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) u)
1249 (THead (Flat Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v
1250 (THead (Bind b) u t))))))))))
1252 \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda
1253 (u: T).(\lambda (H0: (sn3 c u)).(sn3_ind c (\lambda (t: T).(\forall (t0:
1254 T).(\forall (v: T).((sn3 (CHead c (Bind b) t) (THead (Flat Appl) (lift (S O)
1255 O v) t0)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t t0)))))))
1256 (\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
1257 (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall
1258 (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
1259 (\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) t2) (THead (Flat
1260 Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t2
1261 t))))))))))).(\lambda (t: T).(\lambda (v: T).(\lambda (H3: (sn3 (CHead c
1262 (Bind b) t1) (THead (Flat Appl) (lift (S O) O v) t))).(insert_eq T (THead
1263 (Flat Appl) (lift (S O) O v) t) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1)
1264 t0)) (\lambda (_: T).(sn3 c (THead (Flat Appl) v (THead (Bind b) t1 t))))
1265 (\lambda (y: T).(\lambda (H4: (sn3 (CHead c (Bind b) t1) y)).(unintro T t
1266 (\lambda (t0: T).((eq T y (THead (Flat Appl) (lift (S O) O v) t0)) \to (sn3 c
1267 (THead (Flat Appl) v (THead (Bind b) t1 t0))))) (unintro T v (\lambda (t0:
1268 T).(\forall (x: T).((eq T y (THead (Flat Appl) (lift (S O) O t0) x)) \to (sn3
1269 c (THead (Flat Appl) t0 (THead (Bind b) t1 x)))))) (sn3_ind (CHead c (Bind b)
1270 t1) (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat
1271 Appl) (lift (S O) O x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b)
1272 t1 x0))))))) (\lambda (t2: T).(\lambda (H5: ((\forall (t3: T).((((eq T t2 t3)
1273 \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3
1274 (CHead c (Bind b) t1) t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t2
1275 t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to
1276 (\forall (x: T).(\forall (x0: T).((eq T t3 (THead (Flat Appl) (lift (S O) O
1277 x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) t1
1278 x0))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H7: (eq T t2 (THead
1279 (Flat Appl) (lift (S O) O x) x0))).(let H8 \def (eq_ind T t2 (\lambda (t0:
1280 T).(\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3
1281 (CHead c (Bind b) t1) t0 t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3
1282 (THead (Flat Appl) (lift (S O) O x1) x2)) \to (sn3 c (THead (Flat Appl) x1
1283 (THead (Bind b) t1 x2)))))))))) H6 (THead (Flat Appl) (lift (S O) O x) x0)
1284 H7) in (let H9 \def (eq_ind T t2 (\lambda (t0: T).(\forall (t3: T).((((eq T
1285 t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t0 t3) \to
1286 (sn3 (CHead c (Bind b) t1) t3))))) H5 (THead (Flat Appl) (lift (S O) O x) x0)
1287 H7) in (sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind b) t1 x0)) (\lambda
1288 (t3: T).(\lambda (H10: (((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0))
1289 t3) \to (\forall (P: Prop).P)))).(\lambda (H11: (pr2 c (THead (Flat Appl) x
1290 (THead (Bind b) t1 x0)) t3)).(let H12 \def (pr2_gen_appl c x (THead (Bind b)
1291 t1 x0) t3 H11) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
1292 t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x
1293 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))))
1294 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
1295 T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1296 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
1297 Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1298 (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1299 T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0)
1300 u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_:
1301 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
1302 b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda
1303 (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead
1304 (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
1305 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
1306 b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
1307 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1308 (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
1309 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
1310 (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
1311 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2)))))))) (sn3 c t3)
1312 (\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
1313 (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
1314 (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0)
1315 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
1316 (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
1317 (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))) (sn3 c
1318 t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H14: (eq T t3 (THead (Flat
1319 Appl) x1 x2))).(\lambda (H15: (pr2 c x x1)).(\lambda (H16: (pr2 c (THead
1320 (Bind b) t1 x0) x2)).(let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T
1321 (THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P)))
1322 H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat Appl) x1 x2)
1323 (\lambda (t0: T).(sn3 c t0)) (let H_x \def (pr3_gen_bind b H c t1 x0 x2) in
1324 (let H18 \def (H_x (pr3_pr2 c (THead (Bind b) t1 x0) x2 H16)) in (or_ind
1325 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2
1326 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
1327 T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4)))) (pr3 (CHead c (Bind
1328 b) t1) x0 (lift (S O) O x2)) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (H19:
1329 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2
1330 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
1331 T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4))))).(ex3_2_ind T T
1332 (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 t4)))) (\lambda
1333 (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3
1334 (CHead c (Bind b) t1) x0 t4))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda
1335 (x3: T).(\lambda (x4: T).(\lambda (H20: (eq T x2 (THead (Bind b) x3
1336 x4))).(\lambda (H21: (pr3 c t1 x3)).(\lambda (H22: (pr3 (CHead c (Bind b) t1)
1337 x0 x4)).(let H23 \def (eq_ind T x2 (\lambda (t0: T).((eq T (THead (Flat Appl)
1338 x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 t0)) \to (\forall (P:
1339 Prop).P))) H17 (THead (Bind b) x3 x4) H20) in (eq_ind_r T (THead (Bind b) x3
1340 x4) (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 t0))) (let H_x0 \def
1341 (term_dec t1 x3) in (let H24 \def H_x0 in (or_ind (eq T t1 x3) ((eq T t1 x3)
1342 \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind b) x3
1343 x4))) (\lambda (H25: (eq T t1 x3)).(let H26 \def (eq_ind_r T x3 (\lambda (t0:
1344 T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1
1345 (THead (Bind b) t0 x4))) \to (\forall (P: Prop).P))) H23 t1 H25) in (let H27
1346 \def (eq_ind_r T x3 (\lambda (t0: T).(pr3 c t1 t0)) H21 t1 H25) in (eq_ind T
1347 t1 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 (THead (Bind b) t0 x4))))
1348 (let H_x1 \def (term_dec x0 x4) in (let H28 \def H_x1 in (or_ind (eq T x0 x4)
1349 ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead
1350 (Bind b) t1 x4))) (\lambda (H29: (eq T x0 x4)).(let H30 \def (eq_ind_r T x4
1351 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead
1352 (Flat Appl) x1 (THead (Bind b) t1 t0))) \to (\forall (P: Prop).P))) H26 x0
1353 H29) in (let H31 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b)
1354 t1) x0 t0)) H22 x0 H29) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat
1355 Appl) x1 (THead (Bind b) t1 t0)))) (let H_x2 \def (term_dec x x1) in (let H32
1356 \def H_x2 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3
1357 c (THead (Flat Appl) x1 (THead (Bind b) t1 x0))) (\lambda (H33: (eq T x
1358 x1)).(let H34 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl)
1359 x (THead (Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to
1360 (\forall (P: Prop).P))) H30 x H33) in (let H35 \def (eq_ind_r T x1 (\lambda
1361 (t0: T).(pr2 c x t0)) H15 x H33) in (eq_ind T x (\lambda (t0: T).(sn3 c
1362 (THead (Flat Appl) t0 (THead (Bind b) t1 x0)))) (H34 (refl_equal T (THead
1363 (Flat Appl) x (THead (Bind b) t1 x0))) (sn3 c (THead (Flat Appl) x (THead
1364 (Bind b) t1 x0)))) x1 H33)))) (\lambda (H33: (((eq T x x1) \to (\forall (P:
1365 Prop).P)))).(H8 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H34: (eq T
1366 (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
1367 x0))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: T).(match e
1368 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
1369 (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
1370 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
1371 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
1372 \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
1373 lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
1374 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
1375 t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
1376 (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
1377 (THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1378 t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
1379 \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
1380 (lift (S O) O x1) x0) H34) in (let H36 \def (eq_ind_r T x1 (\lambda (t0:
1381 T).((eq T x t0) \to (\forall (P0: Prop).P0))) H33 x (lift_inj x x1 (S O) O
1382 H35)) in (let H37 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
1383 (lift_inj x x1 (S O) O H35)) in (H36 (refl_equal T x) P)))))) (pr3_flat
1384 (CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
1385 (Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
1386 (pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl) x1 x0
1387 (refl_equal T (THead (Flat Appl) (lift (S O) O x1) x0)))) H32))) x4 H29))))
1388 (\lambda (H29: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H8 (THead (Flat
1389 Appl) (lift (S O) O x1) x4) (\lambda (H30: (eq T (THead (Flat Appl) (lift (S
1390 O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P:
1391 Prop).(let H31 \def (f_equal T T (\lambda (e: T).(match e in T return
1392 (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
1393 \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
1394 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
1395 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
1396 \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
1397 lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
1398 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
1399 t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
1400 (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
1401 (THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1402 t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
1403 \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
1404 (lift (S O) O x1) x4) H30) in ((let H32 \def (f_equal T T (\lambda (e:
1405 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
1406 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
1407 Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H30) in
1408 (\lambda (H33: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H34 \def
1409 (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0)))
1410 H29 x0 H32) in (let H35 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T (THead
1411 (Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 (THead (Bind b)
1412 t1 t0))) \to (\forall (P0: Prop).P0))) H26 x0 H32) in (let H36 \def (eq_ind_r
1413 T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) H22 x0 H32) in (let
1414 H37 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead
1415 (Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to (\forall
1416 (P0: Prop).P0))) H35 x (lift_inj x x1 (S O) O H33)) in (let H38 \def
1417 (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O
1418 H33)) in (H34 (refl_equal T x0) P)))))))) H31)))) (pr3_flat (CHead c (Bind b)
1419 t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S
1420 O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15))
1421 x0 x4 H22 Appl) x1 x4 (refl_equal T (THead (Flat Appl) (lift (S O) O x1)
1422 x4)))) H28))) x3 H25)))) (\lambda (H25: (((eq T t1 x3) \to (\forall (P:
1423 Prop).P)))).(H2 x3 H25 H21 x4 x1 (sn3_cpr3_trans c t1 x3 H21 (Bind b) (THead
1424 (Flat Appl) (lift (S O) O x1) x4) (let H_x1 \def (term_dec x0 x4) in (let H26
1425 \def H_x1 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P))
1426 (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x4)) (\lambda
1427 (H27: (eq T x0 x4)).(let H28 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead
1428 c (Bind b) t1) x0 t0)) H22 x0 H27) in (eq_ind T x0 (\lambda (t0: T).(sn3
1429 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) t0))) (let H_x2
1430 \def (term_dec x x1) in (let H29 \def H_x2 in (or_ind (eq T x x1) ((eq T x
1431 x1) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl)
1432 (lift (S O) O x1) x0)) (\lambda (H30: (eq T x x1)).(let H31 \def (eq_ind_r T
1433 x1 (\lambda (t0: T).(pr2 c x t0)) H15 x H30) in (eq_ind T x (\lambda (t0:
1434 T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0)))
1435 (sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9)
1436 x1 H30))) (\lambda (H30: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9
1437 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H31: (eq T (THead (Flat
1438 Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
1439 x0))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e
1440 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
1441 (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
1442 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
1443 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
1444 \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
1445 lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
1446 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
1447 t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
1448 (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
1449 (THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1450 t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
1451 \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
1452 (lift (S O) O x1) x0) H31) in (let H33 \def (eq_ind_r T x1 (\lambda (t0:
1453 T).((eq T x t0) \to (\forall (P0: Prop).P0))) H30 x (lift_inj x x1 (S O) O
1454 H32)) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
1455 (lift_inj x x1 (S O) O H32)) in (H33 (refl_equal T x) P)))))) (pr3_flat
1456 (CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
1457 (Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
1458 (pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl)))
1459 H29))) x4 H27))) (\lambda (H27: (((eq T x0 x4) \to (\forall (P:
1460 Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x1) x4) (\lambda (H28: (eq T
1461 (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
1462 x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e
1463 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
1464 (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
1465 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
1466 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
1467 \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
1468 lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
1469 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
1470 t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
1471 (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
1472 (THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1473 t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
1474 \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
1475 (lift (S O) O x1) x4) H28) in ((let H30 \def (f_equal T T (\lambda (e:
1476 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
1477 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
1478 Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H28) in
1479 (\lambda (H31: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H32 \def
1480 (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0)))
1481 H27 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c
1482 (Bind b) t1) x0 t0)) H22 x0 H30) in (let H34 \def (eq_ind_r T x1 (\lambda
1483 (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H31)) in (H32 (refl_equal
1484 T x0) P)))))) H29)))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift
1485 (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c
1486 c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x4 H22 Appl))) H26))))))
1487 H24))) x2 H20))))))) H19)) (\lambda (H19: (pr3 (CHead c (Bind b) t1) x0 (lift
1488 (S O) O x2))).(sn3_gen_lift (CHead c (Bind b) t1) (THead (Flat Appl) x1 x2)
1489 (S O) O (eq_ind_r T (THead (Flat Appl) (lift (S O) O x1) (lift (S O) (s (Flat
1490 Appl) O) x2)) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) t0)) (sn3_pr3_trans
1491 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x0) (let H_x0 \def
1492 (term_dec x x1) in (let H20 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to
1493 (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S
1494 O) O x1) x0)) (\lambda (H21: (eq T x x1)).(let H22 \def (eq_ind_r T x1
1495 (\lambda (t0: T).(pr2 c x t0)) H15 x H21) in (eq_ind T x (\lambda (t0:
1496 T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0)))
1497 (sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9)
1498 x1 H21))) (\lambda (H21: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9
1499 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H22: (eq T (THead (Flat
1500 Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
1501 x0))).(\lambda (P: Prop).(let H23 \def (f_equal T T (\lambda (e: T).(match e
1502 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
1503 (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
1504 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
1505 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
1506 \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
1507 lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (TLRef _) \Rightarrow
1508 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
1509 t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
1510 (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
1511 (THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1512 t4))]) in lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (THead _ t0 _)
1513 \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
1514 (lift (S O) O x1) x0) H22) in (let H24 \def (eq_ind_r T x1 (\lambda (t0:
1515 T).((eq T x t0) \to (\forall (P0: Prop).P0))) H21 x (lift_inj x x1 (S O) O
1516 H23)) in (let H25 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
1517 (lift_inj x x1 (S O) O H23)) in (H24 (refl_equal T x) P)))))) (pr3_flat
1518 (CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
1519 (Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
1520 (pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl)))
1521 H20))) (THead (Flat Appl) (lift (S O) O x1) (lift (S O) O x2)) (pr3_thin_dx
1522 (CHead c (Bind b) t1) x0 (lift (S O) O x2) H19 (lift (S O) O x1) Appl)) (lift
1523 (S O) O (THead (Flat Appl) x1 x2)) (lift_head (Flat Appl) x1 x2 (S O) O)) c
1524 (drop_drop (Bind b) O c c (drop_refl c) t1))) H18))) t3 H14))))))) H13))
1525 (\lambda (H13: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1526 T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1
1527 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
1528 T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
1529 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda
1530 (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0:
1531 T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))))).(ex4_4_ind T T T T (\lambda
1532 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
1533 b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
1534 T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4))))))
1535 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x
1536 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4:
1537 T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) z1 t4)))))))
1538 (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
1539 T).(\lambda (H14: (eq T (THead (Bind b) t1 x0) (THead (Bind Abst) x1
1540 x2))).(\lambda (H15: (eq T t3 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c
1541 x x3)).(\lambda (H17: ((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind
1542 b0) u0) x2 x4))))).(let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T (THead
1543 (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) H10
1544 (THead (Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4)
1545 (\lambda (t0: T).(sn3 c t0)) (let H19 \def (f_equal T B (\lambda (e:
1546 T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
1547 (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
1548 (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
1549 b])])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in ((let H20
1550 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
1551 with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _)
1552 \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in
1553 ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
1554 T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
1555 t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14)
1556 in (\lambda (_: (eq T t1 x1)).(\lambda (H23: (eq B b Abst)).(let H24 \def
1557 (eq_ind_r T x2 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead
1558 c (Bind b0) u0) t0 x4)))) H17 x0 H21) in (let H25 \def (eq_ind B b (\lambda
1559 (b0: B).((eq T (THead (Flat Appl) x (THead (Bind b0) t1 x0)) (THead (Bind
1560 Abbr) x3 x4)) \to (\forall (P: Prop).P))) H18 Abst H23) in (let H26 \def
1561 (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T (THead (Flat Appl)
1562 (lift (S O) O x) x0) t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind
1563 b0) t1) (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (sn3 (CHead c (Bind
1564 b0) t1) t4))))) H9 Abst H23) in (let H27 \def (eq_ind B b (\lambda (b0:
1565 B).(\forall (t4: T).((((eq T (THead (Flat Appl) (lift (S O) O x) x0) t4) \to
1566 (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) (THead (Flat Appl)
1567 (lift (S O) O x) x0) t4) \to (\forall (x5: T).(\forall (x6: T).((eq T t4
1568 (THead (Flat Appl) (lift (S O) O x5) x6)) \to (sn3 c (THead (Flat Appl) x5
1569 (THead (Bind b0) t1 x6)))))))))) H8 Abst H23) in (let H28 \def (eq_ind B b
1570 (\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P)))
1571 \to ((pr3 c t1 t4) \to (\forall (t0: T).(\forall (v0: T).((sn3 (CHead c (Bind
1572 b0) t4) (THead (Flat Appl) (lift (S O) O v0) t0)) \to (sn3 c (THead (Flat
1573 Appl) v0 (THead (Bind b0) t4 t0)))))))))) H2 Abst H23) in (let H29 \def
1574 (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H Abst H23) in (let H30
1575 \def (match (H29 (refl_equal B Abst)) in False return (\lambda (_:
1576 False).(sn3 c (THead (Bind Abbr) x3 x4))) with []) in H30)))))))))) H20))
1577 H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 B T T T T T (\lambda (b0:
1578 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1579 (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
1580 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b)
1581 t1 x0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_:
1582 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
1583 t3 (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
1584 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1585 T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
1586 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
1587 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
1588 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1
1589 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda
1590 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0
1591 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1592 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind
1593 b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1594 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b0) y2 (THead
1595 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1596 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x
1597 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1598 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b0:
1599 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1600 (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2))))))) (sn3 c t3) (\lambda (x1:
1601 B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
1602 T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15: (eq T
1603 (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T t3
1604 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
1605 (H17: (pr2 c x x5)).(\lambda (H18: (pr2 c x2 x6)).(\lambda (H19: (pr2 (CHead
1606 c (Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t0: T).((eq T
1607 (THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P)))
1608 H10 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) H16) in
1609 (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))
1610 (\lambda (t0: T).(sn3 c t0)) (let H21 \def (f_equal T B (\lambda (e:
1611 T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
1612 (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
1613 (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
1614 b])])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in ((let H22 \def
1615 (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
1616 [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _)
1617 \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in
1618 ((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
1619 T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
1620 t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in
1621 (\lambda (H24: (eq T t1 x2)).(\lambda (H25: (eq B b x1)).(let H26 \def
1622 (eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H19 x0
1623 H23) in (let H27 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H18 t1
1624 H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b0: B).(pr2 (CHead c (Bind b0)
1625 x6) x0 x4)) H26 b H25) in (eq_ind B b (\lambda (b0: B).(sn3 c (THead (Bind
1626 b0) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (sn3_pr3_trans c (THead
1627 (Bind b) t1 (THead (Flat Appl) (lift (S O) O x5) x4)) (sn3_bind b c t1
1628 (sn3_sing c t1 H1) (THead (Flat Appl) (lift (S O) O x5) x4) (let H_x \def
1629 (term_dec x x5) in (let H29 \def H_x in (or_ind (eq T x x5) ((eq T x x5) \to
1630 (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S
1631 O) O x5) x4)) (\lambda (H30: (eq T x x5)).(let H31 \def (eq_ind_r T x5
1632 (\lambda (t0: T).(pr2 c x t0)) H17 x H30) in (eq_ind T x (\lambda (t0:
1633 T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x4))) (let
1634 H_x0 \def (term_dec x0 x4) in (let H32 \def H_x0 in (or_ind (eq T x0 x4) ((eq
1635 T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat
1636 Appl) (lift (S O) O x) x4)) (\lambda (H33: (eq T x0 x4)).(let H34 \def
1637 (eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0
1638 H33) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat
1639 Appl) (lift (S O) O x) t0))) (sn3_sing (CHead c (Bind b) t1) (THead (Flat
1640 Appl) (lift (S O) O x) x0) H9) x4 H33))) (\lambda (H33: (((eq T x0 x4) \to
1641 (\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x) x4) (\lambda
1642 (H34: (eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
1643 (S O) O x) x4))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e:
1644 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
1645 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
1646 Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x) x4) H34) in
1647 (let H36 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0:
1648 Prop).P0))) H33 x0 H35) in (let H37 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2
1649 (CHead c (Bind b) x6) x0 t0)) H28 x0 H35) in (H36 (refl_equal T x0) P))))))
1650 (pr3_pr3_pr3_t c t1 x6 (pr3_pr2 c t1 x6 H27) (THead (Flat Appl) (lift (S O) O
1651 x) x0) (THead (Flat Appl) (lift (S O) O x) x4) (Bind b) (pr3_pr2 (CHead c
1652 (Bind b) x6) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
1653 (S O) O x) x4) (pr2_thin_dx (CHead c (Bind b) x6) x0 x4 H28 (lift (S O) O x)
1654 Appl))))) H32))) x5 H30))) (\lambda (H30: (((eq T x x5) \to (\forall (P:
1655 Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x5) x4) (\lambda (H31: (eq T
1656 (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5)
1657 x4))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e
1658 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
1659 (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
1660 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
1661 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
1662 \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
1663 lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (TLRef _) \Rightarrow
1664 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
1665 t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
1666 (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
1667 (THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1668 t4))]) in lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (THead _ t0 _)
1669 \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
1670 (lift (S O) O x5) x4) H31) in ((let H33 \def (f_equal T T (\lambda (e:
1671 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
1672 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
1673 Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) x4) H31) in
1674 (\lambda (H34: (eq T (lift (S O) O x) (lift (S O) O x5))).(let H35 \def
1675 (eq_ind_r T x5 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
1676 H30 x (lift_inj x x5 (S O) O H34)) in (let H36 \def (eq_ind_r T x5 (\lambda
1677 (t0: T).(pr2 c x t0)) H17 x (lift_inj x x5 (S O) O H34)) in (let H37 \def
1678 (eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0
1679 H33) in (H35 (refl_equal T x) P)))))) H32)))) (pr3_pr3_pr3_t c t1 x6 (pr3_pr2
1680 c t1 x6 H27) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
1681 (S O) O x5) x4) (Bind b) (pr3_flat (CHead c (Bind b) x6) (lift (S O) O x)
1682 (lift (S O) O x5) (pr3_lift (CHead c (Bind b) x6) c (S O) O (drop_drop (Bind
1683 b) O c c (drop_refl c) x6) x x5 (pr3_pr2 c x x5 H17)) x0 x4 (pr3_pr2 (CHead c
1684 (Bind b) x6) x0 x4 H28) Appl)))) H29)))) (THead (Bind b) x6 (THead (Flat
1685 Appl) (lift (S O) O x5) x4)) (pr3_pr2 c (THead (Bind b) t1 (THead (Flat Appl)
1686 (lift (S O) O x5) x4)) (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O
1687 x5) x4)) (pr2_head_1 c t1 x6 H27 (Bind b) (THead (Flat Appl) (lift (S O) O
1688 x5) x4)))) x1 H25))))))) H22)) H21)) t3 H16)))))))))))))) H13))
1689 H12)))))))))))))) y H4))))) H3))))))) u H0))))).
1691 theorem sn3_appl_appl:
1692 \forall (v1: T).(\forall (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in
1693 (\forall (c: C).((sn3 c u1) \to (\forall (v2: T).((sn3 c v2) \to (((\forall
1694 (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to
1695 (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2
1698 \lambda (v1: T).(\lambda (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in
1699 (\lambda (c: C).(\lambda (H: (sn3 c (THead (Flat Appl) v1 t1))).(insert_eq T
1700 (THead (Flat Appl) v1 t1) (\lambda (t: T).(sn3 c t)) (\lambda (t: T).(\forall
1701 (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2)
1702 \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to
1703 (sn3 c (THead (Flat Appl) v2 t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c
1704 y)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Flat Appl) v1 t)) \to
1705 (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso
1706 y u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2))))))
1707 \to (sn3 c (THead (Flat Appl) v2 y))))))) (unintro T v1 (\lambda (t:
1708 T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to (\forall (v2:
1709 T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso y u2) \to
1710 (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c
1711 (THead (Flat Appl) v2 y)))))))) (sn3_ind c (\lambda (t: T).(\forall (x:
1712 T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (v2:
1713 T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to
1714 (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c
1715 (THead (Flat Appl) v2 t))))))))) (\lambda (t2: T).(\lambda (H1: ((\forall
1716 (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to
1717 (sn3 c t3)))))).(\lambda (H2: ((\forall (t3: T).((((eq T t2 t3) \to (\forall
1718 (P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x: T).(\forall (x0: T).((eq T
1719 t3 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall
1720 (u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to
1721 (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2
1722 t3))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t2
1723 (THead (Flat Appl) x x0))).(\lambda (v2: T).(\lambda (H4: (sn3 c
1724 v2)).(sn3_ind c (\lambda (t: T).(((\forall (u2: T).((pr3 c t2 u2) \to ((((iso
1725 t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t u2))))))
1726 \to (sn3 c (THead (Flat Appl) t t2)))) (\lambda (t0: T).(\lambda (H5:
1727 ((\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0
1728 t3) \to (sn3 c t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t0 t3) \to
1729 (\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to (((\forall (u2: T).((pr3 c t2
1730 u2) \to ((((iso t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat
1731 Appl) t3 u2)))))) \to (sn3 c (THead (Flat Appl) t3 t2)))))))).(\lambda (H7:
1732 ((\forall (u2: T).((pr3 c t2 u2) \to ((((iso t2 u2) \to (\forall (P:
1733 Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2))))))).(let H8 \def (eq_ind T
1734 t2 (\lambda (t: T).(\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to
1735 (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H7 (THead
1736 (Flat Appl) x x0) H3) in (let H9 \def (eq_ind T t2 (\lambda (t: T).(\forall
1737 (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to
1738 (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to (\forall (P:
1739 Prop).P))) \to (sn3 c (THead (Flat Appl) t3 u2)))))) \to (sn3 c (THead (Flat
1740 Appl) t3 t))))))) H6 (THead (Flat Appl) x x0) H3) in (let H10 \def (eq_ind T
1741 t2 (\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P:
1742 Prop).P))) \to ((pr3 c t t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3
1743 (THead (Flat Appl) x1 x2)) \to (\forall (v3: T).((sn3 c v3) \to (((\forall
1744 (u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to
1745 (sn3 c (THead (Flat Appl) v3 u2)))))) \to (sn3 c (THead (Flat Appl) v3
1746 t3)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H11 \def (eq_ind T t2
1747 (\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P: Prop).P)))
1748 \to ((pr3 c t t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in
1749 (eq_ind_r T (THead (Flat Appl) x x0) (\lambda (t: T).(sn3 c (THead (Flat
1750 Appl) t0 t))) (sn3_pr2_intro c (THead (Flat Appl) t0 (THead (Flat Appl) x
1751 x0)) (\lambda (t3: T).(\lambda (H12: (((eq T (THead (Flat Appl) t0 (THead
1752 (Flat Appl) x x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H13: (pr2 c
1753 (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t3)).(let H14 \def
1754 (pr2_gen_appl c t0 (THead (Flat Appl) x x0) t3 H13) in (or3_ind (ex3_2 T T
1755 (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
1756 (\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda
1757 (t4: T).(pr2 c (THead (Flat Appl) x x0) t4)))) (ex4_4 T T T T (\lambda (y1:
1758 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl)
1759 x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
1760 (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
1761 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2)))))
1762 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
1763 (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T
1764 T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1765 (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
1766 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
1767 T (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
1768 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
1769 (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1770 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1771 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_:
1772 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1773 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1774 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
1775 y2) z1 z2)))))))) (sn3 c t3) (\lambda (H15: (ex3_2 T T (\lambda (u2:
1776 T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
1777 T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
1778 (THead (Flat Appl) x x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
1779 (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_:
1780 T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Flat Appl)
1781 x x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H16: (eq T
1782 t3 (THead (Flat Appl) x1 x2))).(\lambda (H17: (pr2 c t0 x1)).(\lambda (H18:
1783 (pr2 c (THead (Flat Appl) x x0) x2)).(let H19 \def (eq_ind T t3 (\lambda (t:
1784 T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P:
1785 Prop).P))) H12 (THead (Flat Appl) x1 x2) H16) in (eq_ind_r T (THead (Flat
1786 Appl) x1 x2) (\lambda (t: T).(sn3 c t)) (let H20 \def (pr2_gen_appl c x x0 x2
1787 H18) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead
1788 (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
1789 (\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4)))) (ex4_4 T T T T (\lambda
1790 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead
1791 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1792 T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
1793 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
1794 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
1795 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T
1796 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1797 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1798 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
1799 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1800 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind
1801 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
1802 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1803 (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
1804 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
1805 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
1806 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c
1807 (THead (Flat Appl) x1 x2)) (\lambda (H21: (ex3_2 T T (\lambda (u2:
1808 T).(\lambda (t4: T).(eq T x2 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
1809 T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c x0
1810 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead
1811 (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
1812 (\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4))) (sn3 c (THead (Flat Appl) x1
1813 x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H22: (eq T x2 (THead (Flat
1814 Appl) x3 x4))).(\lambda (H23: (pr2 c x x3)).(\lambda (H24: (pr2 c x0
1815 x4)).(let H25 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0
1816 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P:
1817 Prop).P))) H19 (THead (Flat Appl) x3 x4) H22) in (eq_ind_r T (THead (Flat
1818 Appl) x3 x4) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H_x \def
1819 (term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) in (let H26
1820 \def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))
1821 ((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) \to (\forall (P:
1822 Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 x4))) (\lambda
1823 (H27: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))).(let H28
1824 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
1825 with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _)
1826 \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27) in
1827 ((let H29 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
1828 T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
1829 t) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27)
1830 in (\lambda (H30: (eq T x x3)).(let H31 \def (eq_ind_r T x4 (\lambda (t:
1831 T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl)
1832 x1 (THead (Flat Appl) x3 t))) \to (\forall (P: Prop).P))) H25 x0 H29) in (let
1833 H32 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) H24 x0 H29) in (eq_ind
1834 T x0 (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 t))))
1835 (let H33 \def (eq_ind_r T x3 (\lambda (t: T).((eq T (THead (Flat Appl) t0
1836 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 (THead (Flat Appl) t x0)))
1837 \to (\forall (P: Prop).P))) H31 x H30) in (let H34 \def (eq_ind_r T x3
1838 (\lambda (t: T).(pr2 c x t)) H23 x H30) in (eq_ind T x (\lambda (t: T).(sn3 c
1839 (THead (Flat Appl) x1 (THead (Flat Appl) t x0)))) (let H_x0 \def (term_dec t0
1840 x1) in (let H35 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to (\forall
1841 (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x x0)))
1842 (\lambda (H36: (eq T t0 x1)).(let H37 \def (eq_ind_r T x1 (\lambda (t:
1843 T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl)
1844 t (THead (Flat Appl) x x0))) \to (\forall (P: Prop).P))) H33 t0 H36) in (let
1845 H38 \def (eq_ind_r T x1 (\lambda (t: T).(pr2 c t0 t)) H17 t0 H36) in (eq_ind
1846 T t0 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (THead (Flat Appl) x x0))))
1847 (H37 (refl_equal T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))) (sn3 c
1848 (THead (Flat Appl) t0 (THead (Flat Appl) x x0)))) x1 H36)))) (\lambda (H36:
1849 (((eq T t0 x1) \to (\forall (P: Prop).P)))).(H9 x1 H36 (pr3_pr2 c t0 x1 H17)
1850 (\lambda (u2: T).(\lambda (H37: (pr3 c (THead (Flat Appl) x x0) u2)).(\lambda
1851 (H38: (((iso (THead (Flat Appl) x x0) u2) \to (\forall (P:
1852 Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H8 u2 H37 H38) (THead
1853 (Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2) (THead (Flat Appl) x1
1854 u2) (pr2_head_1 c t0 x1 H17 (Flat Appl) u2)))))))) H35))) x3 H30))) x4
1855 H29))))) H28))) (\lambda (H27: (((eq T (THead (Flat Appl) x x0) (THead (Flat
1856 Appl) x3 x4)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat Appl) x3 x4) H27
1857 (pr3_flat c x x3 (pr3_pr2 c x x3 H23) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x3 x4
1858 (refl_equal T (THead (Flat Appl) x3 x4)) x1 (sn3_pr3_trans c t0 (sn3_sing c
1859 t0 H5) x1 (pr3_pr2 c t0 x1 H17)) (\lambda (u2: T).(\lambda (H28: (pr3 c
1860 (THead (Flat Appl) x3 x4) u2)).(\lambda (H29: (((iso (THead (Flat Appl) x3
1861 x4) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0
1862 u2) (H8 u2 (pr3_sing c (THead (Flat Appl) x x4) (THead (Flat Appl) x x0)
1863 (pr2_thin_dx c x0 x4 H24 x Appl) u2 (pr3_sing c (THead (Flat Appl) x3 x4)
1864 (THead (Flat Appl) x x4) (pr2_head_1 c x x3 H23 (Flat Appl) x4) u2 H28))
1865 (\lambda (H30: (iso (THead (Flat Appl) x x0) u2)).(\lambda (P: Prop).(H29
1866 (iso_trans (THead (Flat Appl) x3 x4) (THead (Flat Appl) x x0) (iso_head x3 x
1867 x4 x0 (Flat Appl)) u2 H30) P)))) (THead (Flat Appl) x1 u2) (pr3_pr2 c (THead
1868 (Flat Appl) t0 u2) (THead (Flat Appl) x1 u2) (pr2_head_1 c t0 x1 H17 (Flat
1869 Appl) u2)))))))) H26))) x2 H22))))))) H21)) (\lambda (H21: (ex4_4 T T T T
1870 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
1871 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1872 T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
1873 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
1874 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
1875 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T
1876 T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
1877 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1878 T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
1879 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
1880 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
1881 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c (THead (Flat
1882 Appl) x1 x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda
1883 (x6: T).(\lambda (H22: (eq T x0 (THead (Bind Abst) x3 x4))).(\lambda (H23:
1884 (eq T x2 (THead (Bind Abbr) x5 x6))).(\lambda (H24: (pr2 c x x5)).(\lambda
1885 (H25: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x4
1886 x6))))).(let H26 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl)
1887 t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P:
1888 Prop).P))) H19 (THead (Bind Abbr) x5 x6) H23) in (eq_ind_r T (THead (Bind
1889 Abbr) x5 x6) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H27 \def
1890 (eq_ind T x0 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl)
1891 x t)) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6))) \to (\forall (P:
1892 Prop).P))) H26 (THead (Bind Abst) x3 x4) H22) in (let H28 \def (eq_ind T x0
1893 (\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
1894 (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c
1895 t4))))) H11 (THead (Bind Abst) x3 x4) H22) in (let H29 \def (eq_ind T x0
1896 (\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
1897 (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall
1898 (x7: T).(\forall (x8: T).((eq T t4 (THead (Flat Appl) x7 x8)) \to (\forall
1899 (v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2)
1900 \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to
1901 (sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind Abst) x3 x4)
1902 H22) in (let H30 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c
1903 (THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to
1904 (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead
1905 (Bind Abst) x3 x4) H22) in (let H31 \def (eq_ind T x0 (\lambda (t:
1906 T).(\forall (t4: T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c
1907 t0 t4) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso
1908 (THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead
1909 (Flat Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x
1910 t)))))))) H9 (THead (Bind Abst) x3 x4) H22) in (sn3_pr3_trans c (THead (Flat
1911 Appl) t0 (THead (Bind Abbr) x5 x6)) (H30 (THead (Bind Abbr) x5 x6) (pr3_sing
1912 c (THead (Bind Abbr) x x4) (THead (Flat Appl) x (THead (Bind Abst) x3 x4))
1913 (pr2_free c (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind
1914 Abbr) x x4) (pr0_beta x3 x x (pr0_refl x) x4 x4 (pr0_refl x4))) (THead (Bind
1915 Abbr) x5 x6) (pr3_head_12 c x x5 (pr3_pr2 c x x5 H24) (Bind Abbr) x4 x6
1916 (pr3_pr2 (CHead c (Bind Abbr) x5) x4 x6 (H25 Abbr x5)))) (\lambda (H32: (iso
1917 (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind Abbr) x5
1918 x6))).(\lambda (P: Prop).(let H33 \def (match H32 in iso return (\lambda (t:
1919 T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x
1920 (THead (Bind Abst) x3 x4))) \to ((eq T t4 (THead (Bind Abbr) x5 x6)) \to
1921 P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H33: (eq T (TSort n1)
1922 (THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H34: (eq T (TSort
1923 n2) (THead (Bind Abbr) x5 x6))).((let H35 \def (eq_ind T (TSort n1) (\lambda
1924 (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
1925 True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
1926 (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T
1927 (TSort n2) (THead (Bind Abbr) x5 x6)) \to P) H35)) H34))) | (iso_lref i1 i2)
1928 \Rightarrow (\lambda (H33: (eq T (TLRef i1) (THead (Flat Appl) x (THead (Bind
1929 Abst) x3 x4)))).(\lambda (H34: (eq T (TLRef i2) (THead (Bind Abbr) x5
1930 x6))).((let H35 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T
1931 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
1932 \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x
1933 (THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T (TLRef i2) (THead (Bind
1934 Abbr) x5 x6)) \to P) H35)) H34))) | (iso_head v4 v5 t4 t5 k) \Rightarrow
1935 (\lambda (H33: (eq T (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst)
1936 x3 x4)))).(\lambda (H34: (eq T (THead k v5 t5) (THead (Bind Abbr) x5
1937 x6))).((let H35 \def (f_equal T T (\lambda (e: T).(match e in T return
1938 (\lambda (_: T).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4
1939 | (THead _ _ t) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead
1940 (Bind Abst) x3 x4)) H33) in ((let H36 \def (f_equal T T (\lambda (e:
1941 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v4 |
1942 (TLRef _) \Rightarrow v4 | (THead _ t _) \Rightarrow t])) (THead k v4 t4)
1943 (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in ((let H37 \def
1944 (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
1945 [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
1946 \Rightarrow k0])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) x3
1947 x4)) H33) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x) \to ((eq T
1948 t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead k0 v5 t5) (THead (Bind Abbr)
1949 x5 x6)) \to P)))) (\lambda (H38: (eq T v4 x)).(eq_ind T x (\lambda (_:
1950 T).((eq T t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead (Flat Appl) v5 t5)
1951 (THead (Bind Abbr) x5 x6)) \to P))) (\lambda (H39: (eq T t4 (THead (Bind
1952 Abst) x3 x4))).(eq_ind T (THead (Bind Abst) x3 x4) (\lambda (_: T).((eq T
1953 (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6)) \to P)) (\lambda (H40:
1954 (eq T (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6))).(let H41 \def
1955 (eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return
1956 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
1957 \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
1958 (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
1959 True])])) I (THead (Bind Abbr) x5 x6) H40) in (False_ind P H41))) t4 (sym_eq
1960 T t4 (THead (Bind Abst) x3 x4) H39))) v4 (sym_eq T v4 x H38))) k (sym_eq K k
1961 (Flat Appl) H37))) H36)) H35)) H34)))]) in (H33 (refl_equal T (THead (Flat
1962 Appl) x (THead (Bind Abst) x3 x4))) (refl_equal T (THead (Bind Abbr) x5
1963 x6))))))) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6)) (pr3_pr2 c (THead
1964 (Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (THead (Flat Appl) x1 (THead (Bind
1965 Abbr) x5 x6)) (pr2_head_1 c t0 x1 H17 (Flat Appl) (THead (Bind Abbr) x5
1966 x6))))))))) x2 H23)))))))))) H21)) (\lambda (H21: (ex6_6 B T T T T T (\lambda
1967 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1968 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1969 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
1970 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1971 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind
1972 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
1973 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1974 (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
1975 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
1976 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
1977 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind
1978 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1979 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1980 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1981 (_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
1982 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
1983 x2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
1984 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1985 T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
1986 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
1987 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
1988 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
1989 (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3: B).(\lambda (x4: T).(\lambda
1990 (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H22:
1991 (not (eq B x3 Abst))).(\lambda (H23: (eq T x0 (THead (Bind x3) x4
1992 x5))).(\lambda (H24: (eq T x2 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S
1993 O) O x7) x6)))).(\lambda (H25: (pr2 c x x7)).(\lambda (H26: (pr2 c x4
1994 x8)).(\lambda (H27: (pr2 (CHead c (Bind x3) x8) x5 x6)).(let H28 \def (eq_ind
1995 T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))
1996 (THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H19 (THead (Bind x3) x8
1997 (THead (Flat Appl) (lift (S O) O x7) x6)) H24) in (eq_ind_r T (THead (Bind
1998 x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (\lambda (t: T).(sn3 c
1999 (THead (Flat Appl) x1 t))) (let H29 \def (eq_ind T x0 (\lambda (t: T).((eq T
2000 (THead (Flat Appl) t0 (THead (Flat Appl) x t)) (THead (Flat Appl) x1 (THead
2001 (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))) \to (\forall (P:
2002 Prop).P))) H28 (THead (Bind x3) x4 x5) H23) in (let H30 \def (eq_ind T x0
2003 (\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
2004 (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c
2005 t4))))) H11 (THead (Bind x3) x4 x5) H23) in (let H31 \def (eq_ind T x0
2006 (\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
2007 (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall
2008 (x9: T).(\forall (x10: T).((eq T t4 (THead (Flat Appl) x9 x10)) \to (\forall
2009 (v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2)
2010 \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to
2011 (sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind x3) x4 x5) H23)
2012 in (let H32 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c (THead
2013 (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P:
2014 Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead (Bind x3) x4
2015 x5) H23) in (let H33 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4:
2016 T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t4) \to
2017 (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead
2018 (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat
2019 Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x
2020 t)))))))) H9 (THead (Bind x3) x4 x5) H23) in (sn3_pr3_trans c (THead (Flat
2021 Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (H32
2022 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (pr3_sing c
2023 (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) x5)) (THead (Flat
2024 Appl) x (THead (Bind x3) x4 x5)) (pr2_free c (THead (Flat Appl) x (THead
2025 (Bind x3) x4 x5)) (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x)
2026 x5)) (pr0_upsilon x3 H22 x x (pr0_refl x) x4 x4 (pr0_refl x4) x5 x5 (pr0_refl
2027 x5))) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))
2028 (pr3_head_12 c x4 x8 (pr3_pr2 c x4 x8 H26) (Bind x3) (THead (Flat Appl) (lift
2029 (S O) O x) x5) (THead (Flat Appl) (lift (S O) O x7) x6) (pr3_head_12 (CHead c
2030 (Bind x3) x8) (lift (S O) O x) (lift (S O) O x7) (pr3_lift (CHead c (Bind x3)
2031 x8) c (S O) O (drop_drop (Bind x3) O c c (drop_refl c) x8) x x7 (pr3_pr2 c x
2032 x7 H25)) (Flat Appl) x5 x6 (pr3_pr2 (CHead (CHead c (Bind x3) x8) (Flat Appl)
2033 (lift (S O) O x7)) x5 x6 (pr2_cflat (CHead c (Bind x3) x8) x5 x6 H27 Appl
2034 (lift (S O) O x7)))))) (\lambda (H34: (iso (THead (Flat Appl) x (THead (Bind
2035 x3) x4 x5)) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
2036 x6)))).(\lambda (P: Prop).(let H35 \def (match H34 in iso return (\lambda (t:
2037 T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x
2038 (THead (Bind x3) x4 x5))) \to ((eq T t4 (THead (Bind x3) x8 (THead (Flat
2039 Appl) (lift (S O) O x7) x6))) \to P))))) with [(iso_sort n1 n2) \Rightarrow
2040 (\lambda (H35: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind x3) x4
2041 x5)))).(\lambda (H36: (eq T (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl)
2042 (lift (S O) O x7) x6)))).((let H37 \def (eq_ind T (TSort n1) (\lambda (e:
2043 T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
2044 True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
2045 (THead (Flat Appl) x (THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T
2046 (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to
2047 P) H37)) H36))) | (iso_lref i1 i2) \Rightarrow (\lambda (H35: (eq T (TLRef
2048 i1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T
2049 (TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
2050 x6)))).((let H37 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T
2051 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
2052 \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x
2053 (THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T (TLRef i2) (THead (Bind
2054 x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H37)) H36))) |
2055 (iso_head v4 v5 t4 t5 k) \Rightarrow (\lambda (H35: (eq T (THead k v4 t4)
2056 (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T (THead k
2057 v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let
2058 H37 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
2059 with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t)
2060 \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4
2061 x5)) H35) in ((let H38 \def (f_equal T T (\lambda (e: T).(match e in T return
2062 (\lambda (_: T).T) with [(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4
2063 | (THead _ t _) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead
2064 (Bind x3) x4 x5)) H35) in ((let H39 \def (f_equal T K (\lambda (e: T).(match
2065 e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
2066 \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k v4 t4) (THead (Flat
2067 Appl) x (THead (Bind x3) x4 x5)) H35) in (eq_ind K (Flat Appl) (\lambda (k0:
2068 K).((eq T v4 x) \to ((eq T t4 (THead (Bind x3) x4 x5)) \to ((eq T (THead k0
2069 v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to
2070 P)))) (\lambda (H40: (eq T v4 x)).(eq_ind T x (\lambda (_: T).((eq T t4
2071 (THead (Bind x3) x4 x5)) \to ((eq T (THead (Flat Appl) v5 t5) (THead (Bind
2072 x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P))) (\lambda (H41: (eq
2073 T t4 (THead (Bind x3) x4 x5))).(eq_ind T (THead (Bind x3) x4 x5) (\lambda (_:
2074 T).((eq T (THead (Flat Appl) v5 t5) (THead (Bind x3) x8 (THead (Flat Appl)
2075 (lift (S O) O x7) x6))) \to P)) (\lambda (H42: (eq T (THead (Flat Appl) v5
2076 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(let H43
2077 \def (eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return
2078 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
2079 \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
2080 (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
2081 True])])) I (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))
2082 H42) in (False_ind P H43))) t4 (sym_eq T t4 (THead (Bind x3) x4 x5) H41))) v4
2083 (sym_eq T v4 x H40))) k (sym_eq K k (Flat Appl) H39))) H38)) H37)) H36)))])
2084 in (H35 (refl_equal T (THead (Flat Appl) x (THead (Bind x3) x4 x5)))
2085 (refl_equal T (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
2086 x6)))))))) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift
2087 (S O) O x7) x6))) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x3) x8 (THead
2088 (Flat Appl) (lift (S O) O x7) x6))) (THead (Flat Appl) x1 (THead (Bind x3) x8
2089 (THead (Flat Appl) (lift (S O) O x7) x6))) (pr2_head_1 c t0 x1 H17 (Flat
2090 Appl) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))))))))))
2091 x2 H24)))))))))))))) H21)) H20)) t3 H16))))))) H15)) (\lambda (H15: (ex4_4 T
2092 T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
2093 (THead (Flat Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2094 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
2095 Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2096 (_: T).(pr2 c t0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
2097 T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
2098 z1 t4))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
2099 (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind Abst) y1
2100 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
2101 T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
2102 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) (\lambda (_:
2103 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
2104 (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c t3) (\lambda (x1:
2105 T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H16: (eq T
2106 (THead (Flat Appl) x x0) (THead (Bind Abst) x1 x2))).(\lambda (H17: (eq T t3
2107 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c t0 x3)).(\lambda (_:
2108 ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let
2109 H20 \def (eq_ind T t3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead
2110 (Flat Appl) x x0)) t) \to (\forall (P: Prop).P))) H12 (THead (Bind Abbr) x3
2111 x4) H17) in (eq_ind_r T (THead (Bind Abbr) x3 x4) (\lambda (t: T).(sn3 c t))
2112 (let H21 \def (eq_ind T (THead (Flat Appl) x x0) (\lambda (ee: T).(match ee
2113 in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
2114 _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
2115 (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
2116 True])])) I (THead (Bind Abst) x1 x2) H16) in (False_ind (sn3 c (THead (Bind
2117 Abbr) x3 x4)) H21)) t3 H17)))))))))) H15)) (\lambda (H15: (ex6_6 B T T T T T
2118 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2119 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2120 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2121 (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2122 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2123 (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2124 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2125 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_:
2126 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2127 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2128 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
2129 y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_:
2130 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
2131 b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2132 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead
2133 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
2134 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
2135 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
2136 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2137 (_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
2138 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
2139 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
2140 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3)
2141 (\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda
2142 (x5: T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H17:
2143 (eq T (THead (Flat Appl) x x0) (THead (Bind x1) x2 x3))).(\lambda (H18: (eq T
2144 t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
2145 (_: (pr2 c t0 x5)).(\lambda (_: (pr2 c x2 x6)).(\lambda (_: (pr2 (CHead c
2146 (Bind x1) x6) x3 x4)).(let H22 \def (eq_ind T t3 (\lambda (t: T).((eq T
2147 (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P:
2148 Prop).P))) H12 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))
2149 H18) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5)
2150 x4)) (\lambda (t: T).(sn3 c t)) (let H23 \def (eq_ind T (THead (Flat Appl) x
2151 x0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
2152 _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
2153 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
2154 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x1) x2 x3)
2155 H17) in (False_ind (sn3 c (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O)
2156 O x5) x4))) H23)) t3 H18)))))))))))))) H15)) H14)))))) t2 H3))))))))) v2
2157 H4))))))))) y H0))))) H))))).
2159 theorem sn3_appl_beta:
2160 \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((sn3 c
2161 (THead (Flat Appl) u (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w)
2162 \to (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind Abst) w
2165 \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
2166 (sn3 c (THead (Flat Appl) u (THead (Bind Abbr) v t)))).(\lambda (w:
2167 T).(\lambda (H0: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THead (Bind
2168 Abbr) v t) H) in (let H1 \def H_x in (land_ind (sn3 c u) (sn3 c (THead (Bind
2169 Abbr) v t)) (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind
2170 Abst) w t)))) (\lambda (H2: (sn3 c u)).(\lambda (H3: (sn3 c (THead (Bind
2171 Abbr) v t))).(sn3_appl_appl v (THead (Bind Abst) w t) c (sn3_beta c v t H3 w
2172 H0) u H2 (\lambda (u2: T).(\lambda (H4: (pr3 c (THead (Flat Appl) v (THead
2173 (Bind Abst) w t)) u2)).(\lambda (H5: (((iso (THead (Flat Appl) v (THead (Bind
2174 Abst) w t)) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat
2175 Appl) u (THead (Bind Abbr) v t)) H (THead (Flat Appl) u u2) (pr3_thin_dx c
2176 (THead (Bind Abbr) v t) u2 (pr3_iso_beta v w t c u2 H4 H5) u Appl))))))))
2179 theorem sn3_appl_appls:
2180 \forall (v1: T).(\forall (t1: T).(\forall (vs: TList).(let u1 \def (THeads
2181 (Flat Appl) (TCons v1 vs) t1) in (\forall (c: C).((sn3 c u1) \to (\forall
2182 (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2)
2183 \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to
2184 (sn3 c (THead (Flat Appl) v2 u1))))))))))
2186 \lambda (v1: T).(\lambda (t1: T).(\lambda (vs: TList).(let u1 \def (THeads
2187 (Flat Appl) (TCons v1 vs) t1) in (\lambda (c: C).(\lambda (H: (sn3 c (THead
2188 (Flat Appl) v1 (THeads (Flat Appl) vs t1)))).(\lambda (v2: T).(\lambda (H0:
2189 (sn3 c v2)).(\lambda (H1: ((\forall (u2: T).((pr3 c (THead (Flat Appl) v1
2190 (THeads (Flat Appl) vs t1)) u2) \to ((((iso (THead (Flat Appl) v1 (THeads
2191 (Flat Appl) vs t1)) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat
2192 Appl) v2 u2))))))).(sn3_appl_appl v1 (THeads (Flat Appl) vs t1) c H v2 H0
2195 theorem sn3_appls_lref:
2196 \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (us:
2197 TList).((sns3 c us) \to (sn3 c (THeads (Flat Appl) us (TLRef i)))))))
2199 \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda
2200 (us: TList).(TList_ind (\lambda (t: TList).((sns3 c t) \to (sn3 c (THeads
2201 (Flat Appl) t (TLRef i))))) (\lambda (_: True).(sn3_nf2 c (TLRef i) H))
2202 (\lambda (t: T).(\lambda (t0: TList).(TList_ind (\lambda (t1: TList).((((sns3
2203 c t1) \to (sn3 c (THeads (Flat Appl) t1 (TLRef i))))) \to ((land (sn3 c t)
2204 (sns3 c t1)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (TLRef
2205 i))))))) (\lambda (_: (((sns3 c TNil) \to (sn3 c (THeads (Flat Appl) TNil
2206 (TLRef i)))))).(\lambda (H1: (land (sn3 c t) (sns3 c TNil))).(let H2 \def H1
2207 in (land_ind (sn3 c t) True (sn3 c (THead (Flat Appl) t (THeads (Flat Appl)
2208 TNil (TLRef i)))) (\lambda (H3: (sn3 c t)).(\lambda (_: True).(sn3_appl_lref
2209 c i H t H3))) H2)))) (\lambda (t1: T).(\lambda (t2: TList).(\lambda (_:
2210 (((((sns3 c t2) \to (sn3 c (THeads (Flat Appl) t2 (TLRef i))))) \to ((land
2211 (sn3 c t) (sns3 c t2)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2
2212 (TLRef i)))))))).(\lambda (H1: (((sns3 c (TCons t1 t2)) \to (sn3 c (THeads
2213 (Flat Appl) (TCons t1 t2) (TLRef i)))))).(\lambda (H2: (land (sn3 c t) (sns3
2214 c (TCons t1 t2)))).(let H3 \def H2 in (land_ind (sn3 c t) (land (sn3 c t1)
2215 (sns3 c t2)) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2)
2216 (TLRef i)))) (\lambda (H4: (sn3 c t)).(\lambda (H5: (land (sn3 c t1) (sns3 c
2217 t2))).(land_ind (sn3 c t1) (sns3 c t2) (sn3 c (THead (Flat Appl) t (THeads
2218 (Flat Appl) (TCons t1 t2) (TLRef i)))) (\lambda (H6: (sn3 c t1)).(\lambda
2219 (H7: (sns3 c t2)).(sn3_appl_appls t1 (TLRef i) t2 c (H1 (conj (sn3 c t1)
2220 (sns3 c t2) H6 H7)) t H4 (\lambda (u2: T).(\lambda (H8: (pr3 c (THeads (Flat
2221 Appl) (TCons t1 t2) (TLRef i)) u2)).(\lambda (H9: (((iso (THeads (Flat Appl)
2222 (TCons t1 t2) (TLRef i)) u2) \to (\forall (P: Prop).P)))).(H9
2223 (nf2_iso_appls_lref c i H (TCons t1 t2) u2 H8) (sn3 c (THead (Flat Appl) t
2224 u2))))))))) H5))) H3))))))) t0))) us)))).
2226 theorem sn3_appls_cast:
2227 \forall (c: C).(\forall (vs: TList).(\forall (u: T).((sn3 c (THeads (Flat
2228 Appl) vs u)) \to (\forall (t: T).((sn3 c (THeads (Flat Appl) vs t)) \to (sn3
2229 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))
2231 \lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: TList).(\forall
2232 (u: T).((sn3 c (THeads (Flat Appl) t u)) \to (\forall (t0: T).((sn3 c (THeads
2233 (Flat Appl) t t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Flat Cast) u
2234 t0)))))))) (\lambda (u: T).(\lambda (H: (sn3 c u)).(\lambda (t: T).(\lambda
2235 (H0: (sn3 c t)).(sn3_cast c u H t H0))))) (\lambda (t: T).(\lambda (t0:
2236 TList).(TList_ind (\lambda (t1: TList).(((\forall (u: T).((sn3 c (THeads
2237 (Flat Appl) t1 u)) \to (\forall (t2: T).((sn3 c (THeads (Flat Appl) t1 t2))
2238 \to (sn3 c (THeads (Flat Appl) t1 (THead (Flat Cast) u t2)))))))) \to
2239 (\forall (u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 u))) \to
2240 (\forall (t2: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 t2)))
2241 \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (THead (Flat Cast) u
2242 t2)))))))))) (\lambda (_: ((\forall (u: T).((sn3 c (THeads (Flat Appl) TNil
2243 u)) \to (\forall (t1: T).((sn3 c (THeads (Flat Appl) TNil t1)) \to (sn3 c
2244 (THeads (Flat Appl) TNil (THead (Flat Cast) u t1))))))))).(\lambda (u:
2245 T).(\lambda (H0: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) TNil
2246 u)))).(\lambda (t1: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads
2247 (Flat Appl) TNil t1)))).(sn3_appl_cast c t u H0 t1 H1)))))) (\lambda (t1:
2248 T).(\lambda (t2: TList).(\lambda (_: ((((\forall (u: T).((sn3 c (THeads (Flat
2249 Appl) t2 u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) t2 t3)) \to
2250 (sn3 c (THeads (Flat Appl) t2 (THead (Flat Cast) u t3)))))))) \to (\forall
2251 (u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 u))) \to (\forall
2252 (t3: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 t3))) \to (sn3 c
2253 (THead (Flat Appl) t (THeads (Flat Appl) t2 (THead (Flat Cast) u
2254 t3))))))))))).(\lambda (H0: ((\forall (u: T).((sn3 c (THeads (Flat Appl)
2255 (TCons t1 t2) u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) (TCons t1
2256 t2) t3)) \to (sn3 c (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u
2257 t3))))))))).(\lambda (u: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads
2258 (Flat Appl) (TCons t1 t2) u)))).(\lambda (t3: T).(\lambda (H2: (sn3 c (THead
2259 (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) t3)))).(let H_x \def
2260 (sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) t3) H2) in (let H3
2261 \def H_x in (land_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads (Flat
2262 Appl) t2 t3))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2)
2263 (THead (Flat Cast) u t3)))) (\lambda (_: (sn3 c t)).(\lambda (H5: (sn3 c
2264 (THead (Flat Appl) t1 (THeads (Flat Appl) t2 t3)))).(let H6 \def H5 in (let
2265 H_x0 \def (sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) u) H1) in
2266 (let H7 \def H_x0 in (land_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads
2267 (Flat Appl) t2 u))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1
2268 t2) (THead (Flat Cast) u t3)))) (\lambda (H8: (sn3 c t)).(\lambda (H9: (sn3 c
2269 (THead (Flat Appl) t1 (THeads (Flat Appl) t2 u)))).(let H10 \def H9 in
2270 (sn3_appl_appls t1 (THead (Flat Cast) u t3) t2 c (H0 u H10 t3 H6) t H8
2271 (\lambda (u2: T).(\lambda (H11: (pr3 c (THeads (Flat Appl) (TCons t1 t2)
2272 (THead (Flat Cast) u t3)) u2)).(\lambda (H12: (((iso (THeads (Flat Appl)
2273 (TCons t1 t2) (THead (Flat Cast) u t3)) u2) \to (\forall (P:
2274 Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t (THeads (Flat Appl) (TCons
2275 t1 t2) t3)) H2 (THead (Flat Appl) t u2) (pr3_thin_dx c (THeads (Flat Appl)
2276 (TCons t1 t2) t3) u2 (pr3_iso_appls_cast c u t3 (TCons t1 t2) u2 H11 H12) t
2277 Appl))))))))) H7)))))) H3))))))))))) t0))) vs)).
2279 theorem sn3_appls_bind:
2280 \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u:
2281 T).((sn3 c u) \to (\forall (vs: TList).(\forall (t: T).((sn3 (CHead c (Bind
2282 b) u) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to (sn3 c (THeads (Flat
2283 Appl) vs (THead (Bind b) u t))))))))))
2285 \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda
2286 (u: T).(\lambda (H0: (sn3 c u)).(\lambda (vs: TList).(TList_ind (\lambda (t:
2287 TList).(\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) (lifts
2288 (S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u t0))))))
2289 (\lambda (t: T).(\lambda (H1: (sn3 (CHead c (Bind b) u) t)).(sn3_bind b c u
2290 H0 t H1))) (\lambda (v: T).(\lambda (vs0: TList).(TList_ind (\lambda (t:
2291 TList).(((\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl)
2292 (lifts (S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u
2293 t0)))))) \to (\forall (t0: T).((sn3 (CHead c (Bind b) u) (THead (Flat Appl)
2294 (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t) t0))) \to (sn3 c
2295 (THead (Flat Appl) v (THeads (Flat Appl) t (THead (Bind b) u t0))))))))
2296 (\lambda (_: ((\forall (t: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl)
2297 (lifts (S O) O TNil) t)) \to (sn3 c (THeads (Flat Appl) TNil (THead (Bind b)
2298 u t))))))).(\lambda (t: T).(\lambda (H2: (sn3 (CHead c (Bind b) u) (THead
2299 (Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O TNil)
2300 t)))).(sn3_appl_bind b H c u H0 t v H2)))) (\lambda (t: T).(\lambda (t0:
2301 TList).(\lambda (_: ((((\forall (t1: T).((sn3 (CHead c (Bind b) u) (THeads
2302 (Flat Appl) (lifts (S O) O t0) t1)) \to (sn3 c (THeads (Flat Appl) t0 (THead
2303 (Bind b) u t1)))))) \to (\forall (t1: T).((sn3 (CHead c (Bind b) u) (THead
2304 (Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t0) t1))) \to
2305 (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (THead (Bind b) u
2306 t1))))))))).(\lambda (H2: ((\forall (t1: T).((sn3 (CHead c (Bind b) u)
2307 (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) \to (sn3 c (THeads
2308 (Flat Appl) (TCons t t0) (THead (Bind b) u t1))))))).(\lambda (t1:
2309 T).(\lambda (H3: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O
2310 v) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))).(let H_x \def
2311 (sn3_gen_flat Appl (CHead c (Bind b) u) (lift (S O) O v) (THeads (Flat Appl)
2312 (lifts (S O) O (TCons t t0)) t1) H3) in (let H4 \def H_x in (land_ind (sn3
2313 (CHead c (Bind b) u) (lift (S O) O v)) (sn3 (CHead c (Bind b) u) (THead (Flat
2314 Appl) (lift (S O) O t) (THeads (Flat Appl) (lifts (S O) O t0) t1))) (sn3 c
2315 (THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u
2316 t1)))) (\lambda (H5: (sn3 (CHead c (Bind b) u) (lift (S O) O v))).(\lambda
2317 (H6: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O t) (THeads
2318 (Flat Appl) (lifts (S O) O t0) t1)))).(let H_y \def (sn3_gen_lift (CHead c
2319 (Bind b) u) v (S O) O H5 c) in (sn3_appl_appls t (THead (Bind b) u t1) t0 c
2320 (H2 t1 H6) v (H_y (drop_drop (Bind b) O c c (drop_refl c) u)) (\lambda (u2:
2321 T).(\lambda (H7: (pr3 c (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u
2322 t1)) u2)).(\lambda (H8: (((iso (THeads (Flat Appl) (TCons t t0) (THead (Bind
2323 b) u t1)) u2) \to (\forall (P: Prop).P)))).(let H9 \def (pr3_iso_appls_bind b
2324 H (TCons t t0) u t1 c u2 H7 H8) in (sn3_pr3_trans c (THead (Flat Appl) v
2325 (THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))
2326 (sn3_appl_bind b H c u H0 (THeads (Flat Appl) (lifts (S O) O (TCons t t0))
2327 t1) v H3) (THead (Flat Appl) v u2) (pr3_flat c v v (pr3_refl c v) (THead
2328 (Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) u2 H9
2329 Appl)))))))))) H4))))))))) vs0))) vs)))))).
2331 theorem sn3_appls_beta:
2332 \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (us: TList).((sn3 c
2333 (THeads (Flat Appl) us (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c
2334 w) \to (sn3 c (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst)
2337 \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (us:
2338 TList).(TList_ind (\lambda (t0: TList).((sn3 c (THeads (Flat Appl) t0 (THead
2339 (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat
2340 Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) (\lambda (H:
2341 (sn3 c (THead (Bind Abbr) v t))).(\lambda (w: T).(\lambda (H0: (sn3 c
2342 w)).(sn3_beta c v t H w H0)))) (\lambda (u: T).(\lambda (us0:
2343 TList).(TList_ind (\lambda (t0: TList).((((sn3 c (THeads (Flat Appl) t0
2344 (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads
2345 (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3
2346 c (THead (Flat Appl) u (THeads (Flat Appl) t0 (THead (Bind Abbr) v t)))) \to
2347 (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat
2348 Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))) (\lambda (_:
2349 (((sn3 c (THeads (Flat Appl) TNil (THead (Bind Abbr) v t))) \to (\forall (w:
2350 T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) TNil (THead (Flat Appl) v (THead
2351 (Bind Abst) w t))))))))).(\lambda (H0: (sn3 c (THead (Flat Appl) u (THeads
2352 (Flat Appl) TNil (THead (Bind Abbr) v t))))).(\lambda (w: T).(\lambda (H1:
2353 (sn3 c w)).(sn3_appl_beta c u v t H0 w H1))))) (\lambda (t0: T).(\lambda (t1:
2354 TList).(\lambda (_: (((((sn3 c (THeads (Flat Appl) t1 (THead (Bind Abbr) v
2355 t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) t1 (THead
2356 (Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 c (THead (Flat Appl) u
2357 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)))) \to (\forall (w: T).((sn3 c
2358 w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) t1 (THead (Flat Appl)
2359 v (THead (Bind Abst) w t))))))))))).(\lambda (H0: (((sn3 c (THeads (Flat
2360 Appl) (TCons t0 t1) (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w)
2361 \to (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead
2362 (Bind Abst) w t))))))))).(\lambda (H1: (sn3 c (THead (Flat Appl) u (THeads
2363 (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t))))).(\lambda (w:
2364 T).(\lambda (H2: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THeads
2365 (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t)) H1) in (let H3 \def H_x in
2366 (land_ind (sn3 c u) (sn3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1
2367 (THead (Bind Abbr) v t)))) (sn3 c (THead (Flat Appl) u (THeads (Flat Appl)
2368 (TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H4:
2369 (sn3 c u)).(\lambda (H5: (sn3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1
2370 (THead (Bind Abbr) v t))))).(sn3_appl_appls t0 (THead (Flat Appl) v (THead
2371 (Bind Abst) w t)) t1 c (H0 H5 w H2) u H4 (\lambda (u2: T).(\lambda (H6: (pr3
2372 c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w
2373 t))) u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t0 t1) (THead (Flat
2374 Appl) v (THead (Bind Abst) w t))) u2) \to (\forall (P: Prop).P)))).(let H8
2375 \def (pr3_iso_appls_beta (TCons t0 t1) v w t c u2 H6 H7) in (sn3_pr3_trans c
2376 (THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v
2377 t))) H1 (THead (Flat Appl) u u2) (pr3_thin_dx c (THeads (Flat Appl) (TCons t0
2378 t1) (THead (Bind Abbr) v t)) u2 H8 u Appl))))))))) H3)))))))))) us0))) us)))).
2381 \forall (d: C).(\forall (t: T).((sn3 d t) \to (\forall (c: C).(\forall (h:
2382 nat).(\forall (i: nat).((drop h i c d) \to (sn3 c (lift h i t))))))))
2384 \lambda (d: C).(\lambda (t: T).(\lambda (H: (sn3 d t)).(sn3_ind d (\lambda
2385 (t0: T).(\forall (c: C).(\forall (h: nat).(\forall (i: nat).((drop h i c d)
2386 \to (sn3 c (lift h i t0))))))) (\lambda (t1: T).(\lambda (_: ((\forall (t2:
2387 T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 d t1 t2) \to (sn3 d
2388 t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
2389 Prop).P))) \to ((pr3 d t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall
2390 (i: nat).((drop h i c d) \to (sn3 c (lift h i t2))))))))))).(\lambda (c:
2391 C).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H2: (drop h i c
2392 d)).(sn3_pr2_intro c (lift h i t1) (\lambda (t2: T).(\lambda (H3: (((eq T
2393 (lift h i t1) t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr2 c (lift h i
2394 t1) t2)).(let H5 \def (pr2_gen_lift c t1 t2 h i H4 d H2) in (ex2_ind T
2395 (\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t1 t3))
2396 (sn3 c t2) (\lambda (x: T).(\lambda (H6: (eq T t2 (lift h i x))).(\lambda
2397 (H7: (pr2 d t1 x)).(let H8 \def (eq_ind T t2 (\lambda (t0: T).((eq T (lift h
2398 i t1) t0) \to (\forall (P: Prop).P))) H3 (lift h i x) H6) in (eq_ind_r T
2399 (lift h i x) (\lambda (t0: T).(sn3 c t0)) (H1 x (\lambda (H9: (eq T t1
2400 x)).(\lambda (P: Prop).(let H10 \def (eq_ind_r T x (\lambda (t0: T).((eq T
2401 (lift h i t1) (lift h i t0)) \to (\forall (P0: Prop).P0))) H8 t1 H9) in (let
2402 H11 \def (eq_ind_r T x (\lambda (t0: T).(pr2 d t1 t0)) H7 t1 H9) in (H10
2403 (refl_equal T (lift h i t1)) P))))) (pr3_pr2 d t1 x H7) c h i H2) t2 H6)))))
2404 H5))))))))))))) t H))).
2407 \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
2408 (CHead d (Bind Abbr) v)) \to ((sn3 d v) \to (sn3 c (TLRef i)))))))
2410 \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
2411 (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 d
2412 v)).(sn3_pr2_intro c (TLRef i) (\lambda (t2: T).(\lambda (H1: (((eq T (TLRef
2413 i) t2) \to (\forall (P: Prop).P)))).(\lambda (H2: (pr2 c (TLRef i) t2)).(let
2414 H3 \def (pr2_gen_lref c t2 i H2) in (or_ind (eq T t2 (TLRef i)) (ex2_2 C T
2415 (\lambda (d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u))))
2416 (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S i) O u))))) (sn3 c t2)
2417 (\lambda (H4: (eq T t2 (TLRef i))).(let H5 \def (eq_ind T t2 (\lambda (t:
2418 T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H4) in
2419 (eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c t)) (H5 (refl_equal T (TLRef i))
2420 (sn3 c (TLRef i))) t2 H4))) (\lambda (H4: (ex2_2 C T (\lambda (d0:
2421 C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_:
2422 C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda
2423 (d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_:
2424 C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))) (sn3 c t2) (\lambda (x0:
2425 C).(\lambda (x1: T).(\lambda (H5: (getl i c (CHead x0 (Bind Abbr)
2426 x1))).(\lambda (H6: (eq T t2 (lift (S i) O x1))).(let H7 \def (eq_ind T t2
2427 (\lambda (t: T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (lift (S
2428 i) O x1) H6) in (eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(sn3 c t)) (let
2429 H8 \def (eq_ind C (CHead d (Bind Abbr) v) (\lambda (c0: C).(getl i c c0)) H
2430 (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0
2431 (Bind Abbr) x1) H5)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in
2432 C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
2433 \Rightarrow c0])) (CHead d (Bind Abbr) v) (CHead x0 (Bind Abbr) x1)
2434 (getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 (Bind Abbr) x1) H5)) in
2435 ((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
2436 C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead d
2437 (Bind Abbr) v) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v)
2438 i H (CHead x0 (Bind Abbr) x1) H5)) in (\lambda (H11: (eq C d x0)).(let H12
2439 \def (eq_ind_r T x1 (\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 v
2440 H10) in (eq_ind T v (\lambda (t: T).(sn3 c (lift (S i) O t))) (let H13 \def
2441 (eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) v))) H12 d
2442 H11) in (sn3_lift d v H0 c (S i) O (getl_drop Abbr c d v i H13))) x1 H10))))
2443 H9))) t2 H6)))))) H4)) H3))))))))))).
2445 theorem sn3_appls_abbr:
2446 \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c
2447 (CHead d (Bind Abbr) w)) \to (\forall (vs: TList).((sn3 c (THeads (Flat Appl)
2448 vs (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))
2450 \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda
2451 (H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind
2452 (\lambda (t: TList).((sn3 c (THeads (Flat Appl) t (lift (S i) O w))) \to (sn3
2453 c (THeads (Flat Appl) t (TLRef i))))) (\lambda (H0: (sn3 c (lift (S i) O
2454 w))).(let H_y \def (sn3_gen_lift c w (S i) O H0 d (getl_drop Abbr c d w i H))
2455 in (sn3_abbr c d w i H H_y))) (\lambda (v: T).(\lambda (vs0:
2456 TList).(TList_ind (\lambda (t: TList).((((sn3 c (THeads (Flat Appl) t (lift
2457 (S i) O w))) \to (sn3 c (THeads (Flat Appl) t (TLRef i))))) \to ((sn3 c
2458 (THead (Flat Appl) v (THeads (Flat Appl) t (lift (S i) O w)))) \to (sn3 c
2459 (THead (Flat Appl) v (THeads (Flat Appl) t (TLRef i))))))) (\lambda (_:
2460 (((sn3 c (THeads (Flat Appl) TNil (lift (S i) O w))) \to (sn3 c (THeads (Flat
2461 Appl) TNil (TLRef i)))))).(\lambda (H1: (sn3 c (THead (Flat Appl) v (THeads
2462 (Flat Appl) TNil (lift (S i) O w))))).(sn3_appl_abbr c d w i H v H1)))
2463 (\lambda (t: T).(\lambda (t0: TList).(\lambda (_: (((((sn3 c (THeads (Flat
2464 Appl) t0 (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) t0 (TLRef i)))))
2465 \to ((sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (lift (S i) O w))))
2466 \to (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (TLRef
2467 i)))))))).(\lambda (H1: (((sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i)
2468 O w))) \to (sn3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)))))).(\lambda
2469 (H2: (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (lift (S i)
2470 O w))))).(let H_x \def (sn3_gen_flat Appl c v (THeads (Flat Appl) (TCons t
2471 t0) (lift (S i) O w)) H2) in (let H3 \def H_x in (land_ind (sn3 c v) (sn3 c
2472 (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w)))) (sn3 c (THead
2473 (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (TLRef i)))) (\lambda (H4:
2474 (sn3 c v)).(\lambda (H5: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t0
2475 (lift (S i) O w))))).(sn3_appl_appls t (TLRef i) t0 c (H1 H5) v H4 (\lambda
2476 (u2: T).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t t0) (TLRef i))
2477 u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2) \to
2478 (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) v (THeads (Flat
2479 Appl) (TCons t t0) (lift (S i) O w))) H2 (THead (Flat Appl) v u2)
2480 (pr3_thin_dx c (THeads (Flat Appl) (TCons t t0) (lift (S i) O w)) u2
2481 (pr3_iso_appls_abbr c d w i H (TCons t t0) u2 H6 H7) v Appl))))))))
2482 H3)))))))) vs0))) vs)))))).
2485 \forall (c: C).(\forall (d: C).(\forall (h: nat).(\forall (i: nat).((drop h
2486 i c d) \to (\forall (ts: TList).((sns3 d ts) \to (sns3 c (lifts h i ts))))))))
2488 \lambda (c: C).(\lambda (d: C).(\lambda (h: nat).(\lambda (i: nat).(\lambda
2489 (H: (drop h i c d)).(\lambda (ts: TList).(TList_ind (\lambda (t:
2490 TList).((sns3 d t) \to (sns3 c (lifts h i t)))) (\lambda (H0: True).H0)
2491 (\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: (((sns3 d t0) \to (sns3 c
2492 (lifts h i t0))))).(\lambda (H1: (land (sn3 d t) (sns3 d t0))).(let H2 \def
2493 H1 in (land_ind (sn3 d t) (sns3 d t0) (land (sn3 c (lift h i t)) (sns3 c
2494 (lifts h i t0))) (\lambda (H3: (sn3 d t)).(\lambda (H4: (sns3 d t0)).(conj
2495 (sn3 c (lift h i t)) (sns3 c (lifts h i t0)) (sn3_lift d t H3 c h i H) (H0
2496 H4)))) H2)))))) ts)))))).
projects / helm.git / blob