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 *********************)
25 include "iso/props.ma".
27 theorem sn3_pr3_trans:
28 \forall (c: C).(\forall (t1: T).((sn3 c t1) \to (\forall (t2: T).((pr3 c t1
29 t2) \to (sn3 c t2)))))
31 \lambda (c: C).(\lambda (t1: T).(\lambda (H: (sn3 c t1)).(sn3_ind c (\lambda
32 (t: T).(\forall (t2: T).((pr3 c t t2) \to (sn3 c t2)))) (\lambda (t2:
33 T).(\lambda (H0: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
34 Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H1: ((\forall
35 (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to
36 (\forall (t4: T).((pr3 c t3 t4) \to (sn3 c t4)))))))).(\lambda (t3:
37 T).(\lambda (H2: (pr3 c t2 t3)).(sn3_sing c t3 (\lambda (t0: T).(\lambda (H3:
38 (((eq T t3 t0) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 c t3 t0)).(let
39 H_x \def (term_dec t2 t3) in (let H5 \def H_x in (or_ind (eq T t2 t3) ((eq T
40 t2 t3) \to (\forall (P: Prop).P)) (sn3 c t0) (\lambda (H6: (eq T t2 t3)).(let
41 H7 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t t0)) H4 t2 H6) in (let H8
42 \def (eq_ind_r T t3 (\lambda (t: T).((eq T t t0) \to (\forall (P: Prop).P)))
43 H3 t2 H6) in (let H9 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t2 t)) H2 t2
44 H6) in (H0 t0 H8 H7))))) (\lambda (H6: (((eq T t2 t3) \to (\forall (P:
45 Prop).P)))).(H1 t3 H6 H2 t0 H4)) H5)))))))))))) t1 H))).
47 theorem sn3_pr2_intro:
48 \forall (c: C).(\forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to
49 (\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)))
51 \lambda (c: C).(\lambda (t1: T).(\lambda (H: ((\forall (t2: T).((((eq T t1
52 t2) \to (\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c
53 t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H0: (((eq T t1 t2) \to
54 (\forall (P: Prop).P)))).(\lambda (H1: (pr3 c t1 t2)).(let H2 \def H0 in
55 ((let H3 \def H in (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(((\forall
56 (t3: T).((((eq T t t3) \to (\forall (P: Prop).P))) \to ((pr2 c t t3) \to (sn3
57 c t3))))) \to ((((eq T t t0) \to (\forall (P: Prop).P))) \to (sn3 c t0)))))
58 (\lambda (t: T).(\lambda (H4: ((\forall (t3: T).((((eq T t t3) \to (\forall
59 (P: Prop).P))) \to ((pr2 c t t3) \to (sn3 c t3)))))).(\lambda (H5: (((eq T t
60 t) \to (\forall (P: Prop).P)))).(H4 t H5 (pr2_free c t t (pr0_refl t))))))
61 (\lambda (t3: T).(\lambda (t4: T).(\lambda (H4: (pr2 c t4 t3)).(\lambda (t5:
62 T).(\lambda (H5: (pr3 c t3 t5)).(\lambda (H6: ((((\forall (t6: T).((((eq T t3
63 t6) \to (\forall (P: Prop).P))) \to ((pr2 c t3 t6) \to (sn3 c t6))))) \to
64 ((((eq T t3 t5) \to (\forall (P: Prop).P))) \to (sn3 c t5))))).(\lambda (H7:
65 ((\forall (t6: T).((((eq T t4 t6) \to (\forall (P: Prop).P))) \to ((pr2 c t4
66 t6) \to (sn3 c t6)))))).(\lambda (H8: (((eq T t4 t5) \to (\forall (P:
67 Prop).P)))).(let H_x \def (term_dec t4 t3) in (let H9 \def H_x in (or_ind (eq
68 T t4 t3) ((eq T t4 t3) \to (\forall (P: Prop).P)) (sn3 c t5) (\lambda (H10:
69 (eq T t4 t3)).(let H11 \def (eq_ind T t4 (\lambda (t: T).((eq T t t5) \to
70 (\forall (P: Prop).P))) H8 t3 H10) in (let H12 \def (eq_ind T t4 (\lambda (t:
71 T).(\forall (t6: T).((((eq T t t6) \to (\forall (P: Prop).P))) \to ((pr2 c t
72 t6) \to (sn3 c t6))))) H7 t3 H10) in (let H13 \def (eq_ind T t4 (\lambda (t:
73 T).(pr2 c t t3)) H4 t3 H10) in (H6 H12 H11))))) (\lambda (H10: (((eq T t4 t3)
74 \to (\forall (P: Prop).P)))).(sn3_pr3_trans c t3 (H7 t3 H10 H4) t5 H5))
75 H9))))))))))) t1 t2 H1 H3)) H2)))))))).
78 \forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: T).((sn3 c t) \to
79 (sn3 c (THead (Flat Cast) u t))))))
81 \lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c u)).(sn3_ind c (\lambda
82 (t: T).(\forall (t0: T).((sn3 c t0) \to (sn3 c (THead (Flat Cast) t t0)))))
83 (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
84 (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall
85 (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
86 (\forall (t: T).((sn3 c t) \to (sn3 c (THead (Flat Cast) t2
87 t))))))))).(\lambda (t: T).(\lambda (H2: (sn3 c t)).(sn3_ind c (\lambda (t0:
88 T).(sn3 c (THead (Flat Cast) t1 t0))) (\lambda (t0: T).(\lambda (H3:
89 ((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0
90 t2) \to (sn3 c t2)))))).(\lambda (H4: ((\forall (t2: T).((((eq T t0 t2) \to
91 (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c (THead (Flat Cast) t1
92 t2))))))).(sn3_pr2_intro c (THead (Flat Cast) t1 t0) (\lambda (t2:
93 T).(\lambda (H5: (((eq T (THead (Flat Cast) t1 t0) t2) \to (\forall (P:
94 Prop).P)))).(\lambda (H6: (pr2 c (THead (Flat Cast) t1 t0) t2)).(let H7 \def
95 (pr2_gen_cast c t1 t0 t2 H6) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda
96 (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
97 T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3)))) (pr2 c
98 t0 t2) (sn3 c t2) (\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
99 T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
100 T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0
101 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
102 (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2)))
103 (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3))) (sn3 c t2) (\lambda (x0:
104 T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead (Flat Cast) x0
105 x1))).(\lambda (H10: (pr2 c t1 x0)).(\lambda (H11: (pr2 c t0 x1)).(let H12
106 \def (eq_ind T t2 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) t3) \to
107 (\forall (P: Prop).P))) H5 (THead (Flat Cast) x0 x1) H9) in (eq_ind_r T
108 (THead (Flat Cast) x0 x1) (\lambda (t3: T).(sn3 c t3)) (let H_x \def
109 (term_dec x0 t1) in (let H13 \def H_x in (or_ind (eq T x0 t1) ((eq T x0 t1)
110 \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) x0 x1)) (\lambda (H14:
111 (eq T x0 t1)).(let H15 \def (eq_ind T x0 (\lambda (t3: T).((eq T (THead (Flat
112 Cast) t1 t0) (THead (Flat Cast) t3 x1)) \to (\forall (P: Prop).P))) H12 t1
113 H14) in (let H16 \def (eq_ind T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1
114 H14) in (eq_ind_r T t1 (\lambda (t3: T).(sn3 c (THead (Flat Cast) t3 x1)))
115 (let H_x0 \def (term_dec t0 x1) in (let H17 \def H_x0 in (or_ind (eq T t0 x1)
116 ((eq T t0 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) t1 x1))
117 (\lambda (H18: (eq T t0 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t3:
118 T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat Cast) t1 t3)) \to (\forall
119 (P: Prop).P))) H15 t0 H18) in (let H20 \def (eq_ind_r T x1 (\lambda (t3:
120 T).(pr2 c t0 t3)) H11 t0 H18) in (eq_ind T t0 (\lambda (t3: T).(sn3 c (THead
121 (Flat Cast) t1 t3))) (H19 (refl_equal T (THead (Flat Cast) t1 t0)) (sn3 c
122 (THead (Flat Cast) t1 t0))) x1 H18)))) (\lambda (H18: (((eq T t0 x1) \to
123 (\forall (P: Prop).P)))).(H4 x1 H18 (pr3_pr2 c t0 x1 H11))) H17))) x0 H14))))
124 (\lambda (H14: (((eq T x0 t1) \to (\forall (P: Prop).P)))).(H1 x0 (\lambda
125 (H15: (eq T t1 x0)).(\lambda (P: Prop).(let H16 \def (eq_ind_r T x0 (\lambda
126 (t3: T).((eq T t3 t1) \to (\forall (P0: Prop).P0))) H14 t1 H15) in (let H17
127 \def (eq_ind_r T x0 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead
128 (Flat Cast) t3 x1)) \to (\forall (P0: Prop).P0))) H12 t1 H15) in (let H18
129 \def (eq_ind_r T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1 H15) in (H16
130 (refl_equal T t1) P)))))) (pr3_pr2 c t1 x0 H10) x1 (let H_x0 \def (term_dec
131 t0 x1) in (let H15 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to
132 (\forall (P: Prop).P)) (sn3 c x1) (\lambda (H16: (eq T t0 x1)).(let H17 \def
133 (eq_ind_r T x1 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat
134 Cast) x0 t3)) \to (\forall (P: Prop).P))) H12 t0 H16) in (let H18 \def
135 (eq_ind_r T x1 (\lambda (t3: T).(pr2 c t0 t3)) H11 t0 H16) in (eq_ind T t0
136 (\lambda (t3: T).(sn3 c t3)) (sn3_sing c t0 H3) x1 H16)))) (\lambda (H16:
137 (((eq T t0 x1) \to (\forall (P: Prop).P)))).(H3 x1 H16 (pr3_pr2 c t0 x1
138 H11))) H15))))) H13))) t2 H9))))))) H8)) (\lambda (H8: (pr2 c t0
139 t2)).(sn3_pr3_trans c t0 (sn3_sing c t0 H3) t2 (pr3_pr2 c t0 t2 H8)))
140 H7))))))))) t H2)))))) u H))).
143 \forall (c: C).(\forall (t: T).((sn3 c t) \to (\forall (f: F).(\forall (u:
144 T).(sn3 (CHead c (Flat f) u) t)))))
146 \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(\lambda (f:
147 F).(\lambda (u: T).(sn3_ind c (\lambda (t0: T).(sn3 (CHead c (Flat f) u) t0))
148 (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
149 (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall
150 (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
151 (sn3 (CHead c (Flat f) u) t2)))))).(sn3_pr2_intro (CHead c (Flat f) u) t1
152 (\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) \to (\forall (P:
153 Prop).P)))).(\lambda (H3: (pr2 (CHead c (Flat f) u) t1 t2)).(H1 t2 H2
154 (pr3_pr2 c t1 t2 (pr2_gen_cflat f c u t1 t2 H3)))))))))) t H))))).
157 \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c
158 (THead (Bind b) v t)) \to (sn3 (CHead c (Bind b) v) t)))))
160 \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
161 (sn3 c (THead (Bind b) v t))).(let H_x \def (sn3_gen_bind b c v t H) in (let
162 H0 \def H_x in (and_ind (sn3 c v) (sn3 (CHead c (Bind b) v) t) (sn3 (CHead c
163 (Bind b) v) t) (\lambda (_: (sn3 c v)).(\lambda (H2: (sn3 (CHead c (Bind b)
164 v) t)).H2)) H0))))))).
167 \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1:
168 T).(\forall (t: T).((sn3 (CHead c (Bind b) v1) t) \to (\forall (v2: T).(sn3
169 (CHead c (Bind b) v2) t)))))))
171 \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda
172 (v1: T).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c (Bind b) v1) t)).(\lambda
173 (v2: T).(sn3_ind (CHead c (Bind b) v1) (\lambda (t0: T).(sn3 (CHead c (Bind
174 b) v2) t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2)
175 \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to (sn3
176 (CHead c (Bind b) v1) t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1
177 t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to
178 (sn3 (CHead c (Bind b) v2) t2)))))).(sn3_pr2_intro (CHead c (Bind b) v2) t1
179 (\lambda (t2: T).(\lambda (H3: (((eq T t1 t2) \to (\forall (P:
180 Prop).P)))).(\lambda (H4: (pr2 (CHead c (Bind b) v2) t1 t2)).(H2 t2 H3
181 (pr3_pr2 (CHead c (Bind b) v1) t1 t2 (pr2_change b H c v2 t1 t2 H4
182 v1)))))))))) t H0))))))).
184 theorem sn3_cpr3_trans:
185 \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
186 (k: K).(\forall (t: T).((sn3 (CHead c k u1) t) \to (sn3 (CHead c k u2)
189 \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
190 u2)).(\lambda (k: K).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c k u1)
191 t)).(sn3_ind (CHead c k u1) (\lambda (t0: T).(sn3 (CHead c k u2) t0))
192 (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
193 (P: Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u1)
194 t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
195 Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u2)
196 t2)))))).(sn3_sing (CHead c k u2) t1 (\lambda (t2: T).(\lambda (H3: (((eq T
197 t1 t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 (CHead c k u2) t1
198 t2)).(H2 t2 H3 (pr3_pr3_pr3_t c u1 u2 H t1 t2 k H4))))))))) t H0))))))).
201 \forall (b: B).(\forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t:
202 T).((sn3 (CHead c (Bind b) u) t) \to (sn3 c (THead (Bind b) u t)))))))
204 \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c
205 u)).(sn3_ind c (\lambda (t: T).(\forall (t0: T).((sn3 (CHead c (Bind b) t)
206 t0) \to (sn3 c (THead (Bind b) t t0))))) (\lambda (t1: T).(\lambda (_:
207 ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1
208 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to
209 (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (t: T).((sn3 (CHead c
210 (Bind b) t2) t) \to (sn3 c (THead (Bind b) t2 t))))))))).(\lambda (t:
211 T).(\lambda (H2: (sn3 (CHead c (Bind b) t1) t)).(sn3_ind (CHead c (Bind b)
212 t1) (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (\lambda (t2:
213 T).(\lambda (H3: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
214 Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind b)
215 t1) t3)))))).(\lambda (H4: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
216 Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 c (THead (Bind b)
217 t1 t3))))))).(sn3_sing c (THead (Bind b) t1 t2) (\lambda (t3: T).(\lambda
218 (H5: (((eq T (THead (Bind b) t1 t2) t3) \to (\forall (P: Prop).P)))).(\lambda
219 (H6: (pr3 c (THead (Bind b) t1 t2) t3)).(let H_x \def (bind_dec_not b Abst)
220 in (let H7 \def H_x in (or_ind (eq B b Abst) (not (eq B b Abst)) (sn3 c t3)
221 (\lambda (H8: (eq B b Abst)).(let H9 \def (eq_ind B b (\lambda (b0: B).(pr3 c
222 (THead (Bind b0) t1 t2) t3)) H6 Abst H8) in (let H10 \def (eq_ind B b
223 (\lambda (b0: B).((eq T (THead (Bind b0) t1 t2) t3) \to (\forall (P:
224 Prop).P))) H5 Abst H8) in (let H11 \def (eq_ind B b (\lambda (b0: B).(\forall
225 (t4: T).((((eq T t2 t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind
226 b0) t1) t2 t4) \to (sn3 c (THead (Bind b0) t1 t4)))))) H4 Abst H8) in (let
227 H12 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T t2 t4) \to
228 (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) t2 t4) \to (sn3
229 (CHead c (Bind b0) t1) t4))))) H3 Abst H8) in (let H13 \def (eq_ind B b
230 (\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P)))
231 \to ((pr3 c t1 t4) \to (\forall (t0: T).((sn3 (CHead c (Bind b0) t4) t0) \to
232 (sn3 c (THead (Bind b0) t4 t0)))))))) H1 Abst H8) in (let H14 \def
233 (pr3_gen_abst c t1 t2 t3 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4:
234 T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_:
235 T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall
236 (u0: T).(pr3 (CHead c (Bind b0) u0) t2 t4))))) (sn3 c t3) (\lambda (x0:
237 T).(\lambda (x1: T).(\lambda (H15: (eq T t3 (THead (Bind Abst) x0
238 x1))).(\lambda (H16: (pr3 c t1 x0)).(\lambda (H17: ((\forall (b0: B).(\forall
239 (u0: T).(pr3 (CHead c (Bind b0) u0) t2 x1))))).(let H18 \def (eq_ind T t3
240 (\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) t0) \to (\forall (P:
241 Prop).P))) H10 (THead (Bind Abst) x0 x1) H15) in (eq_ind_r T (THead (Bind
242 Abst) x0 x1) (\lambda (t0: T).(sn3 c t0)) (let H_x0 \def (term_dec t1 x0) in
243 (let H19 \def H_x0 in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P:
244 Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H20: (eq T t1 x0)).(let
245 H21 \def (eq_ind_r T x0 (\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2)
246 (THead (Bind Abst) t0 x1)) \to (\forall (P: Prop).P))) H18 t1 H20) in (let
247 H22 \def (eq_ind_r T x0 (\lambda (t0: T).(pr3 c t1 t0)) H16 t1 H20) in
248 (eq_ind T t1 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t0 x1))) (let H_x1
249 \def (term_dec t2 x1) in (let H23 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2
250 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abst) t1 x1)) (\lambda
251 (H24: (eq T t2 x1)).(let H25 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T
252 (THead (Bind Abst) t1 t2) (THead (Bind Abst) t1 t0)) \to (\forall (P:
253 Prop).P))) H21 t2 H24) in (let H26 \def (eq_ind_r T x1 (\lambda (t0:
254 T).(\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) t2 t0))))
255 H17 t2 H24) in (eq_ind T t2 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t1
256 t0))) (H25 (refl_equal T (THead (Bind Abst) t1 t2)) (sn3 c (THead (Bind Abst)
257 t1 t2))) x1 H24)))) (\lambda (H24: (((eq T t2 x1) \to (\forall (P:
258 Prop).P)))).(H11 x1 H24 (H17 Abst t1))) H23))) x0 H20)))) (\lambda (H20:
259 (((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x1 \def (term_dec t2 x1)
260 in (let H21 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P:
261 Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H22: (eq T t2 x1)).(let
262 H23 \def (eq_ind_r T x1 (\lambda (t0: T).(\forall (b0: B).(\forall (u0:
263 T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) H17 t2 H22) in (eq_ind T t2 (\lambda
264 (t0: T).(sn3 c (THead (Bind Abst) x0 t0))) (H13 x0 H20 H16 t2 (sn3_cpr3_trans
265 c t1 x0 H16 (Bind Abst) t2 (sn3_sing (CHead c (Bind Abst) t1) t2 H12))) x1
266 H22))) (\lambda (H22: (((eq T t2 x1) \to (\forall (P: Prop).P)))).(H13 x0 H20
267 H16 x1 (sn3_cpr3_trans c t1 x0 H16 (Bind Abst) x1 (H12 x1 H22 (H17 Abst
268 t1))))) H21)))) H19))) t3 H15))))))) H14)))))))) (\lambda (H8: (not (eq B b
269 Abst))).(let H_x0 \def (pr3_gen_bind b H8 c t1 t2 t3 H6) in (let H9 \def H_x0
270 in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
271 b) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
272 T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) t2 t4)))) (pr3 (CHead c (Bind
273 b) t1) t2 (lift (S O) O t3)) (sn3 c t3) (\lambda (H10: (ex3_2 T T (\lambda
274 (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2:
275 T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3
276 (CHead c (Bind b) t1) t2 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
277 (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_:
278 T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 (CHead c (Bind b)
279 t1) t2 t4))) (sn3 c t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq
280 T t3 (THead (Bind b) x0 x1))).(\lambda (H12: (pr3 c t1 x0)).(\lambda (H13:
281 (pr3 (CHead c (Bind b) t1) t2 x1)).(let H14 \def (eq_ind T t3 (\lambda (t0:
282 T).((eq T (THead (Bind b) t1 t2) t0) \to (\forall (P: Prop).P))) H5 (THead
283 (Bind b) x0 x1) H11) in (eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0:
284 T).(sn3 c t0)) (let H_x1 \def (term_dec t1 x0) in (let H15 \def H_x1 in
285 (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: Prop).P)) (sn3 c (THead
286 (Bind b) x0 x1)) (\lambda (H16: (eq T t1 x0)).(let H17 \def (eq_ind_r T x0
287 (\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t0 x1)) \to
288 (\forall (P: Prop).P))) H14 t1 H16) in (let H18 \def (eq_ind_r T x0 (\lambda
289 (t0: T).(pr3 c t1 t0)) H12 t1 H16) in (eq_ind T t1 (\lambda (t0: T).(sn3 c
290 (THead (Bind b) t0 x1))) (let H_x2 \def (term_dec t2 x1) in (let H19 \def
291 H_x2 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: Prop).P)) (sn3 c
292 (THead (Bind b) t1 x1)) (\lambda (H20: (eq T t2 x1)).(let H21 \def (eq_ind_r
293 T x1 (\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t1 t0))
294 \to (\forall (P: Prop).P))) H17 t2 H20) in (let H22 \def (eq_ind_r T x1
295 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) t2 t0)) H13 t2 H20) in (eq_ind T
296 t2 (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (H21 (refl_equal T (THead
297 (Bind b) t1 t2)) (sn3 c (THead (Bind b) t1 t2))) x1 H20)))) (\lambda (H20:
298 (((eq T t2 x1) \to (\forall (P: Prop).P)))).(H4 x1 H20 H13)) H19))) x0
299 H16)))) (\lambda (H16: (((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x2
300 \def (term_dec t2 x1) in (let H17 \def H_x2 in (or_ind (eq T t2 x1) ((eq T t2
301 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind b) x0 x1)) (\lambda (H18:
302 (eq T t2 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t0: T).(pr3 (CHead c
303 (Bind b) t1) t2 t0)) H13 t2 H18) in (eq_ind T t2 (\lambda (t0: T).(sn3 c
304 (THead (Bind b) x0 t0))) (H1 x0 H16 H12 t2 (sn3_cpr3_trans c t1 x0 H12 (Bind
305 b) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3))) x1 H18))) (\lambda (H18: (((eq
306 T t2 x1) \to (\forall (P: Prop).P)))).(H1 x0 H16 H12 x1 (sn3_cpr3_trans c t1
307 x0 H12 (Bind b) x1 (H3 x1 H18 H13)))) H17)))) H15))) t3 H11))))))) H10))
308 (\lambda (H10: (pr3 (CHead c (Bind b) t1) t2 (lift (S O) O
309 t3))).(sn3_gen_lift (CHead c (Bind b) t1) t3 (S O) O (sn3_pr3_trans (CHead c
310 (Bind b) t1) t2 (sn3_pr2_intro (CHead c (Bind b) t1) t2 (\lambda (t0:
311 T).(\lambda (H11: (((eq T t2 t0) \to (\forall (P: Prop).P)))).(\lambda (H12:
312 (pr2 (CHead c (Bind b) t1) t2 t0)).(H3 t0 H11 (pr3_pr2 (CHead c (Bind b) t1)
313 t2 t0 H12)))))) (lift (S O) O t3) H10) c (drop_drop (Bind b) O c c (drop_refl
314 c) t1))) H9)))) H7)))))))))) t H2)))))) u H)))).
317 \forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c (THead (Bind Abbr) v
318 t)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead
319 (Bind Abst) w t))))))))
321 \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
322 (Bind Abbr) v t))).(insert_eq T (THead (Bind Abbr) v t) (\lambda (t0: T).(sn3
323 c t0)) (\lambda (_: T).(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat
324 Appl) v (THead (Bind Abst) w t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c
325 y)).(unintro T t (\lambda (t0: T).((eq T y (THead (Bind Abbr) v t0)) \to
326 (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead (Bind Abst)
327 w t0))))))) (unintro T v (\lambda (t0: T).(\forall (x: T).((eq T y (THead
328 (Bind Abbr) t0 x)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat
329 Appl) t0 (THead (Bind Abst) w x)))))))) (sn3_ind c (\lambda (t0: T).(\forall
330 (x: T).(\forall (x0: T).((eq T t0 (THead (Bind Abbr) x x0)) \to (\forall (w:
331 T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) w
332 x0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2)
333 \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda
334 (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3
335 c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Bind Abbr) x
336 x0)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead
337 (Bind Abst) w x0))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3:
338 (eq T t1 (THead (Bind Abbr) x x0))).(\lambda (w: T).(\lambda (H4: (sn3 c
339 w)).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0
340 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (\forall (x1:
341 T).(\forall (x2: T).((eq T t2 (THead (Bind Abbr) x1 x2)) \to (\forall (w0:
342 T).((sn3 c w0) \to (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) w0
343 x2)))))))))))) H2 (THead (Bind Abbr) x x0) H3) in (let H6 \def (eq_ind T t1
344 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P)))
345 \to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead (Bind Abbr) x x0) H3) in
346 (sn3_ind c (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t0
347 x0)))) (\lambda (t2: T).(\lambda (H7: ((\forall (t3: T).((((eq T t2 t3) \to
348 (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H8:
349 ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2
350 t3) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t3
351 x0)))))))).(sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind Abst) t2 x0))
352 (\lambda (t3: T).(\lambda (H9: (((eq T (THead (Flat Appl) x (THead (Bind
353 Abst) t2 x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H10: (pr2 c (THead
354 (Flat Appl) x (THead (Bind Abst) t2 x0)) t3)).(let H11 \def (pr2_gen_appl c x
355 (THead (Bind Abst) t2 x0) t3 H10) in (or3_ind (ex3_2 T T (\lambda (u2:
356 T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
357 T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
358 (THead (Bind Abst) t2 x0) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
359 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0)
360 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
361 T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
362 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
363 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
364 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T
365 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
366 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
367 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
368 (THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
369 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
370 (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
371 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
372 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_:
373 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
374 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
375 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
376 y2) z1 z2)))))))) (sn3 c t3) (\lambda (H12: (ex3_2 T T (\lambda (u2:
377 T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
378 T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
379 (THead (Bind Abst) t2 x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
380 (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_:
381 T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst)
382 t2 x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H13: (eq
383 T t3 (THead (Flat Appl) x1 x2))).(\lambda (H14: (pr2 c x x1)).(\lambda (H15:
384 (pr2 c (THead (Bind Abst) t2 x0) x2)).(let H16 \def (eq_ind T t3 (\lambda
385 (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to
386 (\forall (P: Prop).P))) H9 (THead (Flat Appl) x1 x2) H13) in (eq_ind_r T
387 (THead (Flat Appl) x1 x2) (\lambda (t0: T).(sn3 c t0)) (let H17 \def
388 (pr2_gen_abst c t2 x0 x2 H15) in (ex3_2_ind T T (\lambda (u2: T).(\lambda
389 (t4: T).(eq T x2 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_:
390 T).(pr2 c t2 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
391 (u: T).(pr2 (CHead c (Bind b) u) x0 t4))))) (sn3 c (THead (Flat Appl) x1 x2))
392 (\lambda (x3: T).(\lambda (x4: T).(\lambda (H18: (eq T x2 (THead (Bind Abst)
393 x3 x4))).(\lambda (H19: (pr2 c t2 x3)).(\lambda (H20: ((\forall (b:
394 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 x4))))).(let H21 \def (eq_ind
395 T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0))
396 (THead (Flat Appl) x1 t0)) \to (\forall (P: Prop).P))) H16 (THead (Bind Abst)
397 x3 x4) H18) in (eq_ind_r T (THead (Bind Abst) x3 x4) (\lambda (t0: T).(sn3 c
398 (THead (Flat Appl) x1 t0))) (let H_x \def (term_dec t2 x3) in (let H22 \def
399 H_x in (or_ind (eq T t2 x3) ((eq T t2 x3) \to (\forall (P: Prop).P)) (sn3 c
400 (THead (Flat Appl) x1 (THead (Bind Abst) x3 x4))) (\lambda (H23: (eq T t2
401 x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: T).((eq T (THead (Flat Appl)
402 x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) x1 (THead (Bind Abst) t0
403 x4))) \to (\forall (P: Prop).P))) H21 t2 H23) in (let H25 \def (eq_ind_r T x3
404 (\lambda (t0: T).(pr2 c t2 t0)) H19 t2 H23) in (eq_ind T t2 (\lambda (t0:
405 T).(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t0 x4)))) (let H_x0 \def
406 (term_dec x x1) in (let H26 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to
407 (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t2
408 x4))) (\lambda (H27: (eq T x x1)).(let H28 \def (eq_ind_r T x1 (\lambda (t0:
409 T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (THead (Flat Appl)
410 t0 (THead (Bind Abst) t2 x4))) \to (\forall (P: Prop).P))) H24 x H27) in (let
411 H29 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H27) in (eq_ind
412 T x (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) t2
413 x4)))) (let H_x1 \def (term_dec x0 x4) in (let H30 \def H_x1 in (or_ind (eq T
414 x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x
415 (THead (Bind Abst) t2 x4))) (\lambda (H31: (eq T x0 x4)).(let H32 \def
416 (eq_ind_r T x4 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind
417 Abst) t2 x0)) (THead (Flat Appl) x (THead (Bind Abst) t2 t0))) \to (\forall
418 (P: Prop).P))) H28 x0 H31) in (let H33 \def (eq_ind_r T x4 (\lambda (t0:
419 T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0
420 H31) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead
421 (Bind Abst) t2 t0)))) (H32 (refl_equal T (THead (Flat Appl) x (THead (Bind
422 Abst) t2 x0))) (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)))) x4
423 H31)))) (\lambda (H31: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead
424 (Bind Abbr) x x4) (\lambda (H32: (eq T (THead (Bind Abbr) x x0) (THead (Bind
425 Abbr) x x4))).(\lambda (P: Prop).(let H33 \def (f_equal T T (\lambda (e:
426 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
427 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
428 Abbr) x x0) (THead (Bind Abbr) x x4) H32) in (let H34 \def (eq_ind_r T x4
429 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H31 x0 H33) in
430 (let H35 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u:
431 T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H33) in (H34 (refl_equal T x0)
432 P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4)
433 (pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead
434 (Bind Abbr) x x4)) t2 (sn3_sing c t2 H7))) H30))) x1 H27)))) (\lambda (H27:
435 (((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind Abbr) x1 x4)
436 (\lambda (H28: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x1
437 x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e
438 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _)
439 \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0)
440 (THead (Bind Abbr) x1 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e:
441 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
442 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
443 Abbr) x x0) (THead (Bind Abbr) x1 x4) H28) in (\lambda (H31: (eq T x
444 x1)).(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall
445 (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (let H33 \def
446 (eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
447 H27 x H31) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14
448 x H31) in (H33 (refl_equal T x) P)))))) H29)))) (pr3_head_12 c x x1 (pr3_pr2
449 c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20
450 Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) t2 (sn3_sing c t2
451 H7))) H26))) x3 H23)))) (\lambda (H23: (((eq T t2 x3) \to (\forall (P:
452 Prop).P)))).(let H_x0 \def (term_dec x x1) in (let H24 \def H_x0 in (or_ind
453 (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl)
454 x1 (THead (Bind Abst) x3 x4))) (\lambda (H25: (eq T x x1)).(let H26 \def
455 (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H25) in (eq_ind T x
456 (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4))))
457 (let H_x1 \def (term_dec x0 x4) in (let H27 \def H_x1 in (or_ind (eq T x0 x4)
458 ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x (THead
459 (Bind Abst) x3 x4))) (\lambda (H28: (eq T x0 x4)).(let H29 \def (eq_ind_r T
460 x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
461 x0 t0)))) H20 x0 H28) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat
462 Appl) x (THead (Bind Abst) x3 t0)))) (H8 x3 H23 (pr3_pr2 c t2 x3 H19)) x4
463 H28))) (\lambda (H28: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead
464 (Bind Abbr) x x4) (\lambda (H29: (eq T (THead (Bind Abbr) x x0) (THead (Bind
465 Abbr) x x4))).(\lambda (P: Prop).(let H30 \def (f_equal T T (\lambda (e:
466 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
467 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
468 Abbr) x x0) (THead (Bind Abbr) x x4) H29) in (let H31 \def (eq_ind_r T x4
469 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H28 x0 H30) in
470 (let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u:
471 T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (H31 (refl_equal T x0)
472 P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4)
473 (pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead
474 (Bind Abbr) x x4)) x3 (H7 x3 H23 (pr3_pr2 c t2 x3 H19)))) H27))) x1 H25)))
475 (\lambda (H25: (((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind
476 Abbr) x1 x4) (\lambda (H26: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr)
477 x1 x4))).(\lambda (P: Prop).(let H27 \def (f_equal T T (\lambda (e: T).(match
478 e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _)
479 \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0)
480 (THead (Bind Abbr) x1 x4) H26) in ((let H28 \def (f_equal T T (\lambda (e:
481 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
482 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
483 Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in (\lambda (H29: (eq T x
484 x1)).(let H30 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall
485 (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (let H31 \def
486 (eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
487 H25 x H29) in (let H32 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14
488 x H29) in (H31 (refl_equal T x) P)))))) H27)))) (pr3_head_12 c x x1 (pr3_pr2
489 c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20
490 Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) x3 (H7 x3 H23
491 (pr3_pr2 c t2 x3 H19)))) H24)))) H22))) x2 H18))))))) H17)) t3 H13)))))))
492 H12)) (\lambda (H12: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
493 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead
494 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
495 T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
496 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
497 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
498 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T
499 T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
500 (THead (Bind Abst) t2 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
501 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
502 Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
503 (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
504 T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
505 z1 t4))))))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
506 T).(\lambda (x4: T).(\lambda (H13: (eq T (THead (Bind Abst) t2 x0) (THead
507 (Bind Abst) x1 x2))).(\lambda (H14: (eq T t3 (THead (Bind Abbr) x3
508 x4))).(\lambda (H15: (pr2 c x x3)).(\lambda (H16: ((\forall (b: B).(\forall
509 (u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H17 \def (eq_ind T t3
510 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0)
511 \to (\forall (P: Prop).P))) H9 (THead (Bind Abbr) x3 x4) H14) in (eq_ind_r T
512 (THead (Bind Abbr) x3 x4) (\lambda (t0: T).(sn3 c t0)) (let H18 \def (f_equal
513 T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
514 \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) \Rightarrow t0]))
515 (THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in ((let H19 \def
516 (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
517 [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0)
518 \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in
519 (\lambda (_: (eq T t2 x1)).(let H21 \def (eq_ind_r T x2 (\lambda (t0:
520 T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t0 x4)))) H16 x0
521 H19) in (let H_x \def (term_dec x x3) in (let H22 \def H_x in (or_ind (eq T x
522 x3) ((eq T x x3) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abbr) x3 x4))
523 (\lambda (H23: (eq T x x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0:
524 T).(pr2 c x t0)) H15 x H23) in (eq_ind T x (\lambda (t0: T).(sn3 c (THead
525 (Bind Abbr) t0 x4))) (let H_x0 \def (term_dec x0 x4) in (let H25 \def H_x0 in
526 (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead
527 (Bind Abbr) x x4)) (\lambda (H26: (eq T x0 x4)).(let H27 \def (eq_ind_r T x4
528 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0
529 t0)))) H21 x0 H26) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Bind Abbr)
530 x t0))) (sn3_sing c (THead (Bind Abbr) x x0) H6) x4 H26))) (\lambda (H26:
531 (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x x4)
532 (\lambda (H27: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x
533 x4))).(\lambda (P: Prop).(let H28 \def (f_equal T T (\lambda (e: T).(match e
534 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
535 \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0)
536 (THead (Bind Abbr) x x4) H27) in (let H29 \def (eq_ind_r T x4 (\lambda (t0:
537 T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H26 x0 H28) in (let H30 \def
538 (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c
539 (Bind b) u) x0 t0)))) H21 x0 H28) in (H29 (refl_equal T x0) P)))))) (pr3_pr2
540 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4
541 (Bind Abbr) (H21 Abbr x))))) H25))) x3 H23))) (\lambda (H23: (((eq T x x3)
542 \to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x3 x4) (\lambda (H24: (eq
543 T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x3 x4))).(\lambda (P:
544 Prop).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T return
545 (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x |
546 (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr)
547 x3 x4) H24) in ((let H26 \def (f_equal T T (\lambda (e: T).(match e in T
548 return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
549 \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0)
550 (THead (Bind Abbr) x3 x4) H24) in (\lambda (H27: (eq T x x3)).(let H28 \def
551 (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c
552 (Bind b) u) x0 t0)))) H21 x0 H26) in (let H29 \def (eq_ind_r T x3 (\lambda
553 (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H23 x H27) in (let H30
554 \def (eq_ind_r T x3 (\lambda (t0: T).(pr2 c x t0)) H15 x H27) in (H29
555 (refl_equal T x) P)))))) H25)))) (pr3_head_12 c x x3 (pr3_pr2 c x x3 H15)
556 (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x3) x0 x4 (H21 Abbr x3)))))
557 H22)))))) H18)) t3 H14)))))))))) H12)) (\lambda (H12: (ex6_6 B T T T T T
558 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
559 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
560 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
561 (THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
562 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
563 (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
564 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
565 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_:
566 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
567 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
568 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
569 y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_:
570 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
571 b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
572 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead
573 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
574 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
575 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
576 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
577 (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
578 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
579 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
580 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3)
581 (\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda
582 (x5: T).(\lambda (x6: T).(\lambda (H13: (not (eq B x1 Abst))).(\lambda (H14:
583 (eq T (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3))).(\lambda (H15: (eq
584 T t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
585 (_: (pr2 c x x5)).(\lambda (H17: (pr2 c x2 x6)).(\lambda (H18: (pr2 (CHead c
586 (Bind x1) x6) x3 x4)).(let H19 \def (eq_ind T t3 (\lambda (t0: T).((eq T
587 (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P:
588 Prop).P))) H9 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))
589 H15) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5)
590 x4)) (\lambda (t0: T).(sn3 c t0)) (let H20 \def (f_equal T B (\lambda (e:
591 T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst |
592 (TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return
593 (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
594 Abst])])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in ((let H21
595 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
596 with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _)
597 \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in
598 ((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
599 T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
600 t0) \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14)
601 in (\lambda (H23: (eq T t2 x2)).(\lambda (H24: (eq B Abst x1)).(let H25 \def
602 (eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H18 x0
603 H22) in (let H26 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H17 t2
604 H23) in (let H27 \def (eq_ind_r B x1 (\lambda (b: B).(pr2 (CHead c (Bind b)
605 x6) x0 x4)) H25 Abst H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b:
606 B).(not (eq B b Abst))) H13 Abst H24) in (eq_ind B Abst (\lambda (b: B).(sn3
607 c (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (let H29
608 \def (match (H28 (refl_equal B Abst)) in False return (\lambda (_:
609 False).(sn3 c (THead (Bind Abst) x6 (THead (Flat Appl) (lift (S O) O x5)
610 x4)))) with []) in H29) x1 H24)))))))) H21)) H20)) t3 H15)))))))))))))) H12))
611 H11))))))))) w H4))))))))))) y H0))))) H)))).
613 theorem sn3_appl_lref:
614 \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (v:
615 T).((sn3 c v) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))
617 \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda
618 (v: T).(\lambda (H0: (sn3 c v)).(sn3_ind c (\lambda (t: T).(sn3 c (THead
619 (Flat Appl) t (TLRef i)))) (\lambda (t1: T).(\lambda (_: ((\forall (t2:
620 T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
621 t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
622 Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c (THead (Flat Appl) t2 (TLRef
623 i)))))))).(sn3_pr2_intro c (THead (Flat Appl) t1 (TLRef i)) (\lambda (t2:
624 T).(\lambda (H3: (((eq T (THead (Flat Appl) t1 (TLRef i)) t2) \to (\forall
625 (P: Prop).P)))).(\lambda (H4: (pr2 c (THead (Flat Appl) t1 (TLRef i))
626 t2)).(let H5 \def (pr2_gen_appl c t1 (TLRef i) t2 H4) in (or3_ind (ex3_2 T T
627 (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
628 (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda
629 (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
630 T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1
631 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
632 T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
633 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))) (\lambda (_:
634 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
635 (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
636 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
637 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
638 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
639 (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
640 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
641 t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
642 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
643 T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1:
644 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
645 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
646 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))
647 (sn3 c t2) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
648 t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1
649 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T
650 T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
651 (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda
652 (t3: T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1:
653 T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr2 c t1
654 x0)).(\lambda (H9: (pr2 c (TLRef i) x1)).(let H10 \def (eq_ind T t2 (\lambda
655 (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to (\forall (P: Prop).P)))
656 H3 (THead (Flat Appl) x0 x1) H7) in (eq_ind_r T (THead (Flat Appl) x0 x1)
657 (\lambda (t: T).(sn3 c t)) (let H11 \def (eq_ind_r T x1 (\lambda (t: T).((eq
658 T (THead (Flat Appl) t1 (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P:
659 Prop).P))) H10 (TLRef i) (H x1 H9)) in (let H12 \def (eq_ind_r T x1 (\lambda
660 (t: T).(pr2 c (TLRef i) t)) H9 (TLRef i) (H x1 H9)) in (eq_ind T (TLRef i)
661 (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H_x \def (term_dec t1
662 x0) in (let H13 \def H_x in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall
663 (P: Prop).P)) (sn3 c (THead (Flat Appl) x0 (TLRef i))) (\lambda (H14: (eq T
664 t1 x0)).(let H15 \def (eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat
665 Appl) t1 (TLRef i)) (THead (Flat Appl) t (TLRef i))) \to (\forall (P:
666 Prop).P))) H11 t1 H14) in (let H16 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c
667 t1 t)) H8 t1 H14) in (eq_ind T t1 (\lambda (t: T).(sn3 c (THead (Flat Appl) t
668 (TLRef i)))) (H15 (refl_equal T (THead (Flat Appl) t1 (TLRef i))) (sn3 c
669 (THead (Flat Appl) t1 (TLRef i)))) x0 H14)))) (\lambda (H14: (((eq T t1 x0)
670 \to (\forall (P: Prop).P)))).(H2 x0 H14 (pr3_pr2 c t1 x0 H8))) H13))) x1 (H
671 x1 H9)))) t2 H7))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1:
672 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead
673 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
674 T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
675 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2)))))
676 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
677 (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T
678 T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
679 (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
680 T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))))
681 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1
682 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
683 T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))
684 (sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
685 T).(\lambda (H7: (eq T (TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H8:
686 (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c t1 x2)).(\lambda (_:
687 ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let
688 H11 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i))
689 t) \to (\forall (P: Prop).P))) H3 (THead (Bind Abbr) x2 x3) H8) in (eq_ind_r
690 T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) (let H12 \def (eq_ind
691 T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
692 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
693 \Rightarrow False])) I (THead (Bind Abst) x0 x1) H7) in (False_ind (sn3 c
694 (THead (Bind Abbr) x2 x3)) H12)) t2 H8)))))))))) H6)) (\lambda (H6: (ex6_6 B
695 T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
696 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
697 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
698 (_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
699 (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
700 T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
701 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
702 T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1:
703 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
704 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
705 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
706 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
707 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
708 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
709 T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind b) y1
710 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
711 T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat
712 Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
713 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2)))))))
714 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
715 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
716 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
717 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2) (\lambda (x0: B).(\lambda (x1:
718 T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
719 T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H8: (eq T (TLRef i) (THead
720 (Bind x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x5 (THead (Flat
721 Appl) (lift (S O) O x4) x3)))).(\lambda (_: (pr2 c t1 x4)).(\lambda (_: (pr2
722 c x1 x5)).(\lambda (_: (pr2 (CHead c (Bind x0) x5) x2 x3)).(let H13 \def
723 (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to
724 (\forall (P: Prop).P))) H3 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O)
725 O x4) x3)) H9) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S
726 O) O x4) x3)) (\lambda (t: T).(sn3 c t)) (let H14 \def (eq_ind T (TLRef i)
727 (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
728 \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
729 False])) I (THead (Bind x0) x1 x2) H8) in (False_ind (sn3 c (THead (Bind x0)
730 x5 (THead (Flat Appl) (lift (S O) O x4) x3))) H14)) t2 H9)))))))))))))) H6))
731 H5))))))))) v H0))))).
733 theorem sn3_appl_abbr:
734 \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c
735 (CHead d (Bind Abbr) w)) \to (\forall (v: T).((sn3 c (THead (Flat Appl) v
736 (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))))
738 \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda
739 (H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (v: T).(\lambda (H0: (sn3 c
740 (THead (Flat Appl) v (lift (S i) O w)))).(insert_eq T (THead (Flat Appl) v
741 (lift (S i) O w)) (\lambda (t: T).(sn3 c t)) (\lambda (_: T).(sn3 c (THead
742 (Flat Appl) v (TLRef i)))) (\lambda (y: T).(\lambda (H1: (sn3 c y)).(unintro
743 T v (\lambda (t: T).((eq T y (THead (Flat Appl) t (lift (S i) O w))) \to (sn3
744 c (THead (Flat Appl) t (TLRef i))))) (sn3_ind c (\lambda (t: T).(\forall (x:
745 T).((eq T t (THead (Flat Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat
746 Appl) x (TLRef i)))))) (\lambda (t1: T).(\lambda (H2: ((\forall (t2:
747 T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
748 t2)))))).(\lambda (H3: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
749 Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).((eq T t2 (THead (Flat
750 Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x (TLRef
751 i)))))))))).(\lambda (x: T).(\lambda (H4: (eq T t1 (THead (Flat Appl) x (lift
752 (S i) O w)))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2:
753 T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall
754 (x0: T).((eq T t2 (THead (Flat Appl) x0 (lift (S i) O w))) \to (sn3 c (THead
755 (Flat Appl) x0 (TLRef i))))))))) H3 (THead (Flat Appl) x (lift (S i) O w))
756 H4) in (let H6 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: T).((((eq T t
757 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (sn3 c t2))))) H2
758 (THead (Flat Appl) x (lift (S i) O w)) H4) in (sn3_pr2_intro c (THead (Flat
759 Appl) x (TLRef i)) (\lambda (t2: T).(\lambda (H7: (((eq T (THead (Flat Appl)
760 x (TLRef i)) t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr2 c (THead
761 (Flat Appl) x (TLRef i)) t2)).(let H9 \def (pr2_gen_appl c x (TLRef i) t2 H8)
762 in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
763 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
764 (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T
765 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
766 (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
767 T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))))
768 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x
769 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
770 T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))
771 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
772 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
773 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
774 (_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
775 (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
776 T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
777 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
778 T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
779 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
780 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
781 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))
782 (sn3 c t2) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
783 t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x
784 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T
785 T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
786 (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3:
787 T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1:
788 T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H12: (pr2 c
789 x x0)).(\lambda (H13: (pr2 c (TLRef i) x1)).(let H14 \def (eq_ind T t2
790 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P:
791 Prop).P))) H7 (THead (Flat Appl) x0 x1) H11) in (eq_ind_r T (THead (Flat
792 Appl) x0 x1) (\lambda (t: T).(sn3 c t)) (let H15 \def (pr2_gen_lref c x1 i
793 H13) in (or_ind (eq T x1 (TLRef i)) (ex2_2 C T (\lambda (d0: C).(\lambda (u:
794 T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq
795 T x1 (lift (S i) O u))))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda (H16:
796 (eq T x1 (TLRef i))).(let H17 \def (eq_ind T x1 (\lambda (t: T).((eq T (THead
797 (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P:
798 Prop).P))) H14 (TLRef i) H16) in (eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c
799 (THead (Flat Appl) x0 t))) (let H_x \def (term_dec x x0) in (let H18 \def H_x
800 in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c (THead
801 (Flat Appl) x0 (TLRef i))) (\lambda (H19: (eq T x x0)).(let H20 \def
802 (eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead
803 (Flat Appl) t (TLRef i))) \to (\forall (P: Prop).P))) H17 x H19) in (let H21
804 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H19) in (eq_ind T x
805 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (TLRef i)))) (H20 (refl_equal T
806 (THead (Flat Appl) x (TLRef i))) (sn3 c (THead (Flat Appl) x (TLRef i)))) x0
807 H19)))) (\lambda (H19: (((eq T x x0) \to (\forall (P: Prop).P)))).(H5 (THead
808 (Flat Appl) x0 (lift (S i) O w)) (\lambda (H20: (eq T (THead (Flat Appl) x
809 (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)))).(\lambda (P:
810 Prop).(let H21 \def (f_equal T T (\lambda (e: T).(match e in T return
811 (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x |
812 (THead _ t _) \Rightarrow t])) (THead (Flat Appl) x (lift (S i) O w)) (THead
813 (Flat Appl) x0 (lift (S i) O w)) H20) in (let H22 \def (eq_ind_r T x0
814 (\lambda (t: T).((eq T x t) \to (\forall (P0: Prop).P0))) H19 x H21) in (let
815 H23 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H21) in (H22
816 (refl_equal T x) P)))))) (pr3_pr2 c (THead (Flat Appl) x (lift (S i) O w))
817 (THead (Flat Appl) x0 (lift (S i) O w)) (pr2_head_1 c x x0 H12 (Flat Appl)
818 (lift (S i) O w))) x0 (refl_equal T (THead (Flat Appl) x0 (lift (S i) O
819 w))))) H18))) x1 H16))) (\lambda (H16: (ex2_2 C T (\lambda (d0: C).(\lambda
820 (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
821 T).(eq T x1 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda
822 (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
823 T).(eq T x1 (lift (S i) O u)))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda
824 (x2: C).(\lambda (x3: T).(\lambda (H17: (getl i c (CHead x2 (Bind Abbr)
825 x3))).(\lambda (H18: (eq T x1 (lift (S i) O x3))).(let H19 \def (eq_ind T x1
826 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0
827 t)) \to (\forall (P: Prop).P))) H14 (lift (S i) O x3) H18) in (eq_ind_r T
828 (lift (S i) O x3) (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H20
829 \def (eq_ind C (CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H
830 (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2
831 (Bind Abbr) x3) H17)) in (let H21 \def (f_equal C C (\lambda (e: C).(match e
832 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
833 \Rightarrow c0])) (CHead d (Bind Abbr) w) (CHead x2 (Bind Abbr) x3)
834 (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 (Bind Abbr) x3) H17)) in
835 ((let H22 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
836 C).T) with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d
837 (Bind Abbr) w) (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w)
838 i H (CHead x2 (Bind Abbr) x3) H17)) in (\lambda (H23: (eq C d x2)).(let H24
839 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c (CHead x2 (Bind Abbr) t))) H20
840 w H22) in (eq_ind T w (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 (lift (S
841 i) O t)))) (let H25 \def (eq_ind_r C x2 (\lambda (c0: C).(getl i c (CHead c0
842 (Bind Abbr) w))) H24 d H23) in (let H_x \def (term_dec x x0) in (let H26 \def
843 H_x in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c
844 (THead (Flat Appl) x0 (lift (S i) O w))) (\lambda (H27: (eq T x x0)).(let H28
845 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H27) in (eq_ind T x
846 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (lift (S i) O w)))) (sn3_sing c
847 (THead (Flat Appl) x (lift (S i) O w)) H6) x0 H27))) (\lambda (H27: (((eq T x
848 x0) \to (\forall (P: Prop).P)))).(H6 (THead (Flat Appl) x0 (lift (S i) O w))
849 (\lambda (H28: (eq T (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat
850 Appl) x0 (lift (S i) O w)))).(\lambda (P: Prop).(let H29 \def (f_equal T T
851 (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
852 \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t]))
853 (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O
854 w)) H28) in (let H30 \def (eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to
855 (\forall (P0: Prop).P0))) H27 x H29) in (let H31 \def (eq_ind_r T x0 (\lambda
856 (t: T).(pr2 c x t)) H12 x H29) in (H30 (refl_equal T x) P)))))) (pr3_pr2 c
857 (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O
858 w)) (pr2_head_1 c x x0 H12 (Flat Appl) (lift (S i) O w))))) H26)))) x3
859 H22)))) H21))) x1 H18)))))) H16)) H15)) t2 H11))))))) H10)) (\lambda (H10:
860 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
861 T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
862 (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
863 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
864 T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
865 (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1
866 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
867 T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda
868 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
869 (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
870 T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
871 (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
872 b) u) z1 t3))))))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2:
873 T).(\lambda (x3: T).(\lambda (H11: (eq T (TLRef i) (THead (Bind Abst) x0
874 x1))).(\lambda (H12: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c
875 x x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b)
876 u) x1 x3))))).(let H15 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat
877 Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 (THead (Bind Abbr) x2
878 x3) H12) in (eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t))
879 (let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return
880 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
881 \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0
882 x1) H11) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H16)) t2
883 H12)))))))))) H10)) (\lambda (H10: (ex6_6 B T T T T T (\lambda (b:
884 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
885 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
886 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i)
887 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
888 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
889 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
890 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
891 (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
892 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
893 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
894 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind
895 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
896 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
897 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
898 (_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
899 (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
900 T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
901 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
902 T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
903 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
904 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
905 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
906 (sn3 c t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
907 T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0
908 Abst))).(\lambda (H12: (eq T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda
909 (H13: (eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4)
910 x3)))).(\lambda (_: (pr2 c x x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_:
911 (pr2 (CHead c (Bind x0) x5) x2 x3)).(let H17 \def (eq_ind T t2 (\lambda (t:
912 T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7
913 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) H13) in
914 (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))
915 (\lambda (t: T).(sn3 c t)) (let H18 \def (eq_ind T (TLRef i) (\lambda (ee:
916 T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
917 False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
918 (THead (Bind x0) x1 x2) H12) in (False_ind (sn3 c (THead (Bind x0) x5 (THead
919 (Flat Appl) (lift (S O) O x4) x3))) H18)) t2 H13)))))))))))))) H10))
920 H9))))))))))))) y H1)))) H0))))))).
922 theorem sn3_appl_cast:
923 \forall (c: C).(\forall (v: T).(\forall (u: T).((sn3 c (THead (Flat Appl) v
924 u)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) v t)) \to (sn3 c (THead
925 (Flat Appl) v (THead (Flat Cast) u t))))))))
927 \lambda (c: C).(\lambda (v: T).(\lambda (u: T).(\lambda (H: (sn3 c (THead
928 (Flat Appl) v u))).(insert_eq T (THead (Flat Appl) v u) (\lambda (t: T).(sn3
929 c t)) (\lambda (_: T).(\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to
930 (sn3 c (THead (Flat Appl) v (THead (Flat Cast) u t0)))))) (\lambda (y:
931 T).(\lambda (H0: (sn3 c y)).(unintro T u (\lambda (t: T).((eq T y (THead
932 (Flat Appl) v t)) \to (\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to
933 (sn3 c (THead (Flat Appl) v (THead (Flat Cast) t t0))))))) (unintro T v
934 (\lambda (t: T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to
935 (\forall (t0: T).((sn3 c (THead (Flat Appl) t t0)) \to (sn3 c (THead (Flat
936 Appl) t (THead (Flat Cast) x t0)))))))) (sn3_ind c (\lambda (t: T).(\forall
937 (x: T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (t0:
938 T).((sn3 c (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead
939 (Flat Cast) x0 t0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2:
940 T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
941 t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
942 Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2
943 (THead (Flat Appl) x x0)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) x
944 t)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0
945 t))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1 (THead
946 (Flat Appl) x x0))).(\lambda (t: T).(\lambda (H4: (sn3 c (THead (Flat Appl) x
947 t))).(insert_eq T (THead (Flat Appl) x t) (\lambda (t0: T).(sn3 c t0))
948 (\lambda (_: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 t))))
949 (\lambda (y0: T).(\lambda (H5: (sn3 c y0)).(unintro T t (\lambda (t0: T).((eq
950 T y0 (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead (Flat
951 Cast) x0 t0))))) (sn3_ind c (\lambda (t0: T).(\forall (x1: T).((eq T t0
952 (THead (Flat Appl) x x1)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast)
953 x0 x1)))))) (\lambda (t0: T).(\lambda (H6: ((\forall (t2: T).((((eq T t0 t2)
954 \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2)))))).(\lambda
955 (H7: ((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3
956 c t0 t2) \to (\forall (x1: T).((eq T t2 (THead (Flat Appl) x x1)) \to (sn3 c
957 (THead (Flat Appl) x (THead (Flat Cast) x0 x1)))))))))).(\lambda (x1:
958 T).(\lambda (H8: (eq T t0 (THead (Flat Appl) x x1))).(let H9 \def (eq_ind T
959 t0 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
960 Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).((eq T t3 (THead (Flat
961 Appl) x x2)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0
962 x2))))))))) H7 (THead (Flat Appl) x x1) H8) in (let H10 \def (eq_ind T t0
963 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P)))
964 \to ((pr3 c t2 t3) \to (sn3 c t3))))) H6 (THead (Flat Appl) x x1) H8) in (let
965 H11 \def (eq_ind T t1 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to
966 (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).(\forall (x3:
967 T).((eq T t3 (THead (Flat Appl) x2 x3)) \to (\forall (t4: T).((sn3 c (THead
968 (Flat Appl) x2 t4)) \to (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x3
969 t4)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H12 \def (eq_ind T t1
970 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P)))
971 \to ((pr3 c t2 t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in
972 (sn3_pr2_intro c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (\lambda
973 (t2: T).(\lambda (H13: (((eq T (THead (Flat Appl) x (THead (Flat Cast) x0
974 x1)) t2) \to (\forall (P: Prop).P)))).(\lambda (H14: (pr2 c (THead (Flat
975 Appl) x (THead (Flat Cast) x0 x1)) t2)).(let H15 \def (pr2_gen_appl c x
976 (THead (Flat Cast) x0 x1) t2 H14) in (or3_ind (ex3_2 T T (\lambda (u2:
977 T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
978 T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
979 (THead (Flat Cast) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
980 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1)
981 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
982 T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
983 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
984 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b:
985 B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T
986 T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
987 (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
988 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
989 T (THead (Flat Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b:
990 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
991 (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
992 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
993 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_:
994 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
995 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
996 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
997 y2) z1 z2)))))))) (sn3 c t2) (\lambda (H16: (ex3_2 T T (\lambda (u2:
998 T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
999 T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
1000 (THead (Flat Cast) x0 x1) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
1001 (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
1002 T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Flat Cast)
1003 x0 x1) t3))) (sn3 c t2) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H17: (eq
1004 T t2 (THead (Flat Appl) x2 x3))).(\lambda (H18: (pr2 c x x2)).(\lambda (H19:
1005 (pr2 c (THead (Flat Cast) x0 x1) x3)).(let H20 \def (eq_ind T t2 (\lambda
1006 (t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to
1007 (\forall (P: Prop).P))) H13 (THead (Flat Appl) x2 x3) H17) in (eq_ind_r T
1008 (THead (Flat Appl) x2 x3) (\lambda (t3: T).(sn3 c t3)) (let H21 \def
1009 (pr2_gen_cast c x0 x1 x3 H19) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda
1010 (t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
1011 T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3)))) (pr2 c
1012 x1 x3) (sn3 c (THead (Flat Appl) x2 x3)) (\lambda (H22: (ex3_2 T T (\lambda
1013 (u2: T).(\lambda (t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
1014 T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1
1015 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x3 (THead
1016 (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
1017 (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3))) (sn3 c (THead (Flat Appl) x2
1018 x3)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H23: (eq T x3 (THead (Flat
1019 Cast) x4 x5))).(\lambda (H24: (pr2 c x0 x4)).(\lambda (H25: (pr2 c x1
1020 x5)).(let H26 \def (eq_ind T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x
1021 (THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P:
1022 Prop).P))) H20 (THead (Flat Cast) x4 x5) H23) in (eq_ind_r T (THead (Flat
1023 Cast) x4 x5) (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 t3))) (let H_x
1024 \def (term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) in (let
1025 H27 \def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2
1026 x4)) ((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) \to (\forall
1027 (P: Prop).P)) (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x4 x5)))
1028 (\lambda (H28: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2
1029 x4))).(let H29 \def (f_equal T T (\lambda (e: T).(match e in T return
1030 (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x |
1031 (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x0) (THead (Flat Appl)
1032 x2 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T
1033 return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
1034 \Rightarrow x0 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x0)
1035 (THead (Flat Appl) x2 x4) H28) in (\lambda (H31: (eq T x x2)).(let H32 \def
1036 (eq_ind_r T x4 (\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat
1037 Cast) x0 x1)) (THead (Flat Appl) x2 (THead (Flat Cast) t3 x5))) \to (\forall
1038 (P: Prop).P))) H26 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t3:
1039 T).(pr2 c x0 t3)) H24 x0 H30) in (eq_ind T x0 (\lambda (t3: T).(sn3 c (THead
1040 (Flat Appl) x2 (THead (Flat Cast) t3 x5)))) (let H34 \def (eq_ind_r T x2
1041 (\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1))
1042 (THead (Flat Appl) t3 (THead (Flat Cast) x0 x5))) \to (\forall (P: Prop).P)))
1043 H32 x H31) in (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18
1044 x H31) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead
1045 (Flat Cast) x0 x5)))) (let H_x0 \def (term_dec (THead (Flat Appl) x x1)
1046 (THead (Flat Appl) x x5)) in (let H36 \def H_x0 in (or_ind (eq T (THead (Flat
1047 Appl) x x1) (THead (Flat Appl) x x5)) ((eq T (THead (Flat Appl) x x1) (THead
1048 (Flat Appl) x x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x
1049 (THead (Flat Cast) x0 x5))) (\lambda (H37: (eq T (THead (Flat Appl) x x1)
1050 (THead (Flat Appl) x x5))).(let H38 \def (f_equal T T (\lambda (e: T).(match
1051 e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _)
1052 \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x1)
1053 (THead (Flat Appl) x x5) H37) in (let H39 \def (eq_ind_r T x5 (\lambda (t3:
1054 T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl)
1055 x (THead (Flat Cast) x0 t3))) \to (\forall (P: Prop).P))) H34 x1 H38) in (let
1056 H40 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H38) in
1057 (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast)
1058 x0 t3)))) (H39 (refl_equal T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)))
1059 (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)))) x5 H38))))) (\lambda
1060 (H37: (((eq T (THead (Flat Appl) x x1) (THead (Flat Appl) x x5)) \to (\forall
1061 (P: Prop).P)))).(H9 (THead (Flat Appl) x x5) H37 (pr3_pr2 c (THead (Flat
1062 Appl) x x1) (THead (Flat Appl) x x5) (pr2_thin_dx c x1 x5 H25 x Appl)) x5
1063 (refl_equal T (THead (Flat Appl) x x5)))) H36))) x2 H31))) x4 H30))))) H29)))
1064 (\lambda (H28: (((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4))
1065 \to (\forall (P: Prop).P)))).(let H_x0 \def (term_dec (THead (Flat Appl) x
1066 x1) (THead (Flat Appl) x2 x5)) in (let H29 \def H_x0 in (or_ind (eq T (THead
1067 (Flat Appl) x x1) (THead (Flat Appl) x2 x5)) ((eq T (THead (Flat Appl) x x1)
1068 (THead (Flat Appl) x2 x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat
1069 Appl) x2 (THead (Flat Cast) x4 x5))) (\lambda (H30: (eq T (THead (Flat Appl)
1070 x x1) (THead (Flat Appl) x2 x5))).(let H31 \def (f_equal T T (\lambda (e:
1071 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x |
1072 (TLRef _) \Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl)
1073 x x1) (THead (Flat Appl) x2 x5) H30) in ((let H32 \def (f_equal T T (\lambda
1074 (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1
1075 | (TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat
1076 Appl) x x1) (THead (Flat Appl) x2 x5) H30) in (\lambda (H33: (eq T x
1077 x2)).(let H34 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H32)
1078 in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 (THead (Flat
1079 Cast) x4 t3)))) (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead
1080 (Flat Appl) x x0) (THead (Flat Appl) t3 x4)) \to (\forall (P: Prop).P))) H28
1081 x H33) in (let H36 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18 x
1082 H33) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead
1083 (Flat Cast) x4 x1)))) (H11 (THead (Flat Appl) x x4) H35 (pr3_pr2 c (THead
1084 (Flat Appl) x x0) (THead (Flat Appl) x x4) (pr2_thin_dx c x0 x4 H24 x Appl))
1085 x x4 (refl_equal T (THead (Flat Appl) x x4)) x1 (sn3_sing c (THead (Flat
1086 Appl) x x1) H10)) x2 H33))) x5 H32)))) H31))) (\lambda (H30: (((eq T (THead
1087 (Flat Appl) x x1) (THead (Flat Appl) x2 x5)) \to (\forall (P:
1088 Prop).P)))).(H11 (THead (Flat Appl) x2 x4) H28 (pr3_flat c x x2 (pr3_pr2 c x
1089 x2 H18) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x2 x4 (refl_equal T (THead (Flat
1090 Appl) x2 x4)) x5 (H10 (THead (Flat Appl) x2 x5) H30 (pr3_flat c x x2 (pr3_pr2
1091 c x x2 H18) x1 x5 (pr3_pr2 c x1 x5 H25) Appl)))) H29)))) H27))) x3 H23)))))))
1092 H22)) (\lambda (H22: (pr2 c x1 x3)).(let H_x \def (term_dec (THead (Flat
1093 Appl) x x1) (THead (Flat Appl) x2 x3)) in (let H23 \def H_x in (or_ind (eq T
1094 (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3)) ((eq T (THead (Flat Appl)
1095 x x1) (THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)) (sn3 c (THead
1096 (Flat Appl) x2 x3)) (\lambda (H24: (eq T (THead (Flat Appl) x x1) (THead
1097 (Flat Appl) x2 x3))).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T
1098 return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _)
1099 \Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x1)
1100 (THead (Flat Appl) x2 x3) H24) in ((let H26 \def (f_equal T T (\lambda (e:
1101 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 |
1102 (TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat
1103 Appl) x x1) (THead (Flat Appl) x2 x3) H24) in (\lambda (H27: (eq T x
1104 x2)).(let H28 \def (eq_ind_r T x3 (\lambda (t3: T).(pr2 c x1 t3)) H22 x1 H26)
1105 in (let H29 \def (eq_ind_r T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x
1106 (THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P:
1107 Prop).P))) H20 x1 H26) in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat
1108 Appl) x2 t3))) (let H30 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead
1109 (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) t3 x1)) \to
1110 (\forall (P: Prop).P))) H29 x H27) in (let H31 \def (eq_ind_r T x2 (\lambda
1111 (t3: T).(pr2 c x t3)) H18 x H27) in (eq_ind T x (\lambda (t3: T).(sn3 c
1112 (THead (Flat Appl) t3 x1))) (sn3_sing c (THead (Flat Appl) x x1) H10) x2
1113 H27))) x3 H26))))) H25))) (\lambda (H24: (((eq T (THead (Flat Appl) x x1)
1114 (THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat
1115 Appl) x2 x3) H24 (pr3_flat c x x2 (pr3_pr2 c x x2 H18) x1 x3 (pr3_pr2 c x1 x3
1116 H22) Appl))) H23)))) H21)) t2 H17))))))) H16)) (\lambda (H16: (ex4_4 T T T T
1117 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
1118 (THead (Flat Cast) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1119 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
1120 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1121 (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1122 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1123 u0) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1:
1124 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead
1125 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1126 T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
1127 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
1128 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b:
1129 B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))) (sn3 c t2)
1130 (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda
1131 (H17: (eq T (THead (Flat Cast) x0 x1) (THead (Bind Abst) x2 x3))).(\lambda
1132 (H18: (eq T t2 (THead (Bind Abbr) x4 x5))).(\lambda (_: (pr2 c x
1133 x4)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1134 u0) x3 x5))))).(let H21 \def (eq_ind T t2 (\lambda (t3: T).((eq T (THead
1135 (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P: Prop).P))) H13
1136 (THead (Bind Abbr) x4 x5) H18) in (eq_ind_r T (THead (Bind Abbr) x4 x5)
1137 (\lambda (t3: T).(sn3 c t3)) (let H22 \def (eq_ind T (THead (Flat Cast) x0
1138 x1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
1139 _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
1140 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
1141 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) x2
1142 x3) H17) in (False_ind (sn3 c (THead (Bind Abbr) x4 x5)) H22)) t2
1143 H18)))))))))) H16)) (\lambda (H16: (ex6_6 B T T T T T (\lambda (b:
1144 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1145 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
1146 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat
1147 Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
1148 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
1149 t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
1150 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1151 T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
1152 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
1153 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
1154 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
1155 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1156 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1157 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1158 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead
1159 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1160 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
1161 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
1162 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1163 (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
1164 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
1165 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
1166 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2)
1167 (\lambda (x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda
1168 (x6: T).(\lambda (x7: T).(\lambda (_: (not (eq B x2 Abst))).(\lambda (H18:
1169 (eq T (THead (Flat Cast) x0 x1) (THead (Bind x2) x3 x4))).(\lambda (H19: (eq
1170 T t2 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda
1171 (_: (pr2 c x x6)).(\lambda (_: (pr2 c x3 x7)).(\lambda (_: (pr2 (CHead c
1172 (Bind x2) x7) x4 x5)).(let H23 \def (eq_ind T t2 (\lambda (t3: T).((eq T
1173 (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P:
1174 Prop).P))) H13 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5))
1175 H19) in (eq_ind_r T (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6)
1176 x5)) (\lambda (t3: T).(sn3 c t3)) (let H24 \def (eq_ind T (THead (Flat Cast)
1177 x0 x1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
1178 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
1179 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
1180 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x2) x3 x4)
1181 H18) in (False_ind (sn3 c (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O)
1182 O x6) x5))) H24)) t2 H19)))))))))))))) H16)) H15))))))))))))))) y0 H5))))
1183 H4))))))))) y H0))))) H)))).
1185 theorem sn3_appl_bind:
1186 \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u:
1187 T).((sn3 c u) \to (\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) u)
1188 (THead (Flat Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v
1189 (THead (Bind b) u t))))))))))
1191 \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda
1192 (u: T).(\lambda (H0: (sn3 c u)).(sn3_ind c (\lambda (t: T).(\forall (t0:
1193 T).(\forall (v: T).((sn3 (CHead c (Bind b) t) (THead (Flat Appl) (lift (S O)
1194 O v) t0)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t t0)))))))
1195 (\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
1196 (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall
1197 (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
1198 (\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) t2) (THead (Flat
1199 Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t2
1200 t))))))))))).(\lambda (t: T).(\lambda (v: T).(\lambda (H3: (sn3 (CHead c
1201 (Bind b) t1) (THead (Flat Appl) (lift (S O) O v) t))).(insert_eq T (THead
1202 (Flat Appl) (lift (S O) O v) t) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1)
1203 t0)) (\lambda (_: T).(sn3 c (THead (Flat Appl) v (THead (Bind b) t1 t))))
1204 (\lambda (y: T).(\lambda (H4: (sn3 (CHead c (Bind b) t1) y)).(unintro T t
1205 (\lambda (t0: T).((eq T y (THead (Flat Appl) (lift (S O) O v) t0)) \to (sn3 c
1206 (THead (Flat Appl) v (THead (Bind b) t1 t0))))) (unintro T v (\lambda (t0:
1207 T).(\forall (x: T).((eq T y (THead (Flat Appl) (lift (S O) O t0) x)) \to (sn3
1208 c (THead (Flat Appl) t0 (THead (Bind b) t1 x)))))) (sn3_ind (CHead c (Bind b)
1209 t1) (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat
1210 Appl) (lift (S O) O x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b)
1211 t1 x0))))))) (\lambda (t2: T).(\lambda (H5: ((\forall (t3: T).((((eq T t2 t3)
1212 \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3
1213 (CHead c (Bind b) t1) t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t2
1214 t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to
1215 (\forall (x: T).(\forall (x0: T).((eq T t3 (THead (Flat Appl) (lift (S O) O
1216 x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) t1
1217 x0))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H7: (eq T t2 (THead
1218 (Flat Appl) (lift (S O) O x) x0))).(let H8 \def (eq_ind T t2 (\lambda (t0:
1219 T).(\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3
1220 (CHead c (Bind b) t1) t0 t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3
1221 (THead (Flat Appl) (lift (S O) O x1) x2)) \to (sn3 c (THead (Flat Appl) x1
1222 (THead (Bind b) t1 x2)))))))))) H6 (THead (Flat Appl) (lift (S O) O x) x0)
1223 H7) in (let H9 \def (eq_ind T t2 (\lambda (t0: T).(\forall (t3: T).((((eq T
1224 t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t0 t3) \to
1225 (sn3 (CHead c (Bind b) t1) t3))))) H5 (THead (Flat Appl) (lift (S O) O x) x0)
1226 H7) in (sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind b) t1 x0)) (\lambda
1227 (t3: T).(\lambda (H10: (((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0))
1228 t3) \to (\forall (P: Prop).P)))).(\lambda (H11: (pr2 c (THead (Flat Appl) x
1229 (THead (Bind b) t1 x0)) t3)).(let H12 \def (pr2_gen_appl c x (THead (Bind b)
1230 t1 x0) t3 H11) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
1231 t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x
1232 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))))
1233 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
1234 T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1235 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
1236 Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1237 (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1238 T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0)
1239 u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_:
1240 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
1241 b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda
1242 (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead
1243 (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
1244 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
1245 b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
1246 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1247 (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
1248 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
1249 (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
1250 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2)))))))) (sn3 c t3)
1251 (\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
1252 (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
1253 (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0)
1254 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
1255 (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
1256 (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))) (sn3 c
1257 t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H14: (eq T t3 (THead (Flat
1258 Appl) x1 x2))).(\lambda (H15: (pr2 c x x1)).(\lambda (H16: (pr2 c (THead
1259 (Bind b) t1 x0) x2)).(let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T
1260 (THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P)))
1261 H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat Appl) x1 x2)
1262 (\lambda (t0: T).(sn3 c t0)) (let H_x \def (pr3_gen_bind b H c t1 x0 x2) in
1263 (let H18 \def (H_x (pr3_pr2 c (THead (Bind b) t1 x0) x2 H16)) in (or_ind
1264 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2
1265 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
1266 T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4)))) (pr3 (CHead c (Bind
1267 b) t1) x0 (lift (S O) O x2)) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (H19:
1268 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2
1269 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
1270 T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4))))).(ex3_2_ind T T
1271 (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 t4)))) (\lambda
1272 (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3
1273 (CHead c (Bind b) t1) x0 t4))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda
1274 (x3: T).(\lambda (x4: T).(\lambda (H20: (eq T x2 (THead (Bind b) x3
1275 x4))).(\lambda (H21: (pr3 c t1 x3)).(\lambda (H22: (pr3 (CHead c (Bind b) t1)
1276 x0 x4)).(let H23 \def (eq_ind T x2 (\lambda (t0: T).((eq T (THead (Flat Appl)
1277 x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 t0)) \to (\forall (P:
1278 Prop).P))) H17 (THead (Bind b) x3 x4) H20) in (eq_ind_r T (THead (Bind b) x3
1279 x4) (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 t0))) (let H_x0 \def
1280 (term_dec t1 x3) in (let H24 \def H_x0 in (or_ind (eq T t1 x3) ((eq T t1 x3)
1281 \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind b) x3
1282 x4))) (\lambda (H25: (eq T t1 x3)).(let H26 \def (eq_ind_r T x3 (\lambda (t0:
1283 T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1
1284 (THead (Bind b) t0 x4))) \to (\forall (P: Prop).P))) H23 t1 H25) in (let H27
1285 \def (eq_ind_r T x3 (\lambda (t0: T).(pr3 c t1 t0)) H21 t1 H25) in (eq_ind T
1286 t1 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 (THead (Bind b) t0 x4))))
1287 (let H_x1 \def (term_dec x0 x4) in (let H28 \def H_x1 in (or_ind (eq T x0 x4)
1288 ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead
1289 (Bind b) t1 x4))) (\lambda (H29: (eq T x0 x4)).(let H30 \def (eq_ind_r T x4
1290 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead
1291 (Flat Appl) x1 (THead (Bind b) t1 t0))) \to (\forall (P: Prop).P))) H26 x0
1292 H29) in (let H31 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b)
1293 t1) x0 t0)) H22 x0 H29) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat
1294 Appl) x1 (THead (Bind b) t1 t0)))) (let H_x2 \def (term_dec x x1) in (let H32
1295 \def H_x2 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3
1296 c (THead (Flat Appl) x1 (THead (Bind b) t1 x0))) (\lambda (H33: (eq T x
1297 x1)).(let H34 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl)
1298 x (THead (Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to
1299 (\forall (P: Prop).P))) H30 x H33) in (let H35 \def (eq_ind_r T x1 (\lambda
1300 (t0: T).(pr2 c x t0)) H15 x H33) in (eq_ind T x (\lambda (t0: T).(sn3 c
1301 (THead (Flat Appl) t0 (THead (Bind b) t1 x0)))) (H34 (refl_equal T (THead
1302 (Flat Appl) x (THead (Bind b) t1 x0))) (sn3 c (THead (Flat Appl) x (THead
1303 (Bind b) t1 x0)))) x1 H33)))) (\lambda (H33: (((eq T x x1) \to (\forall (P:
1304 Prop).P)))).(H8 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H34: (eq T
1305 (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
1306 x0))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: T).(match e
1307 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
1308 (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
1309 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
1310 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
1311 \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
1312 lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
1313 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
1314 t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
1315 (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
1316 (THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1317 t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
1318 \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
1319 (lift (S O) O x1) x0) H34) in (let H36 \def (eq_ind_r T x1 (\lambda (t0:
1320 T).((eq T x t0) \to (\forall (P0: Prop).P0))) H33 x (lift_inj x x1 (S O) O
1321 H35)) in (let H37 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
1322 (lift_inj x x1 (S O) O H35)) in (H36 (refl_equal T x) P)))))) (pr3_flat
1323 (CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
1324 (Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
1325 (pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl) x1 x0
1326 (refl_equal T (THead (Flat Appl) (lift (S O) O x1) x0)))) H32))) x4 H29))))
1327 (\lambda (H29: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H8 (THead (Flat
1328 Appl) (lift (S O) O x1) x4) (\lambda (H30: (eq T (THead (Flat Appl) (lift (S
1329 O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P:
1330 Prop).(let H31 \def (f_equal T T (\lambda (e: T).(match e in T return
1331 (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
1332 \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
1333 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
1334 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
1335 \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
1336 lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
1337 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
1338 t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
1339 (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
1340 (THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1341 t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
1342 \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
1343 (lift (S O) O x1) x4) H30) in ((let H32 \def (f_equal T T (\lambda (e:
1344 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
1345 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
1346 Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H30) in
1347 (\lambda (H33: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H34 \def
1348 (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0)))
1349 H29 x0 H32) in (let H35 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T (THead
1350 (Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 (THead (Bind b)
1351 t1 t0))) \to (\forall (P0: Prop).P0))) H26 x0 H32) in (let H36 \def (eq_ind_r
1352 T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) H22 x0 H32) in (let
1353 H37 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead
1354 (Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to (\forall
1355 (P0: Prop).P0))) H35 x (lift_inj x x1 (S O) O H33)) in (let H38 \def
1356 (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O
1357 H33)) in (H34 (refl_equal T x0) P)))))))) H31)))) (pr3_flat (CHead c (Bind b)
1358 t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S
1359 O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15))
1360 x0 x4 H22 Appl) x1 x4 (refl_equal T (THead (Flat Appl) (lift (S O) O x1)
1361 x4)))) H28))) x3 H25)))) (\lambda (H25: (((eq T t1 x3) \to (\forall (P:
1362 Prop).P)))).(H2 x3 H25 H21 x4 x1 (sn3_cpr3_trans c t1 x3 H21 (Bind b) (THead
1363 (Flat Appl) (lift (S O) O x1) x4) (let H_x1 \def (term_dec x0 x4) in (let H26
1364 \def H_x1 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P))
1365 (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x4)) (\lambda
1366 (H27: (eq T x0 x4)).(let H28 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead
1367 c (Bind b) t1) x0 t0)) H22 x0 H27) in (eq_ind T x0 (\lambda (t0: T).(sn3
1368 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) t0))) (let H_x2
1369 \def (term_dec x x1) in (let H29 \def H_x2 in (or_ind (eq T x x1) ((eq T x
1370 x1) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl)
1371 (lift (S O) O x1) x0)) (\lambda (H30: (eq T x x1)).(let H31 \def (eq_ind_r T
1372 x1 (\lambda (t0: T).(pr2 c x t0)) H15 x H30) in (eq_ind T x (\lambda (t0:
1373 T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0)))
1374 (sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9)
1375 x1 H30))) (\lambda (H30: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9
1376 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H31: (eq T (THead (Flat
1377 Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
1378 x0))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e
1379 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
1380 (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
1381 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
1382 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
1383 \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
1384 lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
1385 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
1386 t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
1387 (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
1388 (THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1389 t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
1390 \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
1391 (lift (S O) O x1) x0) H31) in (let H33 \def (eq_ind_r T x1 (\lambda (t0:
1392 T).((eq T x t0) \to (\forall (P0: Prop).P0))) H30 x (lift_inj x x1 (S O) O
1393 H32)) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
1394 (lift_inj x x1 (S O) O H32)) in (H33 (refl_equal T x) P)))))) (pr3_flat
1395 (CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
1396 (Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
1397 (pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl)))
1398 H29))) x4 H27))) (\lambda (H27: (((eq T x0 x4) \to (\forall (P:
1399 Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x1) x4) (\lambda (H28: (eq T
1400 (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
1401 x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e
1402 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
1403 (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
1404 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
1405 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
1406 \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
1407 lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
1408 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
1409 t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
1410 (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
1411 (THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1412 t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
1413 \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
1414 (lift (S O) O x1) x4) H28) in ((let H30 \def (f_equal T T (\lambda (e:
1415 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
1416 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
1417 Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H28) in
1418 (\lambda (H31: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H32 \def
1419 (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0)))
1420 H27 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c
1421 (Bind b) t1) x0 t0)) H22 x0 H30) in (let H34 \def (eq_ind_r T x1 (\lambda
1422 (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H31)) in (H32 (refl_equal
1423 T x0) P)))))) H29)))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift
1424 (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c
1425 c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x4 H22 Appl))) H26))))))
1426 H24))) x2 H20))))))) H19)) (\lambda (H19: (pr3 (CHead c (Bind b) t1) x0 (lift
1427 (S O) O x2))).(sn3_gen_lift (CHead c (Bind b) t1) (THead (Flat Appl) x1 x2)
1428 (S O) O (eq_ind_r T (THead (Flat Appl) (lift (S O) O x1) (lift (S O) (s (Flat
1429 Appl) O) x2)) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) t0)) (sn3_pr3_trans
1430 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x0) (let H_x0 \def
1431 (term_dec x x1) in (let H20 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to
1432 (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S
1433 O) O x1) x0)) (\lambda (H21: (eq T x x1)).(let H22 \def (eq_ind_r T x1
1434 (\lambda (t0: T).(pr2 c x t0)) H15 x H21) in (eq_ind T x (\lambda (t0:
1435 T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0)))
1436 (sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9)
1437 x1 H21))) (\lambda (H21: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9
1438 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H22: (eq T (THead (Flat
1439 Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
1440 x0))).(\lambda (P: Prop).(let H23 \def (f_equal T T (\lambda (e: T).(match e
1441 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
1442 (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
1443 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
1444 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
1445 \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
1446 lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (TLRef _) \Rightarrow
1447 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
1448 t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
1449 (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
1450 (THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1451 t4))]) in lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (THead _ t0 _)
1452 \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
1453 (lift (S O) O x1) x0) H22) in (let H24 \def (eq_ind_r T x1 (\lambda (t0:
1454 T).((eq T x t0) \to (\forall (P0: Prop).P0))) H21 x (lift_inj x x1 (S O) O
1455 H23)) in (let H25 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
1456 (lift_inj x x1 (S O) O H23)) in (H24 (refl_equal T x) P)))))) (pr3_flat
1457 (CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
1458 (Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
1459 (pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl)))
1460 H20))) (THead (Flat Appl) (lift (S O) O x1) (lift (S O) O x2)) (pr3_thin_dx
1461 (CHead c (Bind b) t1) x0 (lift (S O) O x2) H19 (lift (S O) O x1) Appl)) (lift
1462 (S O) O (THead (Flat Appl) x1 x2)) (lift_head (Flat Appl) x1 x2 (S O) O)) c
1463 (drop_drop (Bind b) O c c (drop_refl c) t1))) H18))) t3 H14))))))) H13))
1464 (\lambda (H13: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1465 T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1
1466 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
1467 T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
1468 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda
1469 (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0:
1470 T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))))).(ex4_4_ind T T T T (\lambda
1471 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
1472 b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
1473 T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4))))))
1474 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x
1475 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4:
1476 T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) z1 t4)))))))
1477 (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
1478 T).(\lambda (H14: (eq T (THead (Bind b) t1 x0) (THead (Bind Abst) x1
1479 x2))).(\lambda (H15: (eq T t3 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c
1480 x x3)).(\lambda (H17: ((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind
1481 b0) u0) x2 x4))))).(let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T (THead
1482 (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) H10
1483 (THead (Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4)
1484 (\lambda (t0: T).(sn3 c t0)) (let H19 \def (f_equal T B (\lambda (e:
1485 T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
1486 (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
1487 (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
1488 b])])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in ((let H20
1489 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
1490 with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _)
1491 \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in
1492 ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
1493 T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
1494 t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14)
1495 in (\lambda (_: (eq T t1 x1)).(\lambda (H23: (eq B b Abst)).(let H24 \def
1496 (eq_ind_r T x2 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead
1497 c (Bind b0) u0) t0 x4)))) H17 x0 H21) in (let H25 \def (eq_ind B b (\lambda
1498 (b0: B).((eq T (THead (Flat Appl) x (THead (Bind b0) t1 x0)) (THead (Bind
1499 Abbr) x3 x4)) \to (\forall (P: Prop).P))) H18 Abst H23) in (let H26 \def
1500 (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T (THead (Flat Appl)
1501 (lift (S O) O x) x0) t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind
1502 b0) t1) (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (sn3 (CHead c (Bind
1503 b0) t1) t4))))) H9 Abst H23) in (let H27 \def (eq_ind B b (\lambda (b0:
1504 B).(\forall (t4: T).((((eq T (THead (Flat Appl) (lift (S O) O x) x0) t4) \to
1505 (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) (THead (Flat Appl)
1506 (lift (S O) O x) x0) t4) \to (\forall (x5: T).(\forall (x6: T).((eq T t4
1507 (THead (Flat Appl) (lift (S O) O x5) x6)) \to (sn3 c (THead (Flat Appl) x5
1508 (THead (Bind b0) t1 x6)))))))))) H8 Abst H23) in (let H28 \def (eq_ind B b
1509 (\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P)))
1510 \to ((pr3 c t1 t4) \to (\forall (t0: T).(\forall (v0: T).((sn3 (CHead c (Bind
1511 b0) t4) (THead (Flat Appl) (lift (S O) O v0) t0)) \to (sn3 c (THead (Flat
1512 Appl) v0 (THead (Bind b0) t4 t0)))))))))) H2 Abst H23) in (let H29 \def
1513 (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H Abst H23) in (let H30
1514 \def (match (H29 (refl_equal B Abst)) in False return (\lambda (_:
1515 False).(sn3 c (THead (Bind Abbr) x3 x4))) with []) in H30)))))))))) H20))
1516 H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 B T T T T T (\lambda (b0:
1517 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1518 (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
1519 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b)
1520 t1 x0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_:
1521 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
1522 t3 (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
1523 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1524 T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
1525 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
1526 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
1527 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1
1528 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda
1529 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0
1530 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1531 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind
1532 b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1533 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b0) y2 (THead
1534 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1535 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x
1536 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1537 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b0:
1538 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1539 (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2))))))) (sn3 c t3) (\lambda (x1:
1540 B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
1541 T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15: (eq T
1542 (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T t3
1543 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
1544 (H17: (pr2 c x x5)).(\lambda (H18: (pr2 c x2 x6)).(\lambda (H19: (pr2 (CHead
1545 c (Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t0: T).((eq T
1546 (THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P)))
1547 H10 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) H16) in
1548 (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))
1549 (\lambda (t0: T).(sn3 c t0)) (let H21 \def (f_equal T B (\lambda (e:
1550 T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
1551 (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
1552 (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
1553 b])])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in ((let H22 \def
1554 (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
1555 [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _)
1556 \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in
1557 ((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
1558 T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
1559 t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in
1560 (\lambda (H24: (eq T t1 x2)).(\lambda (H25: (eq B b x1)).(let H26 \def
1561 (eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H19 x0
1562 H23) in (let H27 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H18 t1
1563 H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b0: B).(pr2 (CHead c (Bind b0)
1564 x6) x0 x4)) H26 b H25) in (eq_ind B b (\lambda (b0: B).(sn3 c (THead (Bind
1565 b0) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (sn3_pr3_trans c (THead
1566 (Bind b) t1 (THead (Flat Appl) (lift (S O) O x5) x4)) (sn3_bind b c t1
1567 (sn3_sing c t1 H1) (THead (Flat Appl) (lift (S O) O x5) x4) (let H_x \def
1568 (term_dec x x5) in (let H29 \def H_x in (or_ind (eq T x x5) ((eq T x x5) \to
1569 (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S
1570 O) O x5) x4)) (\lambda (H30: (eq T x x5)).(let H31 \def (eq_ind_r T x5
1571 (\lambda (t0: T).(pr2 c x t0)) H17 x H30) in (eq_ind T x (\lambda (t0:
1572 T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x4))) (let
1573 H_x0 \def (term_dec x0 x4) in (let H32 \def H_x0 in (or_ind (eq T x0 x4) ((eq
1574 T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat
1575 Appl) (lift (S O) O x) x4)) (\lambda (H33: (eq T x0 x4)).(let H34 \def
1576 (eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0
1577 H33) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat
1578 Appl) (lift (S O) O x) t0))) (sn3_sing (CHead c (Bind b) t1) (THead (Flat
1579 Appl) (lift (S O) O x) x0) H9) x4 H33))) (\lambda (H33: (((eq T x0 x4) \to
1580 (\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x) x4) (\lambda
1581 (H34: (eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
1582 (S O) O x) x4))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e:
1583 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
1584 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
1585 Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x) x4) H34) in
1586 (let H36 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0:
1587 Prop).P0))) H33 x0 H35) in (let H37 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2
1588 (CHead c (Bind b) x6) x0 t0)) H28 x0 H35) in (H36 (refl_equal T x0) P))))))
1589 (pr3_pr3_pr3_t c t1 x6 (pr3_pr2 c t1 x6 H27) (THead (Flat Appl) (lift (S O) O
1590 x) x0) (THead (Flat Appl) (lift (S O) O x) x4) (Bind b) (pr3_pr2 (CHead c
1591 (Bind b) x6) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
1592 (S O) O x) x4) (pr2_thin_dx (CHead c (Bind b) x6) x0 x4 H28 (lift (S O) O x)
1593 Appl))))) H32))) x5 H30))) (\lambda (H30: (((eq T x x5) \to (\forall (P:
1594 Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x5) x4) (\lambda (H31: (eq T
1595 (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5)
1596 x4))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e
1597 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
1598 (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
1599 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
1600 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
1601 \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
1602 lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (TLRef _) \Rightarrow
1603 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
1604 t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
1605 (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
1606 (THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1607 t4))]) in lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (THead _ t0 _)
1608 \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
1609 (lift (S O) O x5) x4) H31) in ((let H33 \def (f_equal T T (\lambda (e:
1610 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
1611 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
1612 Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) x4) H31) in
1613 (\lambda (H34: (eq T (lift (S O) O x) (lift (S O) O x5))).(let H35 \def
1614 (eq_ind_r T x5 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
1615 H30 x (lift_inj x x5 (S O) O H34)) in (let H36 \def (eq_ind_r T x5 (\lambda
1616 (t0: T).(pr2 c x t0)) H17 x (lift_inj x x5 (S O) O H34)) in (let H37 \def
1617 (eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0
1618 H33) in (H35 (refl_equal T x) P)))))) H32)))) (pr3_pr3_pr3_t c t1 x6 (pr3_pr2
1619 c t1 x6 H27) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
1620 (S O) O x5) x4) (Bind b) (pr3_flat (CHead c (Bind b) x6) (lift (S O) O x)
1621 (lift (S O) O x5) (pr3_lift (CHead c (Bind b) x6) c (S O) O (drop_drop (Bind
1622 b) O c c (drop_refl c) x6) x x5 (pr3_pr2 c x x5 H17)) x0 x4 (pr3_pr2 (CHead c
1623 (Bind b) x6) x0 x4 H28) Appl)))) H29)))) (THead (Bind b) x6 (THead (Flat
1624 Appl) (lift (S O) O x5) x4)) (pr3_pr2 c (THead (Bind b) t1 (THead (Flat Appl)
1625 (lift (S O) O x5) x4)) (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O
1626 x5) x4)) (pr2_head_1 c t1 x6 H27 (Bind b) (THead (Flat Appl) (lift (S O) O
1627 x5) x4)))) x1 H25))))))) H22)) H21)) t3 H16)))))))))))))) H13))
1628 H12)))))))))))))) y H4))))) H3))))))) u H0))))).
1630 theorem sn3_appl_appl:
1631 \forall (v1: T).(\forall (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in
1632 (\forall (c: C).((sn3 c u1) \to (\forall (v2: T).((sn3 c v2) \to (((\forall
1633 (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to
1634 (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2
1637 \lambda (v1: T).(\lambda (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in
1638 (\lambda (c: C).(\lambda (H: (sn3 c (THead (Flat Appl) v1 t1))).(insert_eq T
1639 (THead (Flat Appl) v1 t1) (\lambda (t: T).(sn3 c t)) (\lambda (t: T).(\forall
1640 (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2)
1641 \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to
1642 (sn3 c (THead (Flat Appl) v2 t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c
1643 y)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Flat Appl) v1 t)) \to
1644 (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso
1645 y u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2))))))
1646 \to (sn3 c (THead (Flat Appl) v2 y))))))) (unintro T v1 (\lambda (t:
1647 T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to (\forall (v2:
1648 T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso y u2) \to
1649 (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c
1650 (THead (Flat Appl) v2 y)))))))) (sn3_ind c (\lambda (t: T).(\forall (x:
1651 T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (v2:
1652 T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to
1653 (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c
1654 (THead (Flat Appl) v2 t))))))))) (\lambda (t2: T).(\lambda (H1: ((\forall
1655 (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to
1656 (sn3 c t3)))))).(\lambda (H2: ((\forall (t3: T).((((eq T t2 t3) \to (\forall
1657 (P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x: T).(\forall (x0: T).((eq T
1658 t3 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall
1659 (u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to
1660 (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2
1661 t3))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t2
1662 (THead (Flat Appl) x x0))).(\lambda (v2: T).(\lambda (H4: (sn3 c
1663 v2)).(sn3_ind c (\lambda (t: T).(((\forall (u2: T).((pr3 c t2 u2) \to ((((iso
1664 t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t u2))))))
1665 \to (sn3 c (THead (Flat Appl) t t2)))) (\lambda (t0: T).(\lambda (H5:
1666 ((\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0
1667 t3) \to (sn3 c t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t0 t3) \to
1668 (\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to (((\forall (u2: T).((pr3 c t2
1669 u2) \to ((((iso t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat
1670 Appl) t3 u2)))))) \to (sn3 c (THead (Flat Appl) t3 t2)))))))).(\lambda (H7:
1671 ((\forall (u2: T).((pr3 c t2 u2) \to ((((iso t2 u2) \to (\forall (P:
1672 Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2))))))).(let H8 \def (eq_ind T
1673 t2 (\lambda (t: T).(\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to
1674 (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H7 (THead
1675 (Flat Appl) x x0) H3) in (let H9 \def (eq_ind T t2 (\lambda (t: T).(\forall
1676 (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to
1677 (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to (\forall (P:
1678 Prop).P))) \to (sn3 c (THead (Flat Appl) t3 u2)))))) \to (sn3 c (THead (Flat
1679 Appl) t3 t))))))) H6 (THead (Flat Appl) x x0) H3) in (let H10 \def (eq_ind T
1680 t2 (\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P:
1681 Prop).P))) \to ((pr3 c t t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3
1682 (THead (Flat Appl) x1 x2)) \to (\forall (v3: T).((sn3 c v3) \to (((\forall
1683 (u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to
1684 (sn3 c (THead (Flat Appl) v3 u2)))))) \to (sn3 c (THead (Flat Appl) v3
1685 t3)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H11 \def (eq_ind T t2
1686 (\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P: Prop).P)))
1687 \to ((pr3 c t t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in
1688 (eq_ind_r T (THead (Flat Appl) x x0) (\lambda (t: T).(sn3 c (THead (Flat
1689 Appl) t0 t))) (sn3_pr2_intro c (THead (Flat Appl) t0 (THead (Flat Appl) x
1690 x0)) (\lambda (t3: T).(\lambda (H12: (((eq T (THead (Flat Appl) t0 (THead
1691 (Flat Appl) x x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H13: (pr2 c
1692 (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t3)).(let H14 \def
1693 (pr2_gen_appl c t0 (THead (Flat Appl) x x0) t3 H13) in (or3_ind (ex3_2 T T
1694 (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
1695 (\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda
1696 (t4: T).(pr2 c (THead (Flat Appl) x x0) t4)))) (ex4_4 T T T T (\lambda (y1:
1697 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl)
1698 x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
1699 (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
1700 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2)))))
1701 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
1702 (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T
1703 T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1704 (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
1705 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
1706 T (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
1707 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
1708 (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1709 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1710 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_:
1711 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1712 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1713 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
1714 y2) z1 z2)))))))) (sn3 c t3) (\lambda (H15: (ex3_2 T T (\lambda (u2:
1715 T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
1716 T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
1717 (THead (Flat Appl) x x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
1718 (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_:
1719 T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Flat Appl)
1720 x x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H16: (eq T
1721 t3 (THead (Flat Appl) x1 x2))).(\lambda (H17: (pr2 c t0 x1)).(\lambda (H18:
1722 (pr2 c (THead (Flat Appl) x x0) x2)).(let H19 \def (eq_ind T t3 (\lambda (t:
1723 T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P:
1724 Prop).P))) H12 (THead (Flat Appl) x1 x2) H16) in (eq_ind_r T (THead (Flat
1725 Appl) x1 x2) (\lambda (t: T).(sn3 c t)) (let H20 \def (pr2_gen_appl c x x0 x2
1726 H18) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead
1727 (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
1728 (\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4)))) (ex4_4 T T T T (\lambda
1729 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead
1730 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1731 T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
1732 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
1733 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
1734 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T
1735 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1736 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1737 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
1738 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1739 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind
1740 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
1741 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1742 (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
1743 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
1744 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
1745 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c
1746 (THead (Flat Appl) x1 x2)) (\lambda (H21: (ex3_2 T T (\lambda (u2:
1747 T).(\lambda (t4: T).(eq T x2 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
1748 T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c x0
1749 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead
1750 (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
1751 (\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4))) (sn3 c (THead (Flat Appl) x1
1752 x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H22: (eq T x2 (THead (Flat
1753 Appl) x3 x4))).(\lambda (H23: (pr2 c x x3)).(\lambda (H24: (pr2 c x0
1754 x4)).(let H25 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0
1755 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P:
1756 Prop).P))) H19 (THead (Flat Appl) x3 x4) H22) in (eq_ind_r T (THead (Flat
1757 Appl) x3 x4) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H_x \def
1758 (term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) in (let H26
1759 \def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))
1760 ((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) \to (\forall (P:
1761 Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 x4))) (\lambda
1762 (H27: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))).(let H28
1763 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
1764 with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _)
1765 \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27) in
1766 ((let H29 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
1767 T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
1768 t) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27)
1769 in (\lambda (H30: (eq T x x3)).(let H31 \def (eq_ind_r T x4 (\lambda (t:
1770 T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl)
1771 x1 (THead (Flat Appl) x3 t))) \to (\forall (P: Prop).P))) H25 x0 H29) in (let
1772 H32 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) H24 x0 H29) in (eq_ind
1773 T x0 (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 t))))
1774 (let H33 \def (eq_ind_r T x3 (\lambda (t: T).((eq T (THead (Flat Appl) t0
1775 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 (THead (Flat Appl) t x0)))
1776 \to (\forall (P: Prop).P))) H31 x H30) in (let H34 \def (eq_ind_r T x3
1777 (\lambda (t: T).(pr2 c x t)) H23 x H30) in (eq_ind T x (\lambda (t: T).(sn3 c
1778 (THead (Flat Appl) x1 (THead (Flat Appl) t x0)))) (let H_x0 \def (term_dec t0
1779 x1) in (let H35 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to (\forall
1780 (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x x0)))
1781 (\lambda (H36: (eq T t0 x1)).(let H37 \def (eq_ind_r T x1 (\lambda (t:
1782 T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl)
1783 t (THead (Flat Appl) x x0))) \to (\forall (P: Prop).P))) H33 t0 H36) in (let
1784 H38 \def (eq_ind_r T x1 (\lambda (t: T).(pr2 c t0 t)) H17 t0 H36) in (eq_ind
1785 T t0 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (THead (Flat Appl) x x0))))
1786 (H37 (refl_equal T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))) (sn3 c
1787 (THead (Flat Appl) t0 (THead (Flat Appl) x x0)))) x1 H36)))) (\lambda (H36:
1788 (((eq T t0 x1) \to (\forall (P: Prop).P)))).(H9 x1 H36 (pr3_pr2 c t0 x1 H17)
1789 (\lambda (u2: T).(\lambda (H37: (pr3 c (THead (Flat Appl) x x0) u2)).(\lambda
1790 (H38: (((iso (THead (Flat Appl) x x0) u2) \to (\forall (P:
1791 Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H8 u2 H37 H38) (THead
1792 (Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2) (THead (Flat Appl) x1
1793 u2) (pr2_head_1 c t0 x1 H17 (Flat Appl) u2)))))))) H35))) x3 H30))) x4
1794 H29))))) H28))) (\lambda (H27: (((eq T (THead (Flat Appl) x x0) (THead (Flat
1795 Appl) x3 x4)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat Appl) x3 x4) H27
1796 (pr3_flat c x x3 (pr3_pr2 c x x3 H23) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x3 x4
1797 (refl_equal T (THead (Flat Appl) x3 x4)) x1 (sn3_pr3_trans c t0 (sn3_sing c
1798 t0 H5) x1 (pr3_pr2 c t0 x1 H17)) (\lambda (u2: T).(\lambda (H28: (pr3 c
1799 (THead (Flat Appl) x3 x4) u2)).(\lambda (H29: (((iso (THead (Flat Appl) x3
1800 x4) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0
1801 u2) (H8 u2 (pr3_sing c (THead (Flat Appl) x x4) (THead (Flat Appl) x x0)
1802 (pr2_thin_dx c x0 x4 H24 x Appl) u2 (pr3_sing c (THead (Flat Appl) x3 x4)
1803 (THead (Flat Appl) x x4) (pr2_head_1 c x x3 H23 (Flat Appl) x4) u2 H28))
1804 (\lambda (H30: (iso (THead (Flat Appl) x x0) u2)).(\lambda (P: Prop).(H29
1805 (iso_trans (THead (Flat Appl) x3 x4) (THead (Flat Appl) x x0) (iso_head x3 x
1806 x4 x0 (Flat Appl)) u2 H30) P)))) (THead (Flat Appl) x1 u2) (pr3_pr2 c (THead
1807 (Flat Appl) t0 u2) (THead (Flat Appl) x1 u2) (pr2_head_1 c t0 x1 H17 (Flat
1808 Appl) u2)))))))) H26))) x2 H22))))))) H21)) (\lambda (H21: (ex4_4 T T T T
1809 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
1810 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1811 T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
1812 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
1813 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
1814 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T
1815 T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
1816 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1817 T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
1818 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
1819 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
1820 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c (THead (Flat
1821 Appl) x1 x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda
1822 (x6: T).(\lambda (H22: (eq T x0 (THead (Bind Abst) x3 x4))).(\lambda (H23:
1823 (eq T x2 (THead (Bind Abbr) x5 x6))).(\lambda (H24: (pr2 c x x5)).(\lambda
1824 (H25: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x4
1825 x6))))).(let H26 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl)
1826 t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P:
1827 Prop).P))) H19 (THead (Bind Abbr) x5 x6) H23) in (eq_ind_r T (THead (Bind
1828 Abbr) x5 x6) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H27 \def
1829 (eq_ind T x0 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl)
1830 x t)) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6))) \to (\forall (P:
1831 Prop).P))) H26 (THead (Bind Abst) x3 x4) H22) in (let H28 \def (eq_ind T x0
1832 (\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
1833 (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c
1834 t4))))) H11 (THead (Bind Abst) x3 x4) H22) in (let H29 \def (eq_ind T x0
1835 (\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
1836 (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall
1837 (x7: T).(\forall (x8: T).((eq T t4 (THead (Flat Appl) x7 x8)) \to (\forall
1838 (v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2)
1839 \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to
1840 (sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind Abst) x3 x4)
1841 H22) in (let H30 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c
1842 (THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to
1843 (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead
1844 (Bind Abst) x3 x4) H22) in (let H31 \def (eq_ind T x0 (\lambda (t:
1845 T).(\forall (t4: T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c
1846 t0 t4) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso
1847 (THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead
1848 (Flat Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x
1849 t)))))))) H9 (THead (Bind Abst) x3 x4) H22) in (sn3_pr3_trans c (THead (Flat
1850 Appl) t0 (THead (Bind Abbr) x5 x6)) (H30 (THead (Bind Abbr) x5 x6) (pr3_sing
1851 c (THead (Bind Abbr) x x4) (THead (Flat Appl) x (THead (Bind Abst) x3 x4))
1852 (pr2_free c (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind
1853 Abbr) x x4) (pr0_beta x3 x x (pr0_refl x) x4 x4 (pr0_refl x4))) (THead (Bind
1854 Abbr) x5 x6) (pr3_head_12 c x x5 (pr3_pr2 c x x5 H24) (Bind Abbr) x4 x6
1855 (pr3_pr2 (CHead c (Bind Abbr) x5) x4 x6 (H25 Abbr x5)))) (\lambda (H32: (iso
1856 (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind Abbr) x5
1857 x6))).(\lambda (P: Prop).(let H33 \def (match H32 in iso return (\lambda (t:
1858 T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x
1859 (THead (Bind Abst) x3 x4))) \to ((eq T t4 (THead (Bind Abbr) x5 x6)) \to
1860 P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H33: (eq T (TSort n1)
1861 (THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H34: (eq T (TSort
1862 n2) (THead (Bind Abbr) x5 x6))).((let H35 \def (eq_ind T (TSort n1) (\lambda
1863 (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
1864 True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
1865 (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T
1866 (TSort n2) (THead (Bind Abbr) x5 x6)) \to P) H35)) H34))) | (iso_lref i1 i2)
1867 \Rightarrow (\lambda (H33: (eq T (TLRef i1) (THead (Flat Appl) x (THead (Bind
1868 Abst) x3 x4)))).(\lambda (H34: (eq T (TLRef i2) (THead (Bind Abbr) x5
1869 x6))).((let H35 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T
1870 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
1871 \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x
1872 (THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T (TLRef i2) (THead (Bind
1873 Abbr) x5 x6)) \to P) H35)) H34))) | (iso_head v4 v5 t4 t5 k) \Rightarrow
1874 (\lambda (H33: (eq T (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst)
1875 x3 x4)))).(\lambda (H34: (eq T (THead k v5 t5) (THead (Bind Abbr) x5
1876 x6))).((let H35 \def (f_equal T T (\lambda (e: T).(match e in T return
1877 (\lambda (_: T).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4
1878 | (THead _ _ t) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead
1879 (Bind Abst) x3 x4)) H33) in ((let H36 \def (f_equal T T (\lambda (e:
1880 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v4 |
1881 (TLRef _) \Rightarrow v4 | (THead _ t _) \Rightarrow t])) (THead k v4 t4)
1882 (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in ((let H37 \def
1883 (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
1884 [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
1885 \Rightarrow k0])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) x3
1886 x4)) H33) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x) \to ((eq T
1887 t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead k0 v5 t5) (THead (Bind Abbr)
1888 x5 x6)) \to P)))) (\lambda (H38: (eq T v4 x)).(eq_ind T x (\lambda (_:
1889 T).((eq T t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead (Flat Appl) v5 t5)
1890 (THead (Bind Abbr) x5 x6)) \to P))) (\lambda (H39: (eq T t4 (THead (Bind
1891 Abst) x3 x4))).(eq_ind T (THead (Bind Abst) x3 x4) (\lambda (_: T).((eq T
1892 (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6)) \to P)) (\lambda (H40:
1893 (eq T (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6))).(let H41 \def
1894 (eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return
1895 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
1896 \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
1897 (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
1898 True])])) I (THead (Bind Abbr) x5 x6) H40) in (False_ind P H41))) t4 (sym_eq
1899 T t4 (THead (Bind Abst) x3 x4) H39))) v4 (sym_eq T v4 x H38))) k (sym_eq K k
1900 (Flat Appl) H37))) H36)) H35)) H34)))]) in (H33 (refl_equal T (THead (Flat
1901 Appl) x (THead (Bind Abst) x3 x4))) (refl_equal T (THead (Bind Abbr) x5
1902 x6))))))) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6)) (pr3_pr2 c (THead
1903 (Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (THead (Flat Appl) x1 (THead (Bind
1904 Abbr) x5 x6)) (pr2_head_1 c t0 x1 H17 (Flat Appl) (THead (Bind Abbr) x5
1905 x6))))))))) x2 H23)))))))))) H21)) (\lambda (H21: (ex6_6 B T T T T T (\lambda
1906 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1907 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1908 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
1909 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1910 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind
1911 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
1912 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1913 (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
1914 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
1915 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
1916 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind
1917 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1918 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1919 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1920 (_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
1921 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
1922 x2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
1923 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1924 T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
1925 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
1926 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
1927 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
1928 (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3: B).(\lambda (x4: T).(\lambda
1929 (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H22:
1930 (not (eq B x3 Abst))).(\lambda (H23: (eq T x0 (THead (Bind x3) x4
1931 x5))).(\lambda (H24: (eq T x2 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S
1932 O) O x7) x6)))).(\lambda (H25: (pr2 c x x7)).(\lambda (H26: (pr2 c x4
1933 x8)).(\lambda (H27: (pr2 (CHead c (Bind x3) x8) x5 x6)).(let H28 \def (eq_ind
1934 T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))
1935 (THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H19 (THead (Bind x3) x8
1936 (THead (Flat Appl) (lift (S O) O x7) x6)) H24) in (eq_ind_r T (THead (Bind
1937 x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (\lambda (t: T).(sn3 c
1938 (THead (Flat Appl) x1 t))) (let H29 \def (eq_ind T x0 (\lambda (t: T).((eq T
1939 (THead (Flat Appl) t0 (THead (Flat Appl) x t)) (THead (Flat Appl) x1 (THead
1940 (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))) \to (\forall (P:
1941 Prop).P))) H28 (THead (Bind x3) x4 x5) H23) in (let H30 \def (eq_ind T x0
1942 (\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
1943 (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c
1944 t4))))) H11 (THead (Bind x3) x4 x5) H23) in (let H31 \def (eq_ind T x0
1945 (\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
1946 (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall
1947 (x9: T).(\forall (x10: T).((eq T t4 (THead (Flat Appl) x9 x10)) \to (\forall
1948 (v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2)
1949 \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to
1950 (sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind x3) x4 x5) H23)
1951 in (let H32 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c (THead
1952 (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P:
1953 Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead (Bind x3) x4
1954 x5) H23) in (let H33 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4:
1955 T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t4) \to
1956 (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead
1957 (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat
1958 Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x
1959 t)))))))) H9 (THead (Bind x3) x4 x5) H23) in (sn3_pr3_trans c (THead (Flat
1960 Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (H32
1961 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (pr3_sing c
1962 (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) x5)) (THead (Flat
1963 Appl) x (THead (Bind x3) x4 x5)) (pr2_free c (THead (Flat Appl) x (THead
1964 (Bind x3) x4 x5)) (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x)
1965 x5)) (pr0_upsilon x3 H22 x x (pr0_refl x) x4 x4 (pr0_refl x4) x5 x5 (pr0_refl
1966 x5))) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))
1967 (pr3_head_12 c x4 x8 (pr3_pr2 c x4 x8 H26) (Bind x3) (THead (Flat Appl) (lift
1968 (S O) O x) x5) (THead (Flat Appl) (lift (S O) O x7) x6) (pr3_head_12 (CHead c
1969 (Bind x3) x8) (lift (S O) O x) (lift (S O) O x7) (pr3_lift (CHead c (Bind x3)
1970 x8) c (S O) O (drop_drop (Bind x3) O c c (drop_refl c) x8) x x7 (pr3_pr2 c x
1971 x7 H25)) (Flat Appl) x5 x6 (pr3_pr2 (CHead (CHead c (Bind x3) x8) (Flat Appl)
1972 (lift (S O) O x7)) x5 x6 (pr2_cflat (CHead c (Bind x3) x8) x5 x6 H27 Appl
1973 (lift (S O) O x7)))))) (\lambda (H34: (iso (THead (Flat Appl) x (THead (Bind
1974 x3) x4 x5)) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
1975 x6)))).(\lambda (P: Prop).(let H35 \def (match H34 in iso return (\lambda (t:
1976 T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x
1977 (THead (Bind x3) x4 x5))) \to ((eq T t4 (THead (Bind x3) x8 (THead (Flat
1978 Appl) (lift (S O) O x7) x6))) \to P))))) with [(iso_sort n1 n2) \Rightarrow
1979 (\lambda (H35: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind x3) x4
1980 x5)))).(\lambda (H36: (eq T (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl)
1981 (lift (S O) O x7) x6)))).((let H37 \def (eq_ind T (TSort n1) (\lambda (e:
1982 T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
1983 True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
1984 (THead (Flat Appl) x (THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T
1985 (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to
1986 P) H37)) H36))) | (iso_lref i1 i2) \Rightarrow (\lambda (H35: (eq T (TLRef
1987 i1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T
1988 (TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
1989 x6)))).((let H37 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T
1990 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
1991 \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x
1992 (THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T (TLRef i2) (THead (Bind
1993 x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H37)) H36))) |
1994 (iso_head v4 v5 t4 t5 k) \Rightarrow (\lambda (H35: (eq T (THead k v4 t4)
1995 (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T (THead k
1996 v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let
1997 H37 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
1998 with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t)
1999 \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4
2000 x5)) H35) in ((let H38 \def (f_equal T T (\lambda (e: T).(match e in T return
2001 (\lambda (_: T).T) with [(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4
2002 | (THead _ t _) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead
2003 (Bind x3) x4 x5)) H35) in ((let H39 \def (f_equal T K (\lambda (e: T).(match
2004 e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
2005 \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k v4 t4) (THead (Flat
2006 Appl) x (THead (Bind x3) x4 x5)) H35) in (eq_ind K (Flat Appl) (\lambda (k0:
2007 K).((eq T v4 x) \to ((eq T t4 (THead (Bind x3) x4 x5)) \to ((eq T (THead k0
2008 v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to
2009 P)))) (\lambda (H40: (eq T v4 x)).(eq_ind T x (\lambda (_: T).((eq T t4
2010 (THead (Bind x3) x4 x5)) \to ((eq T (THead (Flat Appl) v5 t5) (THead (Bind
2011 x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P))) (\lambda (H41: (eq
2012 T t4 (THead (Bind x3) x4 x5))).(eq_ind T (THead (Bind x3) x4 x5) (\lambda (_:
2013 T).((eq T (THead (Flat Appl) v5 t5) (THead (Bind x3) x8 (THead (Flat Appl)
2014 (lift (S O) O x7) x6))) \to P)) (\lambda (H42: (eq T (THead (Flat Appl) v5
2015 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(let H43
2016 \def (eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return
2017 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
2018 \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
2019 (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
2020 True])])) I (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))
2021 H42) in (False_ind P H43))) t4 (sym_eq T t4 (THead (Bind x3) x4 x5) H41))) v4
2022 (sym_eq T v4 x H40))) k (sym_eq K k (Flat Appl) H39))) H38)) H37)) H36)))])
2023 in (H35 (refl_equal T (THead (Flat Appl) x (THead (Bind x3) x4 x5)))
2024 (refl_equal T (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
2025 x6)))))))) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift
2026 (S O) O x7) x6))) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x3) x8 (THead
2027 (Flat Appl) (lift (S O) O x7) x6))) (THead (Flat Appl) x1 (THead (Bind x3) x8
2028 (THead (Flat Appl) (lift (S O) O x7) x6))) (pr2_head_1 c t0 x1 H17 (Flat
2029 Appl) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))))))))))
2030 x2 H24)))))))))))))) H21)) H20)) t3 H16))))))) H15)) (\lambda (H15: (ex4_4 T
2031 T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
2032 (THead (Flat Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2033 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
2034 Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2035 (_: T).(pr2 c t0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
2036 T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
2037 z1 t4))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
2038 (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind Abst) y1
2039 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
2040 T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
2041 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) (\lambda (_:
2042 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
2043 (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c t3) (\lambda (x1:
2044 T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H16: (eq T
2045 (THead (Flat Appl) x x0) (THead (Bind Abst) x1 x2))).(\lambda (H17: (eq T t3
2046 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c t0 x3)).(\lambda (_:
2047 ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let
2048 H20 \def (eq_ind T t3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead
2049 (Flat Appl) x x0)) t) \to (\forall (P: Prop).P))) H12 (THead (Bind Abbr) x3
2050 x4) H17) in (eq_ind_r T (THead (Bind Abbr) x3 x4) (\lambda (t: T).(sn3 c t))
2051 (let H21 \def (eq_ind T (THead (Flat Appl) x x0) (\lambda (ee: T).(match ee
2052 in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
2053 _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
2054 (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
2055 True])])) I (THead (Bind Abst) x1 x2) H16) in (False_ind (sn3 c (THead (Bind
2056 Abbr) x3 x4)) H21)) t3 H17)))))))))) H15)) (\lambda (H15: (ex6_6 B T T T T T
2057 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2058 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2059 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2060 (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2061 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2062 (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2063 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2064 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_:
2065 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2066 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2067 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
2068 y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_:
2069 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
2070 b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2071 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead
2072 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
2073 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
2074 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
2075 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2076 (_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
2077 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
2078 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
2079 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3)
2080 (\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda
2081 (x5: T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H17:
2082 (eq T (THead (Flat Appl) x x0) (THead (Bind x1) x2 x3))).(\lambda (H18: (eq T
2083 t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
2084 (_: (pr2 c t0 x5)).(\lambda (_: (pr2 c x2 x6)).(\lambda (_: (pr2 (CHead c
2085 (Bind x1) x6) x3 x4)).(let H22 \def (eq_ind T t3 (\lambda (t: T).((eq T
2086 (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P:
2087 Prop).P))) H12 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))
2088 H18) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5)
2089 x4)) (\lambda (t: T).(sn3 c t)) (let H23 \def (eq_ind T (THead (Flat Appl) x
2090 x0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
2091 _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
2092 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
2093 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x1) x2 x3)
2094 H17) in (False_ind (sn3 c (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O)
2095 O x5) x4))) H23)) t3 H18)))))))))))))) H15)) H14)))))) t2 H3))))))))) v2
2096 H4))))))))) y H0))))) H))))).
2098 theorem sn3_appl_beta:
2099 \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((sn3 c
2100 (THead (Flat Appl) u (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w)
2101 \to (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind Abst) w
2104 \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
2105 (sn3 c (THead (Flat Appl) u (THead (Bind Abbr) v t)))).(\lambda (w:
2106 T).(\lambda (H0: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THead (Bind
2107 Abbr) v t) H) in (let H1 \def H_x in (and_ind (sn3 c u) (sn3 c (THead (Bind
2108 Abbr) v t)) (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind
2109 Abst) w t)))) (\lambda (H2: (sn3 c u)).(\lambda (H3: (sn3 c (THead (Bind
2110 Abbr) v t))).(sn3_appl_appl v (THead (Bind Abst) w t) c (sn3_beta c v t H3 w
2111 H0) u H2 (\lambda (u2: T).(\lambda (H4: (pr3 c (THead (Flat Appl) v (THead
2112 (Bind Abst) w t)) u2)).(\lambda (H5: (((iso (THead (Flat Appl) v (THead (Bind
2113 Abst) w t)) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat
2114 Appl) u (THead (Bind Abbr) v t)) H (THead (Flat Appl) u u2) (pr3_thin_dx c
2115 (THead (Bind Abbr) v t) u2 (pr3_iso_beta v w t c u2 H4 H5) u Appl))))))))
2118 theorem sn3_appl_appls:
2119 \forall (v1: T).(\forall (t1: T).(\forall (vs: TList).(let u1 \def (THeads
2120 (Flat Appl) (TCons v1 vs) t1) in (\forall (c: C).((sn3 c u1) \to (\forall
2121 (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2)
2122 \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to
2123 (sn3 c (THead (Flat Appl) v2 u1))))))))))
2125 \lambda (v1: T).(\lambda (t1: T).(\lambda (vs: TList).(let u1 \def (THeads
2126 (Flat Appl) (TCons v1 vs) t1) in (\lambda (c: C).(\lambda (H: (sn3 c (THead
2127 (Flat Appl) v1 (THeads (Flat Appl) vs t1)))).(\lambda (v2: T).(\lambda (H0:
2128 (sn3 c v2)).(\lambda (H1: ((\forall (u2: T).((pr3 c (THead (Flat Appl) v1
2129 (THeads (Flat Appl) vs t1)) u2) \to ((((iso (THead (Flat Appl) v1 (THeads
2130 (Flat Appl) vs t1)) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat
2131 Appl) v2 u2))))))).(sn3_appl_appl v1 (THeads (Flat Appl) vs t1) c H v2 H0
2134 theorem sn3_appls_lref:
2135 \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (us:
2136 TList).((sns3 c us) \to (sn3 c (THeads (Flat Appl) us (TLRef i)))))))
2138 \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda
2139 (us: TList).(TList_ind (\lambda (t: TList).((sns3 c t) \to (sn3 c (THeads
2140 (Flat Appl) t (TLRef i))))) (\lambda (_: True).(sn3_nf2 c (TLRef i) H))
2141 (\lambda (t: T).(\lambda (t0: TList).(TList_ind (\lambda (t1: TList).((((sns3
2142 c t1) \to (sn3 c (THeads (Flat Appl) t1 (TLRef i))))) \to ((land (sn3 c t)
2143 (sns3 c t1)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (TLRef
2144 i))))))) (\lambda (_: (((sns3 c TNil) \to (sn3 c (THeads (Flat Appl) TNil
2145 (TLRef i)))))).(\lambda (H1: (land (sn3 c t) (sns3 c TNil))).(let H2 \def H1
2146 in (and_ind (sn3 c t) True (sn3 c (THead (Flat Appl) t (THeads (Flat Appl)
2147 TNil (TLRef i)))) (\lambda (H3: (sn3 c t)).(\lambda (_: True).(sn3_appl_lref
2148 c i H t H3))) H2)))) (\lambda (t1: T).(\lambda (t2: TList).(\lambda (_:
2149 (((((sns3 c t2) \to (sn3 c (THeads (Flat Appl) t2 (TLRef i))))) \to ((land
2150 (sn3 c t) (sns3 c t2)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2
2151 (TLRef i)))))))).(\lambda (H1: (((sns3 c (TCons t1 t2)) \to (sn3 c (THeads
2152 (Flat Appl) (TCons t1 t2) (TLRef i)))))).(\lambda (H2: (land (sn3 c t) (sns3
2153 c (TCons t1 t2)))).(let H3 \def H2 in (and_ind (sn3 c t) (land (sn3 c t1)
2154 (sns3 c t2)) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2)
2155 (TLRef i)))) (\lambda (H4: (sn3 c t)).(\lambda (H5: (land (sn3 c t1) (sns3 c
2156 t2))).(and_ind (sn3 c t1) (sns3 c t2) (sn3 c (THead (Flat Appl) t (THeads
2157 (Flat Appl) (TCons t1 t2) (TLRef i)))) (\lambda (H6: (sn3 c t1)).(\lambda
2158 (H7: (sns3 c t2)).(sn3_appl_appls t1 (TLRef i) t2 c (H1 (conj (sn3 c t1)
2159 (sns3 c t2) H6 H7)) t H4 (\lambda (u2: T).(\lambda (H8: (pr3 c (THeads (Flat
2160 Appl) (TCons t1 t2) (TLRef i)) u2)).(\lambda (H9: (((iso (THeads (Flat Appl)
2161 (TCons t1 t2) (TLRef i)) u2) \to (\forall (P: Prop).P)))).(H9
2162 (nf2_iso_appls_lref c i H (TCons t1 t2) u2 H8) (sn3 c (THead (Flat Appl) t
2163 u2))))))))) H5))) H3))))))) t0))) us)))).
2165 theorem sn3_appls_cast:
2166 \forall (c: C).(\forall (vs: TList).(\forall (u: T).((sn3 c (THeads (Flat
2167 Appl) vs u)) \to (\forall (t: T).((sn3 c (THeads (Flat Appl) vs t)) \to (sn3
2168 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))
2170 \lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: TList).(\forall
2171 (u: T).((sn3 c (THeads (Flat Appl) t u)) \to (\forall (t0: T).((sn3 c (THeads
2172 (Flat Appl) t t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Flat Cast) u
2173 t0)))))))) (\lambda (u: T).(\lambda (H: (sn3 c u)).(\lambda (t: T).(\lambda
2174 (H0: (sn3 c t)).(sn3_cast c u H t H0))))) (\lambda (t: T).(\lambda (t0:
2175 TList).(TList_ind (\lambda (t1: TList).(((\forall (u: T).((sn3 c (THeads
2176 (Flat Appl) t1 u)) \to (\forall (t2: T).((sn3 c (THeads (Flat Appl) t1 t2))
2177 \to (sn3 c (THeads (Flat Appl) t1 (THead (Flat Cast) u t2)))))))) \to
2178 (\forall (u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 u))) \to
2179 (\forall (t2: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 t2)))
2180 \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (THead (Flat Cast) u
2181 t2)))))))))) (\lambda (_: ((\forall (u: T).((sn3 c (THeads (Flat Appl) TNil
2182 u)) \to (\forall (t1: T).((sn3 c (THeads (Flat Appl) TNil t1)) \to (sn3 c
2183 (THeads (Flat Appl) TNil (THead (Flat Cast) u t1))))))))).(\lambda (u:
2184 T).(\lambda (H0: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) TNil
2185 u)))).(\lambda (t1: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads
2186 (Flat Appl) TNil t1)))).(sn3_appl_cast c t u H0 t1 H1)))))) (\lambda (t1:
2187 T).(\lambda (t2: TList).(\lambda (_: ((((\forall (u: T).((sn3 c (THeads (Flat
2188 Appl) t2 u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) t2 t3)) \to
2189 (sn3 c (THeads (Flat Appl) t2 (THead (Flat Cast) u t3)))))))) \to (\forall
2190 (u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 u))) \to (\forall
2191 (t3: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 t3))) \to (sn3 c
2192 (THead (Flat Appl) t (THeads (Flat Appl) t2 (THead (Flat Cast) u
2193 t3))))))))))).(\lambda (H0: ((\forall (u: T).((sn3 c (THeads (Flat Appl)
2194 (TCons t1 t2) u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) (TCons t1
2195 t2) t3)) \to (sn3 c (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u
2196 t3))))))))).(\lambda (u: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads
2197 (Flat Appl) (TCons t1 t2) u)))).(\lambda (t3: T).(\lambda (H2: (sn3 c (THead
2198 (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) t3)))).(let H_x \def
2199 (sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) t3) H2) in (let H3
2200 \def H_x in (and_ind (sn3 c t) (sn3 c (THeads (Flat Appl) (TCons t1 t2) t3))
2201 (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) (THead (Flat
2202 Cast) u t3)))) (\lambda (_: (sn3 c t)).(\lambda (H5: (sn3 c (THeads (Flat
2203 Appl) (TCons t1 t2) t3))).(let H6 \def H5 in (let H_x0 \def (sn3_gen_flat
2204 Appl c t (THeads (Flat Appl) (TCons t1 t2) u) H1) in (let H7 \def H_x0 in
2205 (and_ind (sn3 c t) (sn3 c (THeads (Flat Appl) (TCons t1 t2) u)) (sn3 c (THead
2206 (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u t3))))
2207 (\lambda (H8: (sn3 c t)).(\lambda (H9: (sn3 c (THeads (Flat Appl) (TCons t1
2208 t2) u))).(let H10 \def H9 in (sn3_appl_appls t1 (THead (Flat Cast) u t3) t2 c
2209 (H0 u H10 t3 H6) t H8 (\lambda (u2: T).(\lambda (H11: (pr3 c (THeads (Flat
2210 Appl) (TCons t1 t2) (THead (Flat Cast) u t3)) u2)).(\lambda (H12: (((iso
2211 (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u t3)) u2) \to (\forall
2212 (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t (THeads (Flat Appl)
2213 (TCons t1 t2) t3)) H2 (THead (Flat Appl) t u2) (pr3_thin_dx c (THeads (Flat
2214 Appl) (TCons t1 t2) t3) u2 (pr3_iso_appls_cast c u t3 (TCons t1 t2) u2 H11
2215 H12) t Appl))))))))) H7)))))) H3))))))))))) t0))) vs)).
2217 theorem sn3_appls_bind:
2218 \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u:
2219 T).((sn3 c u) \to (\forall (vs: TList).(\forall (t: T).((sn3 (CHead c (Bind
2220 b) u) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to (sn3 c (THeads (Flat
2221 Appl) vs (THead (Bind b) u t))))))))))
2223 \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda
2224 (u: T).(\lambda (H0: (sn3 c u)).(\lambda (vs: TList).(TList_ind (\lambda (t:
2225 TList).(\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) (lifts
2226 (S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u t0))))))
2227 (\lambda (t: T).(\lambda (H1: (sn3 (CHead c (Bind b) u) t)).(sn3_bind b c u
2228 H0 t H1))) (\lambda (v: T).(\lambda (vs0: TList).(TList_ind (\lambda (t:
2229 TList).(((\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl)
2230 (lifts (S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u
2231 t0)))))) \to (\forall (t0: T).((sn3 (CHead c (Bind b) u) (THead (Flat Appl)
2232 (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t) t0))) \to (sn3 c
2233 (THead (Flat Appl) v (THeads (Flat Appl) t (THead (Bind b) u t0))))))))
2234 (\lambda (_: ((\forall (t: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl)
2235 (lifts (S O) O TNil) t)) \to (sn3 c (THeads (Flat Appl) TNil (THead (Bind b)
2236 u t))))))).(\lambda (t: T).(\lambda (H2: (sn3 (CHead c (Bind b) u) (THead
2237 (Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O TNil)
2238 t)))).(sn3_appl_bind b H c u H0 t v H2)))) (\lambda (t: T).(\lambda (t0:
2239 TList).(\lambda (_: ((((\forall (t1: T).((sn3 (CHead c (Bind b) u) (THeads
2240 (Flat Appl) (lifts (S O) O t0) t1)) \to (sn3 c (THeads (Flat Appl) t0 (THead
2241 (Bind b) u t1)))))) \to (\forall (t1: T).((sn3 (CHead c (Bind b) u) (THead
2242 (Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t0) t1))) \to
2243 (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (THead (Bind b) u
2244 t1))))))))).(\lambda (H2: ((\forall (t1: T).((sn3 (CHead c (Bind b) u)
2245 (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) \to (sn3 c (THeads
2246 (Flat Appl) (TCons t t0) (THead (Bind b) u t1))))))).(\lambda (t1:
2247 T).(\lambda (H3: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O
2248 v) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))).(let H_x \def
2249 (sn3_gen_flat Appl (CHead c (Bind b) u) (lift (S O) O v) (THeads (Flat Appl)
2250 (lifts (S O) O (TCons t t0)) t1) H3) in (let H4 \def H_x in (and_ind (sn3
2251 (CHead c (Bind b) u) (lift (S O) O v)) (sn3 (CHead c (Bind b) u) (THeads
2252 (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) (sn3 c (THead (Flat Appl) v
2253 (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u t1)))) (\lambda (H5: (sn3
2254 (CHead c (Bind b) u) (lift (S O) O v))).(\lambda (H6: (sn3 (CHead c (Bind b)
2255 u) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1))).(let H_y \def
2256 (sn3_gen_lift (CHead c (Bind b) u) v (S O) O H5 c) in (sn3_appl_appls t
2257 (THead (Bind b) u t1) t0 c (H2 t1 H6) v (H_y (drop_drop (Bind b) O c c
2258 (drop_refl c) u)) (\lambda (u2: T).(\lambda (H7: (pr3 c (THeads (Flat Appl)
2259 (TCons t t0) (THead (Bind b) u t1)) u2)).(\lambda (H8: (((iso (THeads (Flat
2260 Appl) (TCons t t0) (THead (Bind b) u t1)) u2) \to (\forall (P:
2261 Prop).P)))).(let H9 \def (pr3_iso_appls_bind b H (TCons t t0) u t1 c u2 H7
2262 H8) in (sn3_pr3_trans c (THead (Flat Appl) v (THead (Bind b) u (THeads (Flat
2263 Appl) (lifts (S O) O (TCons t t0)) t1))) (sn3_appl_bind b H c u H0 (THeads
2264 (Flat Appl) (lifts (S O) O (TCons t t0)) t1) v H3) (THead (Flat Appl) v u2)
2265 (pr3_flat c v v (pr3_refl c v) (THead (Bind b) u (THeads (Flat Appl) (lifts
2266 (S O) O (TCons t t0)) t1)) u2 H9 Appl)))))))))) H4))))))))) vs0))) vs)))))).
2268 theorem sn3_appls_beta:
2269 \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (us: TList).((sn3 c
2270 (THeads (Flat Appl) us (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c
2271 w) \to (sn3 c (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst)
2274 \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (us:
2275 TList).(TList_ind (\lambda (t0: TList).((sn3 c (THeads (Flat Appl) t0 (THead
2276 (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat
2277 Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) (\lambda (H:
2278 (sn3 c (THead (Bind Abbr) v t))).(\lambda (w: T).(\lambda (H0: (sn3 c
2279 w)).(sn3_beta c v t H w H0)))) (\lambda (u: T).(\lambda (us0:
2280 TList).(TList_ind (\lambda (t0: TList).((((sn3 c (THeads (Flat Appl) t0
2281 (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads
2282 (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3
2283 c (THead (Flat Appl) u (THeads (Flat Appl) t0 (THead (Bind Abbr) v t)))) \to
2284 (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat
2285 Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))) (\lambda (_:
2286 (((sn3 c (THeads (Flat Appl) TNil (THead (Bind Abbr) v t))) \to (\forall (w:
2287 T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) TNil (THead (Flat Appl) v (THead
2288 (Bind Abst) w t))))))))).(\lambda (H0: (sn3 c (THead (Flat Appl) u (THeads
2289 (Flat Appl) TNil (THead (Bind Abbr) v t))))).(\lambda (w: T).(\lambda (H1:
2290 (sn3 c w)).(sn3_appl_beta c u v t H0 w H1))))) (\lambda (t0: T).(\lambda (t1:
2291 TList).(\lambda (_: (((((sn3 c (THeads (Flat Appl) t1 (THead (Bind Abbr) v
2292 t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) t1 (THead
2293 (Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 c (THead (Flat Appl) u
2294 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)))) \to (\forall (w: T).((sn3 c
2295 w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) t1 (THead (Flat Appl)
2296 v (THead (Bind Abst) w t))))))))))).(\lambda (H0: (((sn3 c (THeads (Flat
2297 Appl) (TCons t0 t1) (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w)
2298 \to (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead
2299 (Bind Abst) w t))))))))).(\lambda (H1: (sn3 c (THead (Flat Appl) u (THeads
2300 (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t))))).(\lambda (w:
2301 T).(\lambda (H2: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THeads
2302 (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t)) H1) in (let H3 \def H_x in
2303 (and_ind (sn3 c u) (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Bind
2304 Abbr) v t))) (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1)
2305 (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H4: (sn3 c
2306 u)).(\lambda (H5: (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr)
2307 v t)))).(sn3_appl_appls t0 (THead (Flat Appl) v (THead (Bind Abst) w t)) t1 c
2308 (H0 H5 w H2) u H4 (\lambda (u2: T).(\lambda (H6: (pr3 c (THeads (Flat Appl)
2309 (TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w t))) u2)).(\lambda
2310 (H7: (((iso (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead
2311 (Bind Abst) w t))) u2) \to (\forall (P: Prop).P)))).(let H8 \def
2312 (pr3_iso_appls_beta (TCons t0 t1) v w t c u2 H6 H7) in (sn3_pr3_trans c
2313 (THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v
2314 t))) H1 (THead (Flat Appl) u u2) (pr3_thin_dx c (THeads (Flat Appl) (TCons t0
2315 t1) (THead (Bind Abbr) v t)) u2 H8 u Appl))))))))) H3)))))))))) us0))) us)))).
2318 \forall (d: C).(\forall (t: T).((sn3 d t) \to (\forall (c: C).(\forall (h:
2319 nat).(\forall (i: nat).((drop h i c d) \to (sn3 c (lift h i t))))))))
2321 \lambda (d: C).(\lambda (t: T).(\lambda (H: (sn3 d t)).(sn3_ind d (\lambda
2322 (t0: T).(\forall (c: C).(\forall (h: nat).(\forall (i: nat).((drop h i c d)
2323 \to (sn3 c (lift h i t0))))))) (\lambda (t1: T).(\lambda (_: ((\forall (t2:
2324 T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 d t1 t2) \to (sn3 d
2325 t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
2326 Prop).P))) \to ((pr3 d t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall
2327 (i: nat).((drop h i c d) \to (sn3 c (lift h i t2))))))))))).(\lambda (c:
2328 C).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H2: (drop h i c
2329 d)).(sn3_pr2_intro c (lift h i t1) (\lambda (t2: T).(\lambda (H3: (((eq T
2330 (lift h i t1) t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr2 c (lift h i
2331 t1) t2)).(let H5 \def (pr2_gen_lift c t1 t2 h i H4 d H2) in (ex2_ind T
2332 (\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t1 t3))
2333 (sn3 c t2) (\lambda (x: T).(\lambda (H6: (eq T t2 (lift h i x))).(\lambda
2334 (H7: (pr2 d t1 x)).(let H8 \def (eq_ind T t2 (\lambda (t0: T).((eq T (lift h
2335 i t1) t0) \to (\forall (P: Prop).P))) H3 (lift h i x) H6) in (eq_ind_r T
2336 (lift h i x) (\lambda (t0: T).(sn3 c t0)) (H1 x (\lambda (H9: (eq T t1
2337 x)).(\lambda (P: Prop).(let H10 \def (eq_ind_r T x (\lambda (t0: T).((eq T
2338 (lift h i t1) (lift h i t0)) \to (\forall (P0: Prop).P0))) H8 t1 H9) in (let
2339 H11 \def (eq_ind_r T x (\lambda (t0: T).(pr2 d t1 t0)) H7 t1 H9) in (H10
2340 (refl_equal T (lift h i t1)) P))))) (pr3_pr2 d t1 x H7) c h i H2) t2 H6)))))
2341 H5))))))))))))) t H))).
2344 \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
2345 (CHead d (Bind Abbr) v)) \to ((sn3 d v) \to (sn3 c (TLRef i)))))))
2347 \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
2348 (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 d
2349 v)).(sn3_pr2_intro c (TLRef i) (\lambda (t2: T).(\lambda (H1: (((eq T (TLRef
2350 i) t2) \to (\forall (P: Prop).P)))).(\lambda (H2: (pr2 c (TLRef i) t2)).(let
2351 H3 \def (pr2_gen_lref c t2 i H2) in (or_ind (eq T t2 (TLRef i)) (ex2_2 C T
2352 (\lambda (d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u))))
2353 (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S i) O u))))) (sn3 c t2)
2354 (\lambda (H4: (eq T t2 (TLRef i))).(let H5 \def (eq_ind T t2 (\lambda (t:
2355 T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H4) in
2356 (eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c t)) (H5 (refl_equal T (TLRef i))
2357 (sn3 c (TLRef i))) t2 H4))) (\lambda (H4: (ex2_2 C T (\lambda (d0:
2358 C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_:
2359 C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda
2360 (d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_:
2361 C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))) (sn3 c t2) (\lambda (x0:
2362 C).(\lambda (x1: T).(\lambda (H5: (getl i c (CHead x0 (Bind Abbr)
2363 x1))).(\lambda (H6: (eq T t2 (lift (S i) O x1))).(let H7 \def (eq_ind T t2
2364 (\lambda (t: T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (lift (S
2365 i) O x1) H6) in (eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(sn3 c t)) (let
2366 H8 \def (eq_ind C (CHead d (Bind Abbr) v) (\lambda (c0: C).(getl i c c0)) H
2367 (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0
2368 (Bind Abbr) x1) H5)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in
2369 C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
2370 \Rightarrow c0])) (CHead d (Bind Abbr) v) (CHead x0 (Bind Abbr) x1)
2371 (getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 (Bind Abbr) x1) H5)) in
2372 ((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
2373 C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead d
2374 (Bind Abbr) v) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v)
2375 i H (CHead x0 (Bind Abbr) x1) H5)) in (\lambda (H11: (eq C d x0)).(let H12
2376 \def (eq_ind_r T x1 (\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 v
2377 H10) in (eq_ind T v (\lambda (t: T).(sn3 c (lift (S i) O t))) (let H13 \def
2378 (eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) v))) H12 d
2379 H11) in (sn3_lift d v H0 c (S i) O (getl_drop Abbr c d v i H13))) x1 H10))))
2380 H9))) t2 H6)))))) H4)) H3))))))))))).
2382 theorem sn3_appls_abbr:
2383 \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c
2384 (CHead d (Bind Abbr) w)) \to (\forall (vs: TList).((sn3 c (THeads (Flat Appl)
2385 vs (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))
2387 \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda
2388 (H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind
2389 (\lambda (t: TList).((sn3 c (THeads (Flat Appl) t (lift (S i) O w))) \to (sn3
2390 c (THeads (Flat Appl) t (TLRef i))))) (\lambda (H0: (sn3 c (lift (S i) O
2391 w))).(let H_y \def (sn3_gen_lift c w (S i) O H0 d (getl_drop Abbr c d w i H))
2392 in (sn3_abbr c d w i H H_y))) (\lambda (v: T).(\lambda (vs0:
2393 TList).(TList_ind (\lambda (t: TList).((((sn3 c (THeads (Flat Appl) t (lift
2394 (S i) O w))) \to (sn3 c (THeads (Flat Appl) t (TLRef i))))) \to ((sn3 c
2395 (THead (Flat Appl) v (THeads (Flat Appl) t (lift (S i) O w)))) \to (sn3 c
2396 (THead (Flat Appl) v (THeads (Flat Appl) t (TLRef i))))))) (\lambda (_:
2397 (((sn3 c (THeads (Flat Appl) TNil (lift (S i) O w))) \to (sn3 c (THeads (Flat
2398 Appl) TNil (TLRef i)))))).(\lambda (H1: (sn3 c (THead (Flat Appl) v (THeads
2399 (Flat Appl) TNil (lift (S i) O w))))).(sn3_appl_abbr c d w i H v H1)))
2400 (\lambda (t: T).(\lambda (t0: TList).(\lambda (_: (((((sn3 c (THeads (Flat
2401 Appl) t0 (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) t0 (TLRef i)))))
2402 \to ((sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (lift (S i) O w))))
2403 \to (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (TLRef
2404 i)))))))).(\lambda (H1: (((sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i)
2405 O w))) \to (sn3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)))))).(\lambda
2406 (H2: (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (lift (S i)
2407 O w))))).(let H_x \def (sn3_gen_flat Appl c v (THeads (Flat Appl) (TCons t
2408 t0) (lift (S i) O w)) H2) in (let H3 \def H_x in (and_ind (sn3 c v) (sn3 c
2409 (THeads (Flat Appl) (TCons t t0) (lift (S i) O w))) (sn3 c (THead (Flat Appl)
2410 v (THeads (Flat Appl) (TCons t t0) (TLRef i)))) (\lambda (H4: (sn3 c
2411 v)).(\lambda (H5: (sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i) O
2412 w)))).(sn3_appl_appls t (TLRef i) t0 c (H1 H5) v H4 (\lambda (u2: T).(\lambda
2413 (H6: (pr3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2)).(\lambda (H7:
2414 (((iso (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2) \to (\forall (P:
2415 Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) v (THeads (Flat Appl) (TCons
2416 t t0) (lift (S i) O w))) H2 (THead (Flat Appl) v u2) (pr3_thin_dx c (THeads
2417 (Flat Appl) (TCons t t0) (lift (S i) O w)) u2 (pr3_iso_appls_abbr c d w i H
2418 (TCons t t0) u2 H6 H7) v Appl)))))))) H3)))))))) vs0))) vs)))))).
2421 \forall (c: C).(\forall (d: C).(\forall (h: nat).(\forall (i: nat).((drop h
2422 i c d) \to (\forall (ts: TList).((sns3 d ts) \to (sns3 c (lifts h i ts))))))))
2424 \lambda (c: C).(\lambda (d: C).(\lambda (h: nat).(\lambda (i: nat).(\lambda
2425 (H: (drop h i c d)).(\lambda (ts: TList).(TList_ind (\lambda (t:
2426 TList).((sns3 d t) \to (sns3 c (lifts h i t)))) (\lambda (H0: True).H0)
2427 (\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: (((sns3 d t0) \to (sns3 c
2428 (lifts h i t0))))).(\lambda (H1: (land (sn3 d t) (sns3 d t0))).(let H2 \def
2429 H1 in (and_ind (sn3 d t) (sns3 d t0) (land (sn3 c (lift h i t)) (sns3 c
2430 (lifts h i t0))) (\lambda (H3: (sn3 d t)).(\lambda (H4: (sns3 d t0)).(conj
2431 (sn3 c (lift h i t)) (sns3 c (lifts h i t0)) (sn3_lift d t H3 c h i H) (H0
2432 H4)))) H2)))))) ts)))))).
2434 theorem sn3_gen_def:
2435 \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
2436 (CHead d (Bind Abbr) v)) \to ((sn3 c (TLRef i)) \to (sn3 d v))))))
2438 \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
2439 (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 c (TLRef
2440 i))).(sn3_gen_lift c v (S i) O (sn3_pr3_trans c (TLRef i) H0 (lift (S i) O v)
2441 (pr3_pr2 c (TLRef i) (lift (S i) O v) (pr2_delta c d v i H (TLRef i) (TLRef
2442 i) (pr0_refl (TLRef i)) (lift (S i) O v) (subst0_lref v i)))) d (getl_drop
2443 Abbr c d v i H))))))).
2446 \forall (v: T).(\forall (t: T).(\forall (i: nat).(((\forall (w: T).(ex T
2447 (\lambda (u: T).(subst0 i w t u))))) \to (\forall (c: C).(\forall (d:
2448 C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to (sn3 d v))))))))
2450 \lambda (v: T).(\lambda (t: T).(\lambda (i: nat).(\lambda (H: ((\forall (w:
2451 T).(ex T (\lambda (u: T).(subst0 i w t u)))))).(let H_x \def (H v) in (let H0
2452 \def H_x in (ex_ind T (\lambda (u: T).(subst0 i v t u)) (\forall (c:
2453 C).(\forall (d: C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to
2454 (sn3 d v))))) (\lambda (x: T).(\lambda (H1: (subst0 i v t x)).(subst0_ind
2455 (\lambda (n: nat).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_: T).(\forall
2456 (c: C).(\forall (d: C).((getl n c (CHead d (Bind Abbr) t0)) \to ((sn3 c t1)
2457 \to (sn3 d t0))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (c:
2458 C).(\lambda (d: C).(\lambda (H2: (getl i0 c (CHead d (Bind Abbr)
2459 v0))).(\lambda (H3: (sn3 c (TLRef i0))).(sn3_gen_def c d v0 i0 H2 H3)))))))
2460 (\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0:
2461 nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c:
2462 C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to
2463 (sn3 d v0))))))).(\lambda (t0: T).(\lambda (k: K).(\lambda (c: C).(\lambda
2464 (d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) v0))).(\lambda (H5: (sn3
2465 c (THead k u1 t0))).(let H_y \def (sn3_gen_head k c u1 t0 H5) in (H3 c d H4
2466 H_y)))))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: T).(\lambda
2467 (t1: T).(\lambda (i0: nat).(\lambda (H2: (subst0 (s k i0) v0 t1 t2)).(\lambda
2468 (H3: ((\forall (c: C).(\forall (d: C).((getl (s k i0) c (CHead d (Bind Abbr)
2469 v0)) \to ((sn3 c t1) \to (sn3 d v0))))))).(\lambda (u: T).(\lambda (c:
2470 C).(\lambda (d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr)
2471 v0))).(\lambda (H5: (sn3 c (THead k u t1))).(K_ind (\lambda (k0: K).((subst0
2472 (s k0 i0) v0 t1 t2) \to (((\forall (c0: C).(\forall (d0: C).((getl (s k0 i0)
2473 c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0)))))) \to ((sn3
2474 c (THead k0 u t1)) \to (sn3 d v0))))) (\lambda (b: B).(\lambda (_: (subst0 (s
2475 (Bind b) i0) v0 t1 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0:
2476 C).((getl (s (Bind b) i0) c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to
2477 (sn3 d0 v0))))))).(\lambda (H8: (sn3 c (THead (Bind b) u t1))).(let H_x0 \def
2478 (sn3_gen_bind b c u t1 H8) in (let H9 \def H_x0 in (and_ind (sn3 c u) (sn3
2479 (CHead c (Bind b) u) t1) (sn3 d v0) (\lambda (_: (sn3 c u)).(\lambda (H11:
2480 (sn3 (CHead c (Bind b) u) t1)).(H7 (CHead c (Bind b) u) d (getl_clear_bind b
2481 (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d (Bind Abbr) v0) i0 H4)
2482 H11))) H9))))))) (\lambda (f: F).(\lambda (_: (subst0 (s (Flat f) i0) v0 t1
2483 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: C).((getl (s (Flat f) i0)
2484 c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0))))))).(\lambda
2485 (H8: (sn3 c (THead (Flat f) u t1))).(let H_x0 \def (sn3_gen_flat f c u t1 H8)
2486 in (let H9 \def H_x0 in (and_ind (sn3 c u) (sn3 c t1) (sn3 d v0) (\lambda (_:
2487 (sn3 c u)).(\lambda (H11: (sn3 c t1)).(H7 c d H4 H11))) H9))))))) k H2 H3
2488 H5))))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda
2489 (i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c:
2490 C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to
2491 (sn3 d v0))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda
2492 (_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: ((\forall (c: C).(\forall (d:
2493 C).((getl (s k i0) c (CHead d (Bind Abbr) v0)) \to ((sn3 c t1) \to (sn3 d
2494 v0))))))).(\lambda (c: C).(\lambda (d: C).(\lambda (H6: (getl i0 c (CHead d
2495 (Bind Abbr) v0))).(\lambda (H7: (sn3 c (THead k u1 t1))).(let H_y \def
2496 (sn3_gen_head k c u1 t1 H7) in (H3 c d H6 H_y))))))))))))))))) i v t x H1)))
projects / helm.git / blob