1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/sn3/props".
27 include "iso/props.ma".
29 theorem sn3_pr3_trans:
30 \forall (c: C).(\forall (t1: T).((sn3 c t1) \to (\forall (t2: T).((pr3 c t1
31 t2) \to (sn3 c t2)))))
33 \lambda (c: C).(\lambda (t1: T).(\lambda (H: (sn3 c t1)).(sn3_ind c (\lambda
34 (t: T).(\forall (t2: T).((pr3 c t t2) \to (sn3 c t2)))) (\lambda (t2:
35 T).(\lambda (H0: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
36 Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H1: ((\forall
37 (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to
38 (\forall (t4: T).((pr3 c t3 t4) \to (sn3 c t4)))))))).(\lambda (t3:
39 T).(\lambda (H2: (pr3 c t2 t3)).(sn3_sing c t3 (\lambda (t0: T).(\lambda (H3:
40 (((eq T t3 t0) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 c t3 t0)).(let
41 H_x \def (term_dec t2 t3) in (let H5 \def H_x in (or_ind (eq T t2 t3) ((eq T
42 t2 t3) \to (\forall (P: Prop).P)) (sn3 c t0) (\lambda (H6: (eq T t2 t3)).(let
43 H7 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t t0)) H4 t2 H6) in (let H8
44 \def (eq_ind_r T t3 (\lambda (t: T).((eq T t t0) \to (\forall (P: Prop).P)))
45 H3 t2 H6) in (let H9 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t2 t)) H2 t2
46 H6) in (H0 t0 H8 H7))))) (\lambda (H6: (((eq T t2 t3) \to (\forall (P:
47 Prop).P)))).(H1 t3 H6 H2 t0 H4)) H5)))))))))))) t1 H))).
49 theorem sn3_pr2_intro:
50 \forall (c: C).(\forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to
51 (\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)))
53 \lambda (c: C).(\lambda (t1: T).(\lambda (H: ((\forall (t2: T).((((eq T t1
54 t2) \to (\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c
55 t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H0: (((eq T t1 t2) \to
56 (\forall (P: Prop).P)))).(\lambda (H1: (pr3 c t1 t2)).(let H2 \def H0 in
57 ((let H3 \def H in (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(((\forall
58 (t3: T).((((eq T t t3) \to (\forall (P: Prop).P))) \to ((pr2 c t t3) \to (sn3
59 c t3))))) \to ((((eq T t t0) \to (\forall (P: Prop).P))) \to (sn3 c t0)))))
60 (\lambda (t: T).(\lambda (H4: ((\forall (t3: T).((((eq T t t3) \to (\forall
61 (P: Prop).P))) \to ((pr2 c t t3) \to (sn3 c t3)))))).(\lambda (H5: (((eq T t
62 t) \to (\forall (P: Prop).P)))).(H4 t H5 (pr2_free c t t (pr0_refl t))))))
63 (\lambda (t3: T).(\lambda (t4: T).(\lambda (H4: (pr2 c t4 t3)).(\lambda (t5:
64 T).(\lambda (H5: (pr3 c t3 t5)).(\lambda (H6: ((((\forall (t6: T).((((eq T t3
65 t6) \to (\forall (P: Prop).P))) \to ((pr2 c t3 t6) \to (sn3 c t6))))) \to
66 ((((eq T t3 t5) \to (\forall (P: Prop).P))) \to (sn3 c t5))))).(\lambda (H7:
67 ((\forall (t6: T).((((eq T t4 t6) \to (\forall (P: Prop).P))) \to ((pr2 c t4
68 t6) \to (sn3 c t6)))))).(\lambda (H8: (((eq T t4 t5) \to (\forall (P:
69 Prop).P)))).(let H_x \def (term_dec t4 t3) in (let H9 \def H_x in (or_ind (eq
70 T t4 t3) ((eq T t4 t3) \to (\forall (P: Prop).P)) (sn3 c t5) (\lambda (H10:
71 (eq T t4 t3)).(let H11 \def (eq_ind T t4 (\lambda (t: T).((eq T t t5) \to
72 (\forall (P: Prop).P))) H8 t3 H10) in (let H12 \def (eq_ind T t4 (\lambda (t:
73 T).(\forall (t6: T).((((eq T t t6) \to (\forall (P: Prop).P))) \to ((pr2 c t
74 t6) \to (sn3 c t6))))) H7 t3 H10) in (let H13 \def (eq_ind T t4 (\lambda (t:
75 T).(pr2 c t t3)) H4 t3 H10) in (H6 H12 H11))))) (\lambda (H10: (((eq T t4 t3)
76 \to (\forall (P: Prop).P)))).(sn3_pr3_trans c t3 (H7 t3 H10 H4) t5 H5))
77 H9))))))))))) t1 t2 H1 H3)) H2)))))))).
80 \forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: T).((sn3 c t) \to
81 (sn3 c (THead (Flat Cast) u t))))))
83 \lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c u)).(sn3_ind c (\lambda
84 (t: T).(\forall (t0: T).((sn3 c t0) \to (sn3 c (THead (Flat Cast) t t0)))))
85 (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
86 (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall
87 (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
88 (\forall (t: T).((sn3 c t) \to (sn3 c (THead (Flat Cast) t2
89 t))))))))).(\lambda (t: T).(\lambda (H2: (sn3 c t)).(sn3_ind c (\lambda (t0:
90 T).(sn3 c (THead (Flat Cast) t1 t0))) (\lambda (t0: T).(\lambda (H3:
91 ((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0
92 t2) \to (sn3 c t2)))))).(\lambda (H4: ((\forall (t2: T).((((eq T t0 t2) \to
93 (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c (THead (Flat Cast) t1
94 t2))))))).(sn3_pr2_intro c (THead (Flat Cast) t1 t0) (\lambda (t2:
95 T).(\lambda (H5: (((eq T (THead (Flat Cast) t1 t0) t2) \to (\forall (P:
96 Prop).P)))).(\lambda (H6: (pr2 c (THead (Flat Cast) t1 t0) t2)).(let H7 \def
97 (pr2_gen_cast c t1 t0 t2 H6) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda
98 (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
99 T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3)))) (pr2 c
100 t0 t2) (sn3 c t2) (\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
101 T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
102 T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0
103 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
104 (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2)))
105 (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3))) (sn3 c t2) (\lambda (x0:
106 T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead (Flat Cast) x0
107 x1))).(\lambda (H10: (pr2 c t1 x0)).(\lambda (H11: (pr2 c t0 x1)).(let H12
108 \def (eq_ind T t2 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) t3) \to
109 (\forall (P: Prop).P))) H5 (THead (Flat Cast) x0 x1) H9) in (eq_ind_r T
110 (THead (Flat Cast) x0 x1) (\lambda (t3: T).(sn3 c t3)) (let H_x \def
111 (term_dec x0 t1) in (let H13 \def H_x in (or_ind (eq T x0 t1) ((eq T x0 t1)
112 \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) x0 x1)) (\lambda (H14:
113 (eq T x0 t1)).(let H15 \def (eq_ind T x0 (\lambda (t3: T).((eq T (THead (Flat
114 Cast) t1 t0) (THead (Flat Cast) t3 x1)) \to (\forall (P: Prop).P))) H12 t1
115 H14) in (let H16 \def (eq_ind T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1
116 H14) in (eq_ind_r T t1 (\lambda (t3: T).(sn3 c (THead (Flat Cast) t3 x1)))
117 (let H_x0 \def (term_dec t0 x1) in (let H17 \def H_x0 in (or_ind (eq T t0 x1)
118 ((eq T t0 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) t1 x1))
119 (\lambda (H18: (eq T t0 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t3:
120 T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat Cast) t1 t3)) \to (\forall
121 (P: Prop).P))) H15 t0 H18) in (let H20 \def (eq_ind_r T x1 (\lambda (t3:
122 T).(pr2 c t0 t3)) H11 t0 H18) in (eq_ind T t0 (\lambda (t3: T).(sn3 c (THead
123 (Flat Cast) t1 t3))) (H19 (refl_equal T (THead (Flat Cast) t1 t0)) (sn3 c
124 (THead (Flat Cast) t1 t0))) x1 H18)))) (\lambda (H18: (((eq T t0 x1) \to
125 (\forall (P: Prop).P)))).(H4 x1 H18 (pr3_pr2 c t0 x1 H11))) H17))) x0 H14))))
126 (\lambda (H14: (((eq T x0 t1) \to (\forall (P: Prop).P)))).(H1 x0 (\lambda
127 (H15: (eq T t1 x0)).(\lambda (P: Prop).(let H16 \def (eq_ind_r T x0 (\lambda
128 (t3: T).((eq T t3 t1) \to (\forall (P0: Prop).P0))) H14 t1 H15) in (let H17
129 \def (eq_ind_r T x0 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead
130 (Flat Cast) t3 x1)) \to (\forall (P0: Prop).P0))) H12 t1 H15) in (let H18
131 \def (eq_ind_r T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1 H15) in (H16
132 (refl_equal T t1) P)))))) (pr3_pr2 c t1 x0 H10) x1 (let H_x0 \def (term_dec
133 t0 x1) in (let H15 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to
134 (\forall (P: Prop).P)) (sn3 c x1) (\lambda (H16: (eq T t0 x1)).(let H17 \def
135 (eq_ind_r T x1 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat
136 Cast) x0 t3)) \to (\forall (P: Prop).P))) H12 t0 H16) in (let H18 \def
137 (eq_ind_r T x1 (\lambda (t3: T).(pr2 c t0 t3)) H11 t0 H16) in (eq_ind T t0
138 (\lambda (t3: T).(sn3 c t3)) (sn3_sing c t0 H3) x1 H16)))) (\lambda (H16:
139 (((eq T t0 x1) \to (\forall (P: Prop).P)))).(H3 x1 H16 (pr3_pr2 c t0 x1
140 H11))) H15))))) H13))) t2 H9))))))) H8)) (\lambda (H8: (pr2 c t0
141 t2)).(sn3_pr3_trans c t0 (sn3_sing c t0 H3) t2 (pr3_pr2 c t0 t2 H8)))
142 H7))))))))) t H2)))))) u H))).
145 \forall (c: C).(\forall (t: T).((sn3 c t) \to (\forall (f: F).(\forall (u:
146 T).(sn3 (CHead c (Flat f) u) t)))))
148 \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(\lambda (f:
149 F).(\lambda (u: T).(sn3_ind c (\lambda (t0: T).(sn3 (CHead c (Flat f) u) t0))
150 (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
151 (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall
152 (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
153 (sn3 (CHead c (Flat f) u) t2)))))).(sn3_pr2_intro (CHead c (Flat f) u) t1
154 (\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) \to (\forall (P:
155 Prop).P)))).(\lambda (H3: (pr2 (CHead c (Flat f) u) t1 t2)).(H1 t2 H2
156 (pr3_pr2 c t1 t2 (pr2_gen_cflat f c u t1 t2 H3)))))))))) t H))))).
159 \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c
160 (THead (Bind b) v t)) \to (sn3 (CHead c (Bind b) v) t)))))
162 \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
163 (sn3 c (THead (Bind b) v t))).(let H_x \def (sn3_gen_bind b c v t H) in (let
164 H0 \def H_x in (and_ind (sn3 c v) (sn3 (CHead c (Bind b) v) t) (sn3 (CHead c
165 (Bind b) v) t) (\lambda (_: (sn3 c v)).(\lambda (H2: (sn3 (CHead c (Bind b)
166 v) t)).H2)) H0))))))).
169 \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1:
170 T).(\forall (t: T).((sn3 (CHead c (Bind b) v1) t) \to (\forall (v2: T).(sn3
171 (CHead c (Bind b) v2) t)))))))
173 \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda
174 (v1: T).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c (Bind b) v1) t)).(\lambda
175 (v2: T).(sn3_ind (CHead c (Bind b) v1) (\lambda (t0: T).(sn3 (CHead c (Bind
176 b) v2) t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2)
177 \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to (sn3
178 (CHead c (Bind b) v1) t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1
179 t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to
180 (sn3 (CHead c (Bind b) v2) t2)))))).(sn3_pr2_intro (CHead c (Bind b) v2) t1
181 (\lambda (t2: T).(\lambda (H3: (((eq T t1 t2) \to (\forall (P:
182 Prop).P)))).(\lambda (H4: (pr2 (CHead c (Bind b) v2) t1 t2)).(H2 t2 H3
183 (pr3_pr2 (CHead c (Bind b) v1) t1 t2 (pr2_change b H c v2 t1 t2 H4
184 v1)))))))))) t H0))))))).
186 theorem sn3_cpr3_trans:
187 \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
188 (k: K).(\forall (t: T).((sn3 (CHead c k u1) t) \to (sn3 (CHead c k u2)
191 \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
192 u2)).(\lambda (k: K).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c k u1)
193 t)).(sn3_ind (CHead c k u1) (\lambda (t0: T).(sn3 (CHead c k u2) t0))
194 (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
195 (P: Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u1)
196 t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
197 Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u2)
198 t2)))))).(sn3_sing (CHead c k u2) t1 (\lambda (t2: T).(\lambda (H3: (((eq T
199 t1 t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 (CHead c k u2) t1
200 t2)).(H2 t2 H3 (pr3_pr3_pr3_t c u1 u2 H t1 t2 k H4))))))))) t H0))))))).
203 \forall (b: B).(\forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t:
204 T).((sn3 (CHead c (Bind b) u) t) \to (sn3 c (THead (Bind b) u t)))))))
206 \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c
207 u)).(sn3_ind c (\lambda (t: T).(\forall (t0: T).((sn3 (CHead c (Bind b) t)
208 t0) \to (sn3 c (THead (Bind b) t t0))))) (\lambda (t1: T).(\lambda (_:
209 ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1
210 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to
211 (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (t: T).((sn3 (CHead c
212 (Bind b) t2) t) \to (sn3 c (THead (Bind b) t2 t))))))))).(\lambda (t:
213 T).(\lambda (H2: (sn3 (CHead c (Bind b) t1) t)).(sn3_ind (CHead c (Bind b)
214 t1) (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (\lambda (t2:
215 T).(\lambda (H3: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
216 Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind b)
217 t1) t3)))))).(\lambda (H4: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
218 Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 c (THead (Bind b)
219 t1 t3))))))).(sn3_sing c (THead (Bind b) t1 t2) (\lambda (t3: T).(\lambda
220 (H5: (((eq T (THead (Bind b) t1 t2) t3) \to (\forall (P: Prop).P)))).(\lambda
221 (H6: (pr3 c (THead (Bind b) t1 t2) t3)).(let H_x \def (bind_dec_not b Abst)
222 in (let H7 \def H_x in (or_ind (eq B b Abst) (not (eq B b Abst)) (sn3 c t3)
223 (\lambda (H8: (eq B b Abst)).(let H9 \def (eq_ind B b (\lambda (b0: B).(pr3 c
224 (THead (Bind b0) t1 t2) t3)) H6 Abst H8) in (let H10 \def (eq_ind B b
225 (\lambda (b0: B).((eq T (THead (Bind b0) t1 t2) t3) \to (\forall (P:
226 Prop).P))) H5 Abst H8) in (let H11 \def (eq_ind B b (\lambda (b0: B).(\forall
227 (t4: T).((((eq T t2 t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind
228 b0) t1) t2 t4) \to (sn3 c (THead (Bind b0) t1 t4)))))) H4 Abst H8) in (let
229 H12 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T t2 t4) \to
230 (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) t2 t4) \to (sn3
231 (CHead c (Bind b0) t1) t4))))) H3 Abst H8) in (let H13 \def (eq_ind B b
232 (\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P)))
233 \to ((pr3 c t1 t4) \to (\forall (t0: T).((sn3 (CHead c (Bind b0) t4) t0) \to
234 (sn3 c (THead (Bind b0) t4 t0)))))))) H1 Abst H8) in (let H14 \def
235 (pr3_gen_abst c t1 t2 t3 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4:
236 T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_:
237 T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall
238 (u0: T).(pr3 (CHead c (Bind b0) u0) t2 t4))))) (sn3 c t3) (\lambda (x0:
239 T).(\lambda (x1: T).(\lambda (H15: (eq T t3 (THead (Bind Abst) x0
240 x1))).(\lambda (H16: (pr3 c t1 x0)).(\lambda (H17: ((\forall (b0: B).(\forall
241 (u0: T).(pr3 (CHead c (Bind b0) u0) t2 x1))))).(let H18 \def (eq_ind T t3
242 (\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) t0) \to (\forall (P:
243 Prop).P))) H10 (THead (Bind Abst) x0 x1) H15) in (eq_ind_r T (THead (Bind
244 Abst) x0 x1) (\lambda (t0: T).(sn3 c t0)) (let H_x0 \def (term_dec t1 x0) in
245 (let H19 \def H_x0 in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P:
246 Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H20: (eq T t1 x0)).(let
247 H21 \def (eq_ind_r T x0 (\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2)
248 (THead (Bind Abst) t0 x1)) \to (\forall (P: Prop).P))) H18 t1 H20) in (let
249 H22 \def (eq_ind_r T x0 (\lambda (t0: T).(pr3 c t1 t0)) H16 t1 H20) in
250 (eq_ind T t1 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t0 x1))) (let H_x1
251 \def (term_dec t2 x1) in (let H23 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2
252 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abst) t1 x1)) (\lambda
253 (H24: (eq T t2 x1)).(let H25 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T
254 (THead (Bind Abst) t1 t2) (THead (Bind Abst) t1 t0)) \to (\forall (P:
255 Prop).P))) H21 t2 H24) in (let H26 \def (eq_ind_r T x1 (\lambda (t0:
256 T).(\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) t2 t0))))
257 H17 t2 H24) in (eq_ind T t2 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t1
258 t0))) (H25 (refl_equal T (THead (Bind Abst) t1 t2)) (sn3 c (THead (Bind Abst)
259 t1 t2))) x1 H24)))) (\lambda (H24: (((eq T t2 x1) \to (\forall (P:
260 Prop).P)))).(H11 x1 H24 (H17 Abst t1))) H23))) x0 H20)))) (\lambda (H20:
261 (((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x1 \def (term_dec t2 x1)
262 in (let H21 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P:
263 Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H22: (eq T t2 x1)).(let
264 H23 \def (eq_ind_r T x1 (\lambda (t0: T).(\forall (b0: B).(\forall (u0:
265 T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) H17 t2 H22) in (eq_ind T t2 (\lambda
266 (t0: T).(sn3 c (THead (Bind Abst) x0 t0))) (H13 x0 H20 H16 t2 (sn3_cpr3_trans
267 c t1 x0 H16 (Bind Abst) t2 (sn3_sing (CHead c (Bind Abst) t1) t2 H12))) x1
268 H22))) (\lambda (H22: (((eq T t2 x1) \to (\forall (P: Prop).P)))).(H13 x0 H20
269 H16 x1 (sn3_cpr3_trans c t1 x0 H16 (Bind Abst) x1 (H12 x1 H22 (H17 Abst
270 t1))))) H21)))) H19))) t3 H15))))))) H14)))))))) (\lambda (H8: (not (eq B b
271 Abst))).(let H_x0 \def (pr3_gen_bind b H8 c t1 t2 t3 H6) in (let H9 \def H_x0
272 in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
273 b) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
274 T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) t2 t4)))) (pr3 (CHead c (Bind
275 b) t1) t2 (lift (S O) O t3)) (sn3 c t3) (\lambda (H10: (ex3_2 T T (\lambda
276 (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2:
277 T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3
278 (CHead c (Bind b) t1) t2 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
279 (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_:
280 T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 (CHead c (Bind b)
281 t1) t2 t4))) (sn3 c t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq
282 T t3 (THead (Bind b) x0 x1))).(\lambda (H12: (pr3 c t1 x0)).(\lambda (H13:
283 (pr3 (CHead c (Bind b) t1) t2 x1)).(let H14 \def (eq_ind T t3 (\lambda (t0:
284 T).((eq T (THead (Bind b) t1 t2) t0) \to (\forall (P: Prop).P))) H5 (THead
285 (Bind b) x0 x1) H11) in (eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0:
286 T).(sn3 c t0)) (let H_x1 \def (term_dec t1 x0) in (let H15 \def H_x1 in
287 (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: Prop).P)) (sn3 c (THead
288 (Bind b) x0 x1)) (\lambda (H16: (eq T t1 x0)).(let H17 \def (eq_ind_r T x0
289 (\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t0 x1)) \to
290 (\forall (P: Prop).P))) H14 t1 H16) in (let H18 \def (eq_ind_r T x0 (\lambda
291 (t0: T).(pr3 c t1 t0)) H12 t1 H16) in (eq_ind T t1 (\lambda (t0: T).(sn3 c
292 (THead (Bind b) t0 x1))) (let H_x2 \def (term_dec t2 x1) in (let H19 \def
293 H_x2 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: Prop).P)) (sn3 c
294 (THead (Bind b) t1 x1)) (\lambda (H20: (eq T t2 x1)).(let H21 \def (eq_ind_r
295 T x1 (\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t1 t0))
296 \to (\forall (P: Prop).P))) H17 t2 H20) in (let H22 \def (eq_ind_r T x1
297 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) t2 t0)) H13 t2 H20) in (eq_ind T
298 t2 (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (H21 (refl_equal T (THead
299 (Bind b) t1 t2)) (sn3 c (THead (Bind b) t1 t2))) x1 H20)))) (\lambda (H20:
300 (((eq T t2 x1) \to (\forall (P: Prop).P)))).(H4 x1 H20 H13)) H19))) x0
301 H16)))) (\lambda (H16: (((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x2
302 \def (term_dec t2 x1) in (let H17 \def H_x2 in (or_ind (eq T t2 x1) ((eq T t2
303 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind b) x0 x1)) (\lambda (H18:
304 (eq T t2 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t0: T).(pr3 (CHead c
305 (Bind b) t1) t2 t0)) H13 t2 H18) in (eq_ind T t2 (\lambda (t0: T).(sn3 c
306 (THead (Bind b) x0 t0))) (H1 x0 H16 H12 t2 (sn3_cpr3_trans c t1 x0 H12 (Bind
307 b) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3))) x1 H18))) (\lambda (H18: (((eq
308 T t2 x1) \to (\forall (P: Prop).P)))).(H1 x0 H16 H12 x1 (sn3_cpr3_trans c t1
309 x0 H12 (Bind b) x1 (H3 x1 H18 H13)))) H17)))) H15))) t3 H11))))))) H10))
310 (\lambda (H10: (pr3 (CHead c (Bind b) t1) t2 (lift (S O) O
311 t3))).(sn3_gen_lift (CHead c (Bind b) t1) t3 (S O) O (sn3_pr3_trans (CHead c
312 (Bind b) t1) t2 (sn3_pr2_intro (CHead c (Bind b) t1) t2 (\lambda (t0:
313 T).(\lambda (H11: (((eq T t2 t0) \to (\forall (P: Prop).P)))).(\lambda (H12:
314 (pr2 (CHead c (Bind b) t1) t2 t0)).(H3 t0 H11 (pr3_pr2 (CHead c (Bind b) t1)
315 t2 t0 H12)))))) (lift (S O) O t3) H10) c (drop_drop (Bind b) O c c (drop_refl
316 c) t1))) H9)))) H7)))))))))) t H2)))))) u H)))).
319 \forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c (THead (Bind Abbr) v
320 t)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead
321 (Bind Abst) w t))))))))
323 \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
324 (Bind Abbr) v t))).(insert_eq T (THead (Bind Abbr) v t) (\lambda (t0: T).(sn3
325 c t0)) (\lambda (_: T).(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat
326 Appl) v (THead (Bind Abst) w t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c
327 y)).(unintro T t (\lambda (t0: T).((eq T y (THead (Bind Abbr) v t0)) \to
328 (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead (Bind Abst)
329 w t0))))))) (unintro T v (\lambda (t0: T).(\forall (x: T).((eq T y (THead
330 (Bind Abbr) t0 x)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat
331 Appl) t0 (THead (Bind Abst) w x)))))))) (sn3_ind c (\lambda (t0: T).(\forall
332 (x: T).(\forall (x0: T).((eq T t0 (THead (Bind Abbr) x x0)) \to (\forall (w:
333 T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) w
334 x0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2)
335 \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda
336 (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3
337 c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Bind Abbr) x
338 x0)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead
339 (Bind Abst) w x0))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3:
340 (eq T t1 (THead (Bind Abbr) x x0))).(\lambda (w: T).(\lambda (H4: (sn3 c
341 w)).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0
342 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (\forall (x1:
343 T).(\forall (x2: T).((eq T t2 (THead (Bind Abbr) x1 x2)) \to (\forall (w0:
344 T).((sn3 c w0) \to (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) w0
345 x2)))))))))))) H2 (THead (Bind Abbr) x x0) H3) in (let H6 \def (eq_ind T t1
346 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P)))
347 \to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead (Bind Abbr) x x0) H3) in
348 (sn3_ind c (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t0
349 x0)))) (\lambda (t2: T).(\lambda (H7: ((\forall (t3: T).((((eq T t2 t3) \to
350 (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H8:
351 ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2
352 t3) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t3
353 x0)))))))).(sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind Abst) t2 x0))
354 (\lambda (t3: T).(\lambda (H9: (((eq T (THead (Flat Appl) x (THead (Bind
355 Abst) t2 x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H10: (pr2 c (THead
356 (Flat Appl) x (THead (Bind Abst) t2 x0)) t3)).(let H11 \def (pr2_gen_appl c x
357 (THead (Bind Abst) t2 x0) t3 H10) in (or3_ind (ex3_2 T T (\lambda (u2:
358 T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
359 T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
360 (THead (Bind Abst) t2 x0) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
361 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0)
362 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
363 T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
364 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
365 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
366 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T
367 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
368 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
369 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
370 (THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
371 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
372 (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
373 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
374 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_:
375 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
376 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
377 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
378 y2) z1 z2)))))))) (sn3 c t3) (\lambda (H12: (ex3_2 T T (\lambda (u2:
379 T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
380 T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
381 (THead (Bind Abst) t2 x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
382 (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_:
383 T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst)
384 t2 x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H13: (eq
385 T t3 (THead (Flat Appl) x1 x2))).(\lambda (H14: (pr2 c x x1)).(\lambda (H15:
386 (pr2 c (THead (Bind Abst) t2 x0) x2)).(let H16 \def (eq_ind T t3 (\lambda
387 (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to
388 (\forall (P: Prop).P))) H9 (THead (Flat Appl) x1 x2) H13) in (eq_ind_r T
389 (THead (Flat Appl) x1 x2) (\lambda (t0: T).(sn3 c t0)) (let H17 \def
390 (pr2_gen_abst c t2 x0 x2 H15) in (ex3_2_ind T T (\lambda (u2: T).(\lambda
391 (t4: T).(eq T x2 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_:
392 T).(pr2 c t2 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
393 (u: T).(pr2 (CHead c (Bind b) u) x0 t4))))) (sn3 c (THead (Flat Appl) x1 x2))
394 (\lambda (x3: T).(\lambda (x4: T).(\lambda (H18: (eq T x2 (THead (Bind Abst)
395 x3 x4))).(\lambda (H19: (pr2 c t2 x3)).(\lambda (H20: ((\forall (b:
396 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 x4))))).(let H21 \def (eq_ind
397 T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0))
398 (THead (Flat Appl) x1 t0)) \to (\forall (P: Prop).P))) H16 (THead (Bind Abst)
399 x3 x4) H18) in (eq_ind_r T (THead (Bind Abst) x3 x4) (\lambda (t0: T).(sn3 c
400 (THead (Flat Appl) x1 t0))) (let H_x \def (term_dec t2 x3) in (let H22 \def
401 H_x in (or_ind (eq T t2 x3) ((eq T t2 x3) \to (\forall (P: Prop).P)) (sn3 c
402 (THead (Flat Appl) x1 (THead (Bind Abst) x3 x4))) (\lambda (H23: (eq T t2
403 x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: T).((eq T (THead (Flat Appl)
404 x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) x1 (THead (Bind Abst) t0
405 x4))) \to (\forall (P: Prop).P))) H21 t2 H23) in (let H25 \def (eq_ind_r T x3
406 (\lambda (t0: T).(pr2 c t2 t0)) H19 t2 H23) in (eq_ind T t2 (\lambda (t0:
407 T).(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t0 x4)))) (let H_x0 \def
408 (term_dec x x1) in (let H26 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to
409 (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t2
410 x4))) (\lambda (H27: (eq T x x1)).(let H28 \def (eq_ind_r T x1 (\lambda (t0:
411 T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (THead (Flat Appl)
412 t0 (THead (Bind Abst) t2 x4))) \to (\forall (P: Prop).P))) H24 x H27) in (let
413 H29 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H27) in (eq_ind
414 T x (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) t2
415 x4)))) (let H_x1 \def (term_dec x0 x4) in (let H30 \def H_x1 in (or_ind (eq T
416 x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x
417 (THead (Bind Abst) t2 x4))) (\lambda (H31: (eq T x0 x4)).(let H32 \def
418 (eq_ind_r T x4 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind
419 Abst) t2 x0)) (THead (Flat Appl) x (THead (Bind Abst) t2 t0))) \to (\forall
420 (P: Prop).P))) H28 x0 H31) in (let H33 \def (eq_ind_r T x4 (\lambda (t0:
421 T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0
422 H31) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead
423 (Bind Abst) t2 t0)))) (H32 (refl_equal T (THead (Flat Appl) x (THead (Bind
424 Abst) t2 x0))) (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)))) x4
425 H31)))) (\lambda (H31: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead
426 (Bind Abbr) x x4) (\lambda (H32: (eq T (THead (Bind Abbr) x x0) (THead (Bind
427 Abbr) x x4))).(\lambda (P: Prop).(let H33 \def (f_equal T T (\lambda (e:
428 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
429 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
430 Abbr) x x0) (THead (Bind Abbr) x x4) H32) in (let H34 \def (eq_ind_r T x4
431 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H31 x0 H33) in
432 (let H35 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u:
433 T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H33) in (H34 (refl_equal T x0)
434 P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4)
435 (pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead
436 (Bind Abbr) x x4)) t2 (sn3_sing c t2 H7))) H30))) x1 H27)))) (\lambda (H27:
437 (((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind Abbr) x1 x4)
438 (\lambda (H28: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x1
439 x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e
440 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _)
441 \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0)
442 (THead (Bind Abbr) x1 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e:
443 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
444 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
445 Abbr) x x0) (THead (Bind Abbr) x1 x4) H28) in (\lambda (H31: (eq T x
446 x1)).(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall
447 (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (let H33 \def
448 (eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
449 H27 x H31) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14
450 x H31) in (H33 (refl_equal T x) P)))))) H29)))) (pr3_head_12 c x x1 (pr3_pr2
451 c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20
452 Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) t2 (sn3_sing c t2
453 H7))) H26))) x3 H23)))) (\lambda (H23: (((eq T t2 x3) \to (\forall (P:
454 Prop).P)))).(let H_x0 \def (term_dec x x1) in (let H24 \def H_x0 in (or_ind
455 (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl)
456 x1 (THead (Bind Abst) x3 x4))) (\lambda (H25: (eq T x x1)).(let H26 \def
457 (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H25) in (eq_ind T x
458 (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4))))
459 (let H_x1 \def (term_dec x0 x4) in (let H27 \def H_x1 in (or_ind (eq T x0 x4)
460 ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x (THead
461 (Bind Abst) x3 x4))) (\lambda (H28: (eq T x0 x4)).(let H29 \def (eq_ind_r T
462 x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
463 x0 t0)))) H20 x0 H28) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat
464 Appl) x (THead (Bind Abst) x3 t0)))) (H8 x3 H23 (pr3_pr2 c t2 x3 H19)) x4
465 H28))) (\lambda (H28: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead
466 (Bind Abbr) x x4) (\lambda (H29: (eq T (THead (Bind Abbr) x x0) (THead (Bind
467 Abbr) x x4))).(\lambda (P: Prop).(let H30 \def (f_equal T T (\lambda (e:
468 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
469 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
470 Abbr) x x0) (THead (Bind Abbr) x x4) H29) in (let H31 \def (eq_ind_r T x4
471 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H28 x0 H30) in
472 (let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u:
473 T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (H31 (refl_equal T x0)
474 P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4)
475 (pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead
476 (Bind Abbr) x x4)) x3 (H7 x3 H23 (pr3_pr2 c t2 x3 H19)))) H27))) x1 H25)))
477 (\lambda (H25: (((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind
478 Abbr) x1 x4) (\lambda (H26: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr)
479 x1 x4))).(\lambda (P: Prop).(let H27 \def (f_equal T T (\lambda (e: T).(match
480 e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _)
481 \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0)
482 (THead (Bind Abbr) x1 x4) H26) in ((let H28 \def (f_equal T T (\lambda (e:
483 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
484 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
485 Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in (\lambda (H29: (eq T x
486 x1)).(let H30 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall
487 (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (let H31 \def
488 (eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
489 H25 x H29) in (let H32 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14
490 x H29) in (H31 (refl_equal T x) P)))))) H27)))) (pr3_head_12 c x x1 (pr3_pr2
491 c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20
492 Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) x3 (H7 x3 H23
493 (pr3_pr2 c t2 x3 H19)))) H24)))) H22))) x2 H18))))))) H17)) t3 H13)))))))
494 H12)) (\lambda (H12: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
495 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead
496 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
497 T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
498 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
499 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
500 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T
501 T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
502 (THead (Bind Abst) t2 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
503 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
504 Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
505 (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
506 T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
507 z1 t4))))))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
508 T).(\lambda (x4: T).(\lambda (H13: (eq T (THead (Bind Abst) t2 x0) (THead
509 (Bind Abst) x1 x2))).(\lambda (H14: (eq T t3 (THead (Bind Abbr) x3
510 x4))).(\lambda (H15: (pr2 c x x3)).(\lambda (H16: ((\forall (b: B).(\forall
511 (u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H17 \def (eq_ind T t3
512 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0)
513 \to (\forall (P: Prop).P))) H9 (THead (Bind Abbr) x3 x4) H14) in (eq_ind_r T
514 (THead (Bind Abbr) x3 x4) (\lambda (t0: T).(sn3 c t0)) (let H18 \def (f_equal
515 T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
516 \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) \Rightarrow t0]))
517 (THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in ((let H19 \def
518 (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
519 [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0)
520 \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in
521 (\lambda (_: (eq T t2 x1)).(let H21 \def (eq_ind_r T x2 (\lambda (t0:
522 T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t0 x4)))) H16 x0
523 H19) in (let H_x \def (term_dec x x3) in (let H22 \def H_x in (or_ind (eq T x
524 x3) ((eq T x x3) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abbr) x3 x4))
525 (\lambda (H23: (eq T x x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0:
526 T).(pr2 c x t0)) H15 x H23) in (eq_ind T x (\lambda (t0: T).(sn3 c (THead
527 (Bind Abbr) t0 x4))) (let H_x0 \def (term_dec x0 x4) in (let H25 \def H_x0 in
528 (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead
529 (Bind Abbr) x x4)) (\lambda (H26: (eq T x0 x4)).(let H27 \def (eq_ind_r T x4
530 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0
531 t0)))) H21 x0 H26) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Bind Abbr)
532 x t0))) (sn3_sing c (THead (Bind Abbr) x x0) H6) x4 H26))) (\lambda (H26:
533 (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x x4)
534 (\lambda (H27: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x
535 x4))).(\lambda (P: Prop).(let H28 \def (f_equal T T (\lambda (e: T).(match e
536 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
537 \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0)
538 (THead (Bind Abbr) x x4) H27) in (let H29 \def (eq_ind_r T x4 (\lambda (t0:
539 T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H26 x0 H28) in (let H30 \def
540 (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c
541 (Bind b) u) x0 t0)))) H21 x0 H28) in (H29 (refl_equal T x0) P)))))) (pr3_pr2
542 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4
543 (Bind Abbr) (H21 Abbr x))))) H25))) x3 H23))) (\lambda (H23: (((eq T x x3)
544 \to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x3 x4) (\lambda (H24: (eq
545 T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x3 x4))).(\lambda (P:
546 Prop).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T return
547 (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x |
548 (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr)
549 x3 x4) H24) in ((let H26 \def (f_equal T T (\lambda (e: T).(match e in T
550 return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
551 \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0)
552 (THead (Bind Abbr) x3 x4) H24) in (\lambda (H27: (eq T x x3)).(let H28 \def
553 (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c
554 (Bind b) u) x0 t0)))) H21 x0 H26) in (let H29 \def (eq_ind_r T x3 (\lambda
555 (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H23 x H27) in (let H30
556 \def (eq_ind_r T x3 (\lambda (t0: T).(pr2 c x t0)) H15 x H27) in (H29
557 (refl_equal T x) P)))))) H25)))) (pr3_head_12 c x x3 (pr3_pr2 c x x3 H15)
558 (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x3) x0 x4 (H21 Abbr x3)))))
559 H22)))))) H18)) t3 H14)))))))))) H12)) (\lambda (H12: (ex6_6 B T T T T T
560 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
561 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
562 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
563 (THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
564 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
565 (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
566 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
567 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_:
568 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
569 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
570 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
571 y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_:
572 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
573 b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
574 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead
575 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
576 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
577 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
578 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
579 (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
580 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
581 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
582 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3)
583 (\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda
584 (x5: T).(\lambda (x6: T).(\lambda (H13: (not (eq B x1 Abst))).(\lambda (H14:
585 (eq T (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3))).(\lambda (H15: (eq
586 T t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
587 (_: (pr2 c x x5)).(\lambda (H17: (pr2 c x2 x6)).(\lambda (H18: (pr2 (CHead c
588 (Bind x1) x6) x3 x4)).(let H19 \def (eq_ind T t3 (\lambda (t0: T).((eq T
589 (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P:
590 Prop).P))) H9 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))
591 H15) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5)
592 x4)) (\lambda (t0: T).(sn3 c t0)) (let H20 \def (f_equal T B (\lambda (e:
593 T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst |
594 (TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return
595 (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
596 Abst])])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in ((let H21
597 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
598 with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _)
599 \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in
600 ((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
601 T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
602 t0) \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14)
603 in (\lambda (H23: (eq T t2 x2)).(\lambda (H24: (eq B Abst x1)).(let H25 \def
604 (eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H18 x0
605 H22) in (let H26 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H17 t2
606 H23) in (let H27 \def (eq_ind_r B x1 (\lambda (b: B).(pr2 (CHead c (Bind b)
607 x6) x0 x4)) H25 Abst H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b:
608 B).(not (eq B b Abst))) H13 Abst H24) in (eq_ind B Abst (\lambda (b: B).(sn3
609 c (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (let H29
610 \def (match (H28 (refl_equal B Abst)) in False return (\lambda (_:
611 False).(sn3 c (THead (Bind Abst) x6 (THead (Flat Appl) (lift (S O) O x5)
612 x4)))) with []) in H29) x1 H24)))))))) H21)) H20)) t3 H15)))))))))))))) H12))
613 H11))))))))) w H4))))))))))) y H0))))) H)))).
615 theorem sn3_appl_lref:
616 \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (v:
617 T).((sn3 c v) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))
619 \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda
620 (v: T).(\lambda (H0: (sn3 c v)).(sn3_ind c (\lambda (t: T).(sn3 c (THead
621 (Flat Appl) t (TLRef i)))) (\lambda (t1: T).(\lambda (_: ((\forall (t2:
622 T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
623 t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
624 Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c (THead (Flat Appl) t2 (TLRef
625 i)))))))).(sn3_pr2_intro c (THead (Flat Appl) t1 (TLRef i)) (\lambda (t2:
626 T).(\lambda (H3: (((eq T (THead (Flat Appl) t1 (TLRef i)) t2) \to (\forall
627 (P: Prop).P)))).(\lambda (H4: (pr2 c (THead (Flat Appl) t1 (TLRef i))
628 t2)).(let H5 \def (pr2_gen_appl c t1 (TLRef i) t2 H4) in (or3_ind (ex3_2 T T
629 (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
630 (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda
631 (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
632 T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1
633 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
634 T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
635 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))) (\lambda (_:
636 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
637 (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
638 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
639 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
640 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
641 (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
642 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
643 t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
644 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
645 T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1:
646 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
647 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
648 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))
649 (sn3 c t2) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
650 t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1
651 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T
652 T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
653 (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda
654 (t3: T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1:
655 T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr2 c t1
656 x0)).(\lambda (H9: (pr2 c (TLRef i) x1)).(let H10 \def (eq_ind T t2 (\lambda
657 (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to (\forall (P: Prop).P)))
658 H3 (THead (Flat Appl) x0 x1) H7) in (eq_ind_r T (THead (Flat Appl) x0 x1)
659 (\lambda (t: T).(sn3 c t)) (let H11 \def (eq_ind_r T x1 (\lambda (t: T).((eq
660 T (THead (Flat Appl) t1 (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P:
661 Prop).P))) H10 (TLRef i) (H x1 H9)) in (let H12 \def (eq_ind_r T x1 (\lambda
662 (t: T).(pr2 c (TLRef i) t)) H9 (TLRef i) (H x1 H9)) in (eq_ind T (TLRef i)
663 (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H_x \def (term_dec t1
664 x0) in (let H13 \def H_x in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall
665 (P: Prop).P)) (sn3 c (THead (Flat Appl) x0 (TLRef i))) (\lambda (H14: (eq T
666 t1 x0)).(let H15 \def (eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat
667 Appl) t1 (TLRef i)) (THead (Flat Appl) t (TLRef i))) \to (\forall (P:
668 Prop).P))) H11 t1 H14) in (let H16 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c
669 t1 t)) H8 t1 H14) in (eq_ind T t1 (\lambda (t: T).(sn3 c (THead (Flat Appl) t
670 (TLRef i)))) (H15 (refl_equal T (THead (Flat Appl) t1 (TLRef i))) (sn3 c
671 (THead (Flat Appl) t1 (TLRef i)))) x0 H14)))) (\lambda (H14: (((eq T t1 x0)
672 \to (\forall (P: Prop).P)))).(H2 x0 H14 (pr3_pr2 c t1 x0 H8))) H13))) x1 (H
673 x1 H9)))) t2 H7))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1:
674 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead
675 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
676 T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
677 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2)))))
678 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
679 (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T
680 T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
681 (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
682 T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))))
683 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1
684 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
685 T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))
686 (sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
687 T).(\lambda (H7: (eq T (TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H8:
688 (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c t1 x2)).(\lambda (_:
689 ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let
690 H11 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i))
691 t) \to (\forall (P: Prop).P))) H3 (THead (Bind Abbr) x2 x3) H8) in (eq_ind_r
692 T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) (let H12 \def (eq_ind
693 T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
694 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
695 \Rightarrow False])) I (THead (Bind Abst) x0 x1) H7) in (False_ind (sn3 c
696 (THead (Bind Abbr) x2 x3)) H12)) t2 H8)))))))))) H6)) (\lambda (H6: (ex6_6 B
697 T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
698 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
699 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
700 (_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
701 (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
702 T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
703 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
704 T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1:
705 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
706 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
707 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
708 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
709 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
710 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
711 T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind b) y1
712 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
713 T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat
714 Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
715 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2)))))))
716 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
717 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
718 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
719 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2) (\lambda (x0: B).(\lambda (x1:
720 T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
721 T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H8: (eq T (TLRef i) (THead
722 (Bind x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x5 (THead (Flat
723 Appl) (lift (S O) O x4) x3)))).(\lambda (_: (pr2 c t1 x4)).(\lambda (_: (pr2
724 c x1 x5)).(\lambda (_: (pr2 (CHead c (Bind x0) x5) x2 x3)).(let H13 \def
725 (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to
726 (\forall (P: Prop).P))) H3 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O)
727 O x4) x3)) H9) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S
728 O) O x4) x3)) (\lambda (t: T).(sn3 c t)) (let H14 \def (eq_ind T (TLRef i)
729 (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
730 \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
731 False])) I (THead (Bind x0) x1 x2) H8) in (False_ind (sn3 c (THead (Bind x0)
732 x5 (THead (Flat Appl) (lift (S O) O x4) x3))) H14)) t2 H9)))))))))))))) H6))
733 H5))))))))) v H0))))).
735 theorem sn3_appl_abbr:
736 \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c
737 (CHead d (Bind Abbr) w)) \to (\forall (v: T).((sn3 c (THead (Flat Appl) v
738 (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))))
740 \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda
741 (H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (v: T).(\lambda (H0: (sn3 c
742 (THead (Flat Appl) v (lift (S i) O w)))).(insert_eq T (THead (Flat Appl) v
743 (lift (S i) O w)) (\lambda (t: T).(sn3 c t)) (\lambda (_: T).(sn3 c (THead
744 (Flat Appl) v (TLRef i)))) (\lambda (y: T).(\lambda (H1: (sn3 c y)).(unintro
745 T v (\lambda (t: T).((eq T y (THead (Flat Appl) t (lift (S i) O w))) \to (sn3
746 c (THead (Flat Appl) t (TLRef i))))) (sn3_ind c (\lambda (t: T).(\forall (x:
747 T).((eq T t (THead (Flat Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat
748 Appl) x (TLRef i)))))) (\lambda (t1: T).(\lambda (H2: ((\forall (t2:
749 T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
750 t2)))))).(\lambda (H3: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
751 Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).((eq T t2 (THead (Flat
752 Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x (TLRef
753 i)))))))))).(\lambda (x: T).(\lambda (H4: (eq T t1 (THead (Flat Appl) x (lift
754 (S i) O w)))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2:
755 T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall
756 (x0: T).((eq T t2 (THead (Flat Appl) x0 (lift (S i) O w))) \to (sn3 c (THead
757 (Flat Appl) x0 (TLRef i))))))))) H3 (THead (Flat Appl) x (lift (S i) O w))
758 H4) in (let H6 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: T).((((eq T t
759 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (sn3 c t2))))) H2
760 (THead (Flat Appl) x (lift (S i) O w)) H4) in (sn3_pr2_intro c (THead (Flat
761 Appl) x (TLRef i)) (\lambda (t2: T).(\lambda (H7: (((eq T (THead (Flat Appl)
762 x (TLRef i)) t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr2 c (THead
763 (Flat Appl) x (TLRef i)) t2)).(let H9 \def (pr2_gen_appl c x (TLRef i) t2 H8)
764 in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
765 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
766 (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T
767 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
768 (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
769 T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))))
770 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x
771 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
772 T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))
773 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
774 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
775 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
776 (_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
777 (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
778 T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
779 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
780 T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
781 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
782 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
783 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))
784 (sn3 c t2) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
785 t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x
786 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T
787 T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
788 (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3:
789 T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1:
790 T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H12: (pr2 c
791 x x0)).(\lambda (H13: (pr2 c (TLRef i) x1)).(let H14 \def (eq_ind T t2
792 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P:
793 Prop).P))) H7 (THead (Flat Appl) x0 x1) H11) in (eq_ind_r T (THead (Flat
794 Appl) x0 x1) (\lambda (t: T).(sn3 c t)) (let H15 \def (pr2_gen_lref c x1 i
795 H13) in (or_ind (eq T x1 (TLRef i)) (ex2_2 C T (\lambda (d0: C).(\lambda (u:
796 T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq
797 T x1 (lift (S i) O u))))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda (H16:
798 (eq T x1 (TLRef i))).(let H17 \def (eq_ind T x1 (\lambda (t: T).((eq T (THead
799 (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P:
800 Prop).P))) H14 (TLRef i) H16) in (eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c
801 (THead (Flat Appl) x0 t))) (let H_x \def (term_dec x x0) in (let H18 \def H_x
802 in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c (THead
803 (Flat Appl) x0 (TLRef i))) (\lambda (H19: (eq T x x0)).(let H20 \def
804 (eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead
805 (Flat Appl) t (TLRef i))) \to (\forall (P: Prop).P))) H17 x H19) in (let H21
806 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H19) in (eq_ind T x
807 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (TLRef i)))) (H20 (refl_equal T
808 (THead (Flat Appl) x (TLRef i))) (sn3 c (THead (Flat Appl) x (TLRef i)))) x0
809 H19)))) (\lambda (H19: (((eq T x x0) \to (\forall (P: Prop).P)))).(H5 (THead
810 (Flat Appl) x0 (lift (S i) O w)) (\lambda (H20: (eq T (THead (Flat Appl) x
811 (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)))).(\lambda (P:
812 Prop).(let H21 \def (f_equal T T (\lambda (e: T).(match e in T return
813 (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x |
814 (THead _ t _) \Rightarrow t])) (THead (Flat Appl) x (lift (S i) O w)) (THead
815 (Flat Appl) x0 (lift (S i) O w)) H20) in (let H22 \def (eq_ind_r T x0
816 (\lambda (t: T).((eq T x t) \to (\forall (P0: Prop).P0))) H19 x H21) in (let
817 H23 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H21) in (H22
818 (refl_equal T x) P)))))) (pr3_pr2 c (THead (Flat Appl) x (lift (S i) O w))
819 (THead (Flat Appl) x0 (lift (S i) O w)) (pr2_head_1 c x x0 H12 (Flat Appl)
820 (lift (S i) O w))) x0 (refl_equal T (THead (Flat Appl) x0 (lift (S i) O
821 w))))) H18))) x1 H16))) (\lambda (H16: (ex2_2 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)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda
824 (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
825 T).(eq T x1 (lift (S i) O u)))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda
826 (x2: C).(\lambda (x3: T).(\lambda (H17: (getl i c (CHead x2 (Bind Abbr)
827 x3))).(\lambda (H18: (eq T x1 (lift (S i) O x3))).(let H19 \def (eq_ind T x1
828 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0
829 t)) \to (\forall (P: Prop).P))) H14 (lift (S i) O x3) H18) in (eq_ind_r T
830 (lift (S i) O x3) (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H20
831 \def (eq_ind C (CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H
832 (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2
833 (Bind Abbr) x3) H17)) in (let H21 \def (f_equal C C (\lambda (e: C).(match e
834 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
835 \Rightarrow c0])) (CHead d (Bind Abbr) w) (CHead x2 (Bind Abbr) x3)
836 (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 (Bind Abbr) x3) H17)) in
837 ((let H22 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
838 C).T) with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d
839 (Bind Abbr) w) (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w)
840 i H (CHead x2 (Bind Abbr) x3) H17)) in (\lambda (H23: (eq C d x2)).(let H24
841 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c (CHead x2 (Bind Abbr) t))) H20
842 w H22) in (eq_ind T w (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 (lift (S
843 i) O t)))) (let H25 \def (eq_ind_r C x2 (\lambda (c0: C).(getl i c (CHead c0
844 (Bind Abbr) w))) H24 d H23) in (let H_x \def (term_dec x x0) in (let H26 \def
845 H_x in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c
846 (THead (Flat Appl) x0 (lift (S i) O w))) (\lambda (H27: (eq T x x0)).(let H28
847 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H27) in (eq_ind T x
848 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (lift (S i) O w)))) (sn3_sing c
849 (THead (Flat Appl) x (lift (S i) O w)) H6) x0 H27))) (\lambda (H27: (((eq T x
850 x0) \to (\forall (P: Prop).P)))).(H6 (THead (Flat Appl) x0 (lift (S i) O w))
851 (\lambda (H28: (eq T (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat
852 Appl) x0 (lift (S i) O w)))).(\lambda (P: Prop).(let H29 \def (f_equal T T
853 (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
854 \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t]))
855 (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O
856 w)) H28) in (let H30 \def (eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to
857 (\forall (P0: Prop).P0))) H27 x H29) in (let H31 \def (eq_ind_r T x0 (\lambda
858 (t: T).(pr2 c x t)) H12 x H29) in (H30 (refl_equal T x) P)))))) (pr3_pr2 c
859 (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O
860 w)) (pr2_head_1 c x x0 H12 (Flat Appl) (lift (S i) O w))))) H26)))) x3
861 H22)))) H21))) x1 H18)))))) H16)) H15)) t2 H11))))))) H10)) (\lambda (H10:
862 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
863 T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
864 (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
865 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
866 T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
867 (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1
868 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
869 T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda
870 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
871 (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
872 T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
873 (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
874 b) u) z1 t3))))))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2:
875 T).(\lambda (x3: T).(\lambda (H11: (eq T (TLRef i) (THead (Bind Abst) x0
876 x1))).(\lambda (H12: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c
877 x x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b)
878 u) x1 x3))))).(let H15 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat
879 Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 (THead (Bind Abbr) x2
880 x3) H12) in (eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t))
881 (let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return
882 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
883 \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0
884 x1) H11) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H16)) t2
885 H12)))))))))) H10)) (\lambda (H10: (ex6_6 B T T T T T (\lambda (b:
886 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
887 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
888 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i)
889 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
890 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
891 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
892 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
893 (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
894 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
895 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
896 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind
897 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
898 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
899 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
900 (_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
901 (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
902 T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
903 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
904 T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
905 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
906 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
907 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
908 (sn3 c t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
909 T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0
910 Abst))).(\lambda (H12: (eq T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda
911 (H13: (eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4)
912 x3)))).(\lambda (_: (pr2 c x x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_:
913 (pr2 (CHead c (Bind x0) x5) x2 x3)).(let H17 \def (eq_ind T t2 (\lambda (t:
914 T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7
915 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) H13) in
916 (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))
917 (\lambda (t: T).(sn3 c t)) (let H18 \def (eq_ind T (TLRef i) (\lambda (ee:
918 T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
919 False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
920 (THead (Bind x0) x1 x2) H12) in (False_ind (sn3 c (THead (Bind x0) x5 (THead
921 (Flat Appl) (lift (S O) O x4) x3))) H18)) t2 H13)))))))))))))) H10))
922 H9))))))))))))) y H1)))) H0))))))).
924 theorem sn3_appl_cast:
925 \forall (c: C).(\forall (v: T).(\forall (u: T).((sn3 c (THead (Flat Appl) v
926 u)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) v t)) \to (sn3 c (THead
927 (Flat Appl) v (THead (Flat Cast) u t))))))))
929 \lambda (c: C).(\lambda (v: T).(\lambda (u: T).(\lambda (H: (sn3 c (THead
930 (Flat Appl) v u))).(insert_eq T (THead (Flat Appl) v u) (\lambda (t: T).(sn3
931 c t)) (\lambda (_: T).(\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to
932 (sn3 c (THead (Flat Appl) v (THead (Flat Cast) u t0)))))) (\lambda (y:
933 T).(\lambda (H0: (sn3 c y)).(unintro T u (\lambda (t: T).((eq T y (THead
934 (Flat Appl) v t)) \to (\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to
935 (sn3 c (THead (Flat Appl) v (THead (Flat Cast) t t0))))))) (unintro T v
936 (\lambda (t: T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to
937 (\forall (t0: T).((sn3 c (THead (Flat Appl) t t0)) \to (sn3 c (THead (Flat
938 Appl) t (THead (Flat Cast) x t0)))))))) (sn3_ind c (\lambda (t: T).(\forall
939 (x: T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (t0:
940 T).((sn3 c (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead
941 (Flat Cast) x0 t0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2:
942 T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
943 t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
944 Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2
945 (THead (Flat Appl) x x0)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) x
946 t)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0
947 t))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1 (THead
948 (Flat Appl) x x0))).(\lambda (t: T).(\lambda (H4: (sn3 c (THead (Flat Appl) x
949 t))).(insert_eq T (THead (Flat Appl) x t) (\lambda (t0: T).(sn3 c t0))
950 (\lambda (_: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 t))))
951 (\lambda (y0: T).(\lambda (H5: (sn3 c y0)).(unintro T t (\lambda (t0: T).((eq
952 T y0 (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead (Flat
953 Cast) x0 t0))))) (sn3_ind c (\lambda (t0: T).(\forall (x1: T).((eq T t0
954 (THead (Flat Appl) x x1)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast)
955 x0 x1)))))) (\lambda (t0: T).(\lambda (H6: ((\forall (t2: T).((((eq T t0 t2)
956 \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2)))))).(\lambda
957 (H7: ((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3
958 c t0 t2) \to (\forall (x1: T).((eq T t2 (THead (Flat Appl) x x1)) \to (sn3 c
959 (THead (Flat Appl) x (THead (Flat Cast) x0 x1)))))))))).(\lambda (x1:
960 T).(\lambda (H8: (eq T t0 (THead (Flat Appl) x x1))).(let H9 \def (eq_ind T
961 t0 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
962 Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).((eq T t3 (THead (Flat
963 Appl) x x2)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0
964 x2))))))))) H7 (THead (Flat Appl) x x1) H8) in (let H10 \def (eq_ind T t0
965 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P)))
966 \to ((pr3 c t2 t3) \to (sn3 c t3))))) H6 (THead (Flat Appl) x x1) H8) in (let
967 H11 \def (eq_ind T t1 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to
968 (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).(\forall (x3:
969 T).((eq T t3 (THead (Flat Appl) x2 x3)) \to (\forall (t4: T).((sn3 c (THead
970 (Flat Appl) x2 t4)) \to (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x3
971 t4)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H12 \def (eq_ind T t1
972 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P)))
973 \to ((pr3 c t2 t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in
974 (sn3_pr2_intro c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (\lambda
975 (t2: T).(\lambda (H13: (((eq T (THead (Flat Appl) x (THead (Flat Cast) x0
976 x1)) t2) \to (\forall (P: Prop).P)))).(\lambda (H14: (pr2 c (THead (Flat
977 Appl) x (THead (Flat Cast) x0 x1)) t2)).(let H15 \def (pr2_gen_appl c x
978 (THead (Flat Cast) x0 x1) t2 H14) in (or3_ind (ex3_2 T T (\lambda (u2:
979 T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
980 T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
981 (THead (Flat Cast) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
982 (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1)
983 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
984 T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
985 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
986 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b:
987 B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T
988 T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
989 (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
990 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
991 T (THead (Flat Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b:
992 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
993 (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
994 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
995 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_:
996 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
997 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
998 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
999 y2) z1 z2)))))))) (sn3 c t2) (\lambda (H16: (ex3_2 T T (\lambda (u2:
1000 T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
1001 T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
1002 (THead (Flat Cast) x0 x1) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
1003 (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
1004 T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Flat Cast)
1005 x0 x1) t3))) (sn3 c t2) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H17: (eq
1006 T t2 (THead (Flat Appl) x2 x3))).(\lambda (H18: (pr2 c x x2)).(\lambda (H19:
1007 (pr2 c (THead (Flat Cast) x0 x1) x3)).(let H20 \def (eq_ind T t2 (\lambda
1008 (t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to
1009 (\forall (P: Prop).P))) H13 (THead (Flat Appl) x2 x3) H17) in (eq_ind_r T
1010 (THead (Flat Appl) x2 x3) (\lambda (t3: T).(sn3 c t3)) (let H21 \def
1011 (pr2_gen_cast c x0 x1 x3 H19) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda
1012 (t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
1013 T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3)))) (pr2 c
1014 x1 x3) (sn3 c (THead (Flat Appl) x2 x3)) (\lambda (H22: (ex3_2 T T (\lambda
1015 (u2: T).(\lambda (t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
1016 T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1
1017 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x3 (THead
1018 (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
1019 (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3))) (sn3 c (THead (Flat Appl) x2
1020 x3)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H23: (eq T x3 (THead (Flat
1021 Cast) x4 x5))).(\lambda (H24: (pr2 c x0 x4)).(\lambda (H25: (pr2 c x1
1022 x5)).(let H26 \def (eq_ind T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x
1023 (THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P:
1024 Prop).P))) H20 (THead (Flat Cast) x4 x5) H23) in (eq_ind_r T (THead (Flat
1025 Cast) x4 x5) (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 t3))) (let H_x
1026 \def (term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) in (let
1027 H27 \def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2
1028 x4)) ((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) \to (\forall
1029 (P: Prop).P)) (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x4 x5)))
1030 (\lambda (H28: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2
1031 x4))).(let H29 \def (f_equal T T (\lambda (e: T).(match e in T return
1032 (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x |
1033 (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x0) (THead (Flat Appl)
1034 x2 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T
1035 return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
1036 \Rightarrow x0 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x0)
1037 (THead (Flat Appl) x2 x4) H28) in (\lambda (H31: (eq T x x2)).(let H32 \def
1038 (eq_ind_r T x4 (\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat
1039 Cast) x0 x1)) (THead (Flat Appl) x2 (THead (Flat Cast) t3 x5))) \to (\forall
1040 (P: Prop).P))) H26 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t3:
1041 T).(pr2 c x0 t3)) H24 x0 H30) in (eq_ind T x0 (\lambda (t3: T).(sn3 c (THead
1042 (Flat Appl) x2 (THead (Flat Cast) t3 x5)))) (let H34 \def (eq_ind_r T x2
1043 (\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1))
1044 (THead (Flat Appl) t3 (THead (Flat Cast) x0 x5))) \to (\forall (P: Prop).P)))
1045 H32 x H31) in (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18
1046 x H31) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead
1047 (Flat Cast) x0 x5)))) (let H_x0 \def (term_dec (THead (Flat Appl) x x1)
1048 (THead (Flat Appl) x x5)) in (let H36 \def H_x0 in (or_ind (eq T (THead (Flat
1049 Appl) x x1) (THead (Flat Appl) x x5)) ((eq T (THead (Flat Appl) x x1) (THead
1050 (Flat Appl) x x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x
1051 (THead (Flat Cast) x0 x5))) (\lambda (H37: (eq T (THead (Flat Appl) x x1)
1052 (THead (Flat Appl) x x5))).(let H38 \def (f_equal T T (\lambda (e: T).(match
1053 e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _)
1054 \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x1)
1055 (THead (Flat Appl) x x5) H37) in (let H39 \def (eq_ind_r T x5 (\lambda (t3:
1056 T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl)
1057 x (THead (Flat Cast) x0 t3))) \to (\forall (P: Prop).P))) H34 x1 H38) in (let
1058 H40 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H38) in
1059 (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast)
1060 x0 t3)))) (H39 (refl_equal T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)))
1061 (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)))) x5 H38))))) (\lambda
1062 (H37: (((eq T (THead (Flat Appl) x x1) (THead (Flat Appl) x x5)) \to (\forall
1063 (P: Prop).P)))).(H9 (THead (Flat Appl) x x5) H37 (pr3_pr2 c (THead (Flat
1064 Appl) x x1) (THead (Flat Appl) x x5) (pr2_thin_dx c x1 x5 H25 x Appl)) x5
1065 (refl_equal T (THead (Flat Appl) x x5)))) H36))) x2 H31))) x4 H30))))) H29)))
1066 (\lambda (H28: (((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4))
1067 \to (\forall (P: Prop).P)))).(let H_x0 \def (term_dec (THead (Flat Appl) x
1068 x1) (THead (Flat Appl) x2 x5)) in (let H29 \def H_x0 in (or_ind (eq T (THead
1069 (Flat Appl) x x1) (THead (Flat Appl) x2 x5)) ((eq T (THead (Flat Appl) x x1)
1070 (THead (Flat Appl) x2 x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat
1071 Appl) x2 (THead (Flat Cast) x4 x5))) (\lambda (H30: (eq T (THead (Flat Appl)
1072 x x1) (THead (Flat Appl) x2 x5))).(let H31 \def (f_equal T T (\lambda (e:
1073 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x |
1074 (TLRef _) \Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl)
1075 x x1) (THead (Flat Appl) x2 x5) H30) in ((let H32 \def (f_equal T T (\lambda
1076 (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1
1077 | (TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat
1078 Appl) x x1) (THead (Flat Appl) x2 x5) H30) in (\lambda (H33: (eq T x
1079 x2)).(let H34 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H32)
1080 in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 (THead (Flat
1081 Cast) x4 t3)))) (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead
1082 (Flat Appl) x x0) (THead (Flat Appl) t3 x4)) \to (\forall (P: Prop).P))) H28
1083 x H33) in (let H36 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18 x
1084 H33) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead
1085 (Flat Cast) x4 x1)))) (H11 (THead (Flat Appl) x x4) H35 (pr3_pr2 c (THead
1086 (Flat Appl) x x0) (THead (Flat Appl) x x4) (pr2_thin_dx c x0 x4 H24 x Appl))
1087 x x4 (refl_equal T (THead (Flat Appl) x x4)) x1 (sn3_sing c (THead (Flat
1088 Appl) x x1) H10)) x2 H33))) x5 H32)))) H31))) (\lambda (H30: (((eq T (THead
1089 (Flat Appl) x x1) (THead (Flat Appl) x2 x5)) \to (\forall (P:
1090 Prop).P)))).(H11 (THead (Flat Appl) x2 x4) H28 (pr3_flat c x x2 (pr3_pr2 c x
1091 x2 H18) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x2 x4 (refl_equal T (THead (Flat
1092 Appl) x2 x4)) x5 (H10 (THead (Flat Appl) x2 x5) H30 (pr3_flat c x x2 (pr3_pr2
1093 c x x2 H18) x1 x5 (pr3_pr2 c x1 x5 H25) Appl)))) H29)))) H27))) x3 H23)))))))
1094 H22)) (\lambda (H22: (pr2 c x1 x3)).(let H_x \def (term_dec (THead (Flat
1095 Appl) x x1) (THead (Flat Appl) x2 x3)) in (let H23 \def H_x in (or_ind (eq T
1096 (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3)) ((eq T (THead (Flat Appl)
1097 x x1) (THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)) (sn3 c (THead
1098 (Flat Appl) x2 x3)) (\lambda (H24: (eq T (THead (Flat Appl) x x1) (THead
1099 (Flat Appl) x2 x3))).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T
1100 return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _)
1101 \Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x1)
1102 (THead (Flat Appl) x2 x3) H24) in ((let H26 \def (f_equal T T (\lambda (e:
1103 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 |
1104 (TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat
1105 Appl) x x1) (THead (Flat Appl) x2 x3) H24) in (\lambda (H27: (eq T x
1106 x2)).(let H28 \def (eq_ind_r T x3 (\lambda (t3: T).(pr2 c x1 t3)) H22 x1 H26)
1107 in (let H29 \def (eq_ind_r T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x
1108 (THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P:
1109 Prop).P))) H20 x1 H26) in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat
1110 Appl) x2 t3))) (let H30 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead
1111 (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) t3 x1)) \to
1112 (\forall (P: Prop).P))) H29 x H27) in (let H31 \def (eq_ind_r T x2 (\lambda
1113 (t3: T).(pr2 c x t3)) H18 x H27) in (eq_ind T x (\lambda (t3: T).(sn3 c
1114 (THead (Flat Appl) t3 x1))) (sn3_sing c (THead (Flat Appl) x x1) H10) x2
1115 H27))) x3 H26))))) H25))) (\lambda (H24: (((eq T (THead (Flat Appl) x x1)
1116 (THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat
1117 Appl) x2 x3) H24 (pr3_flat c x x2 (pr3_pr2 c x x2 H18) x1 x3 (pr3_pr2 c x1 x3
1118 H22) Appl))) H23)))) H21)) t2 H17))))))) H16)) (\lambda (H16: (ex4_4 T T T T
1119 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
1120 (THead (Flat Cast) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1121 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
1122 Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1123 (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1124 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1125 u0) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1:
1126 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead
1127 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1128 T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
1129 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
1130 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b:
1131 B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))) (sn3 c t2)
1132 (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda
1133 (H17: (eq T (THead (Flat Cast) x0 x1) (THead (Bind Abst) x2 x3))).(\lambda
1134 (H18: (eq T t2 (THead (Bind Abbr) x4 x5))).(\lambda (_: (pr2 c x
1135 x4)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
1136 u0) x3 x5))))).(let H21 \def (eq_ind T t2 (\lambda (t3: T).((eq T (THead
1137 (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P: Prop).P))) H13
1138 (THead (Bind Abbr) x4 x5) H18) in (eq_ind_r T (THead (Bind Abbr) x4 x5)
1139 (\lambda (t3: T).(sn3 c t3)) (let H22 \def (eq_ind T (THead (Flat Cast) x0
1140 x1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
1141 _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
1142 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
1143 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) x2
1144 x3) H17) in (False_ind (sn3 c (THead (Bind Abbr) x4 x5)) H22)) t2
1145 H18)))))))))) H16)) (\lambda (H16: (ex6_6 B T T T T T (\lambda (b:
1146 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1147 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
1148 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat
1149 Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
1150 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
1151 t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
1152 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1153 T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
1154 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
1155 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
1156 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
1157 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
1158 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
1159 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1160 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead
1161 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1162 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
1163 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
1164 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1165 (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
1166 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
1167 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
1168 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2)
1169 (\lambda (x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda
1170 (x6: T).(\lambda (x7: T).(\lambda (_: (not (eq B x2 Abst))).(\lambda (H18:
1171 (eq T (THead (Flat Cast) x0 x1) (THead (Bind x2) x3 x4))).(\lambda (H19: (eq
1172 T t2 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda
1173 (_: (pr2 c x x6)).(\lambda (_: (pr2 c x3 x7)).(\lambda (_: (pr2 (CHead c
1174 (Bind x2) x7) x4 x5)).(let H23 \def (eq_ind T t2 (\lambda (t3: T).((eq T
1175 (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P:
1176 Prop).P))) H13 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5))
1177 H19) in (eq_ind_r T (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6)
1178 x5)) (\lambda (t3: T).(sn3 c t3)) (let H24 \def (eq_ind T (THead (Flat Cast)
1179 x0 x1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
1180 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
1181 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
1182 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x2) x3 x4)
1183 H18) in (False_ind (sn3 c (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O)
1184 O x6) x5))) H24)) t2 H19)))))))))))))) H16)) H15))))))))))))))) y0 H5))))
1185 H4))))))))) y H0))))) H)))).
1187 theorem sn3_appl_bind:
1188 \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u:
1189 T).((sn3 c u) \to (\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) u)
1190 (THead (Flat Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v
1191 (THead (Bind b) u t))))))))))
1193 \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda
1194 (u: T).(\lambda (H0: (sn3 c u)).(sn3_ind c (\lambda (t: T).(\forall (t0:
1195 T).(\forall (v: T).((sn3 (CHead c (Bind b) t) (THead (Flat Appl) (lift (S O)
1196 O v) t0)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t t0)))))))
1197 (\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
1198 (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall
1199 (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
1200 (\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) t2) (THead (Flat
1201 Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t2
1202 t))))))))))).(\lambda (t: T).(\lambda (v: T).(\lambda (H3: (sn3 (CHead c
1203 (Bind b) t1) (THead (Flat Appl) (lift (S O) O v) t))).(insert_eq T (THead
1204 (Flat Appl) (lift (S O) O v) t) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1)
1205 t0)) (\lambda (_: T).(sn3 c (THead (Flat Appl) v (THead (Bind b) t1 t))))
1206 (\lambda (y: T).(\lambda (H4: (sn3 (CHead c (Bind b) t1) y)).(unintro T t
1207 (\lambda (t0: T).((eq T y (THead (Flat Appl) (lift (S O) O v) t0)) \to (sn3 c
1208 (THead (Flat Appl) v (THead (Bind b) t1 t0))))) (unintro T v (\lambda (t0:
1209 T).(\forall (x: T).((eq T y (THead (Flat Appl) (lift (S O) O t0) x)) \to (sn3
1210 c (THead (Flat Appl) t0 (THead (Bind b) t1 x)))))) (sn3_ind (CHead c (Bind b)
1211 t1) (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat
1212 Appl) (lift (S O) O x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b)
1213 t1 x0))))))) (\lambda (t2: T).(\lambda (H5: ((\forall (t3: T).((((eq T t2 t3)
1214 \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3
1215 (CHead c (Bind b) t1) t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t2
1216 t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to
1217 (\forall (x: T).(\forall (x0: T).((eq T t3 (THead (Flat Appl) (lift (S O) O
1218 x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) t1
1219 x0))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H7: (eq T t2 (THead
1220 (Flat Appl) (lift (S O) O x) x0))).(let H8 \def (eq_ind T t2 (\lambda (t0:
1221 T).(\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3
1222 (CHead c (Bind b) t1) t0 t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3
1223 (THead (Flat Appl) (lift (S O) O x1) x2)) \to (sn3 c (THead (Flat Appl) x1
1224 (THead (Bind b) t1 x2)))))))))) H6 (THead (Flat Appl) (lift (S O) O x) x0)
1225 H7) in (let H9 \def (eq_ind T t2 (\lambda (t0: T).(\forall (t3: T).((((eq T
1226 t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t0 t3) \to
1227 (sn3 (CHead c (Bind b) t1) t3))))) H5 (THead (Flat Appl) (lift (S O) O x) x0)
1228 H7) in (sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind b) t1 x0)) (\lambda
1229 (t3: T).(\lambda (H10: (((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0))
1230 t3) \to (\forall (P: Prop).P)))).(\lambda (H11: (pr2 c (THead (Flat Appl) x
1231 (THead (Bind b) t1 x0)) t3)).(let H12 \def (pr2_gen_appl c x (THead (Bind b)
1232 t1 x0) t3 H11) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
1233 t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x
1234 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))))
1235 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
1236 T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
1237 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
1238 Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1239 (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
1240 T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0)
1241 u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_:
1242 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
1243 b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda
1244 (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead
1245 (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
1246 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
1247 b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
1248 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1249 (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
1250 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
1251 (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
1252 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2)))))))) (sn3 c t3)
1253 (\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
1254 (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
1255 (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0)
1256 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
1257 (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
1258 (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))) (sn3 c
1259 t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H14: (eq T t3 (THead (Flat
1260 Appl) x1 x2))).(\lambda (H15: (pr2 c x x1)).(\lambda (H16: (pr2 c (THead
1261 (Bind b) t1 x0) x2)).(let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T
1262 (THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P)))
1263 H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat Appl) x1 x2)
1264 (\lambda (t0: T).(sn3 c t0)) (let H_x \def (pr3_gen_bind b H c t1 x0 x2) in
1265 (let H18 \def (H_x (pr3_pr2 c (THead (Bind b) t1 x0) x2 H16)) in (or_ind
1266 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2
1267 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
1268 T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4)))) (pr3 (CHead c (Bind
1269 b) t1) x0 (lift (S O) O x2)) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (H19:
1270 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2
1271 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
1272 T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4))))).(ex3_2_ind T T
1273 (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 t4)))) (\lambda
1274 (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3
1275 (CHead c (Bind b) t1) x0 t4))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda
1276 (x3: T).(\lambda (x4: T).(\lambda (H20: (eq T x2 (THead (Bind b) x3
1277 x4))).(\lambda (H21: (pr3 c t1 x3)).(\lambda (H22: (pr3 (CHead c (Bind b) t1)
1278 x0 x4)).(let H23 \def (eq_ind T x2 (\lambda (t0: T).((eq T (THead (Flat Appl)
1279 x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 t0)) \to (\forall (P:
1280 Prop).P))) H17 (THead (Bind b) x3 x4) H20) in (eq_ind_r T (THead (Bind b) x3
1281 x4) (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 t0))) (let H_x0 \def
1282 (term_dec t1 x3) in (let H24 \def H_x0 in (or_ind (eq T t1 x3) ((eq T t1 x3)
1283 \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind b) x3
1284 x4))) (\lambda (H25: (eq T t1 x3)).(let H26 \def (eq_ind_r T x3 (\lambda (t0:
1285 T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1
1286 (THead (Bind b) t0 x4))) \to (\forall (P: Prop).P))) H23 t1 H25) in (let H27
1287 \def (eq_ind_r T x3 (\lambda (t0: T).(pr3 c t1 t0)) H21 t1 H25) in (eq_ind T
1288 t1 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 (THead (Bind b) t0 x4))))
1289 (let H_x1 \def (term_dec x0 x4) in (let H28 \def H_x1 in (or_ind (eq T x0 x4)
1290 ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead
1291 (Bind b) t1 x4))) (\lambda (H29: (eq T x0 x4)).(let H30 \def (eq_ind_r T x4
1292 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead
1293 (Flat Appl) x1 (THead (Bind b) t1 t0))) \to (\forall (P: Prop).P))) H26 x0
1294 H29) in (let H31 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b)
1295 t1) x0 t0)) H22 x0 H29) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat
1296 Appl) x1 (THead (Bind b) t1 t0)))) (let H_x2 \def (term_dec x x1) in (let H32
1297 \def H_x2 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3
1298 c (THead (Flat Appl) x1 (THead (Bind b) t1 x0))) (\lambda (H33: (eq T x
1299 x1)).(let H34 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl)
1300 x (THead (Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to
1301 (\forall (P: Prop).P))) H30 x H33) in (let H35 \def (eq_ind_r T x1 (\lambda
1302 (t0: T).(pr2 c x t0)) H15 x H33) in (eq_ind T x (\lambda (t0: T).(sn3 c
1303 (THead (Flat Appl) t0 (THead (Bind b) t1 x0)))) (H34 (refl_equal T (THead
1304 (Flat Appl) x (THead (Bind b) t1 x0))) (sn3 c (THead (Flat Appl) x (THead
1305 (Bind b) t1 x0)))) x1 H33)))) (\lambda (H33: (((eq T x x1) \to (\forall (P:
1306 Prop).P)))).(H8 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H34: (eq T
1307 (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
1308 x0))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: T).(match e
1309 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
1310 (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
1311 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
1312 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
1313 \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
1314 lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
1315 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
1316 t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
1317 (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
1318 (THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1319 t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
1320 \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
1321 (lift (S O) O x1) x0) H34) in (let H36 \def (eq_ind_r T x1 (\lambda (t0:
1322 T).((eq T x t0) \to (\forall (P0: Prop).P0))) H33 x (lift_inj x x1 (S O) O
1323 H35)) in (let H37 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
1324 (lift_inj x x1 (S O) O H35)) in (H36 (refl_equal T x) P)))))) (pr3_flat
1325 (CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
1326 (Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
1327 (pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl) x1 x0
1328 (refl_equal T (THead (Flat Appl) (lift (S O) O x1) x0)))) H32))) x4 H29))))
1329 (\lambda (H29: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H8 (THead (Flat
1330 Appl) (lift (S O) O x1) x4) (\lambda (H30: (eq T (THead (Flat Appl) (lift (S
1331 O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P:
1332 Prop).(let H31 \def (f_equal T T (\lambda (e: T).(match e in T return
1333 (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
1334 \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
1335 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
1336 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
1337 \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
1338 lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
1339 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
1340 t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
1341 (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
1342 (THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1343 t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
1344 \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
1345 (lift (S O) O x1) x4) H30) in ((let H32 \def (f_equal T T (\lambda (e:
1346 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
1347 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
1348 Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H30) in
1349 (\lambda (H33: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H34 \def
1350 (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0)))
1351 H29 x0 H32) in (let H35 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T (THead
1352 (Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 (THead (Bind b)
1353 t1 t0))) \to (\forall (P0: Prop).P0))) H26 x0 H32) in (let H36 \def (eq_ind_r
1354 T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) H22 x0 H32) in (let
1355 H37 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead
1356 (Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to (\forall
1357 (P0: Prop).P0))) H35 x (lift_inj x x1 (S O) O H33)) in (let H38 \def
1358 (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O
1359 H33)) in (H34 (refl_equal T x0) P)))))))) H31)))) (pr3_flat (CHead c (Bind b)
1360 t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S
1361 O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15))
1362 x0 x4 H22 Appl) x1 x4 (refl_equal T (THead (Flat Appl) (lift (S O) O x1)
1363 x4)))) H28))) x3 H25)))) (\lambda (H25: (((eq T t1 x3) \to (\forall (P:
1364 Prop).P)))).(H2 x3 H25 H21 x4 x1 (sn3_cpr3_trans c t1 x3 H21 (Bind b) (THead
1365 (Flat Appl) (lift (S O) O x1) x4) (let H_x1 \def (term_dec x0 x4) in (let H26
1366 \def H_x1 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P))
1367 (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x4)) (\lambda
1368 (H27: (eq T x0 x4)).(let H28 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead
1369 c (Bind b) t1) x0 t0)) H22 x0 H27) in (eq_ind T x0 (\lambda (t0: T).(sn3
1370 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) t0))) (let H_x2
1371 \def (term_dec x x1) in (let H29 \def H_x2 in (or_ind (eq T x x1) ((eq T x
1372 x1) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl)
1373 (lift (S O) O x1) x0)) (\lambda (H30: (eq T x x1)).(let H31 \def (eq_ind_r T
1374 x1 (\lambda (t0: T).(pr2 c x t0)) H15 x H30) in (eq_ind T x (\lambda (t0:
1375 T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0)))
1376 (sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9)
1377 x1 H30))) (\lambda (H30: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9
1378 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H31: (eq T (THead (Flat
1379 Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
1380 x0))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e
1381 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
1382 (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
1383 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
1384 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
1385 \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
1386 lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
1387 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
1388 t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
1389 (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
1390 (THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1391 t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
1392 \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
1393 (lift (S O) O x1) x0) H31) in (let H33 \def (eq_ind_r T x1 (\lambda (t0:
1394 T).((eq T x t0) \to (\forall (P0: Prop).P0))) H30 x (lift_inj x x1 (S O) O
1395 H32)) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
1396 (lift_inj x x1 (S O) O H32)) in (H33 (refl_equal T x) P)))))) (pr3_flat
1397 (CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
1398 (Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
1399 (pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl)))
1400 H29))) x4 H27))) (\lambda (H27: (((eq T x0 x4) \to (\forall (P:
1401 Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x1) x4) (\lambda (H28: (eq T
1402 (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
1403 x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e
1404 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
1405 (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
1406 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
1407 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
1408 \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
1409 lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
1410 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
1411 t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
1412 (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
1413 (THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1414 t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
1415 \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
1416 (lift (S O) O x1) x4) H28) in ((let H30 \def (f_equal T T (\lambda (e:
1417 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
1418 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
1419 Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H28) in
1420 (\lambda (H31: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H32 \def
1421 (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0)))
1422 H27 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c
1423 (Bind b) t1) x0 t0)) H22 x0 H30) in (let H34 \def (eq_ind_r T x1 (\lambda
1424 (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H31)) in (H32 (refl_equal
1425 T x0) P)))))) H29)))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift
1426 (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c
1427 c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x4 H22 Appl))) H26))))))
1428 H24))) x2 H20))))))) H19)) (\lambda (H19: (pr3 (CHead c (Bind b) t1) x0 (lift
1429 (S O) O x2))).(sn3_gen_lift (CHead c (Bind b) t1) (THead (Flat Appl) x1 x2)
1430 (S O) O (eq_ind_r T (THead (Flat Appl) (lift (S O) O x1) (lift (S O) (s (Flat
1431 Appl) O) x2)) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) t0)) (sn3_pr3_trans
1432 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x0) (let H_x0 \def
1433 (term_dec x x1) in (let H20 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to
1434 (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S
1435 O) O x1) x0)) (\lambda (H21: (eq T x x1)).(let H22 \def (eq_ind_r T x1
1436 (\lambda (t0: T).(pr2 c x t0)) H15 x H21) in (eq_ind T x (\lambda (t0:
1437 T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0)))
1438 (sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9)
1439 x1 H21))) (\lambda (H21: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9
1440 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H22: (eq T (THead (Flat
1441 Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
1442 x0))).(\lambda (P: Prop).(let H23 \def (f_equal T T (\lambda (e: T).(match e
1443 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
1444 (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
1445 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
1446 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
1447 \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
1448 lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (TLRef _) \Rightarrow
1449 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
1450 t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
1451 (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
1452 (THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1453 t4))]) in lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (THead _ t0 _)
1454 \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
1455 (lift (S O) O x1) x0) H22) in (let H24 \def (eq_ind_r T x1 (\lambda (t0:
1456 T).((eq T x t0) \to (\forall (P0: Prop).P0))) H21 x (lift_inj x x1 (S O) O
1457 H23)) in (let H25 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
1458 (lift_inj x x1 (S O) O H23)) in (H24 (refl_equal T x) P)))))) (pr3_flat
1459 (CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
1460 (Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
1461 (pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl)))
1462 H20))) (THead (Flat Appl) (lift (S O) O x1) (lift (S O) O x2)) (pr3_thin_dx
1463 (CHead c (Bind b) t1) x0 (lift (S O) O x2) H19 (lift (S O) O x1) Appl)) (lift
1464 (S O) O (THead (Flat Appl) x1 x2)) (lift_head (Flat Appl) x1 x2 (S O) O)) c
1465 (drop_drop (Bind b) O c c (drop_refl c) t1))) H18))) t3 H14))))))) H13))
1466 (\lambda (H13: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1467 T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1
1468 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
1469 T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
1470 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda
1471 (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0:
1472 T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))))).(ex4_4_ind T T T T (\lambda
1473 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
1474 b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
1475 T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4))))))
1476 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x
1477 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4:
1478 T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) z1 t4)))))))
1479 (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
1480 T).(\lambda (H14: (eq T (THead (Bind b) t1 x0) (THead (Bind Abst) x1
1481 x2))).(\lambda (H15: (eq T t3 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c
1482 x x3)).(\lambda (H17: ((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind
1483 b0) u0) x2 x4))))).(let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T (THead
1484 (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) H10
1485 (THead (Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4)
1486 (\lambda (t0: T).(sn3 c t0)) (let H19 \def (f_equal T B (\lambda (e:
1487 T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
1488 (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
1489 (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
1490 b])])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in ((let H20
1491 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
1492 with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _)
1493 \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in
1494 ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
1495 T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
1496 t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14)
1497 in (\lambda (_: (eq T t1 x1)).(\lambda (H23: (eq B b Abst)).(let H24 \def
1498 (eq_ind_r T x2 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead
1499 c (Bind b0) u0) t0 x4)))) H17 x0 H21) in (let H25 \def (eq_ind B b (\lambda
1500 (b0: B).((eq T (THead (Flat Appl) x (THead (Bind b0) t1 x0)) (THead (Bind
1501 Abbr) x3 x4)) \to (\forall (P: Prop).P))) H18 Abst H23) in (let H26 \def
1502 (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T (THead (Flat Appl)
1503 (lift (S O) O x) x0) t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind
1504 b0) t1) (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (sn3 (CHead c (Bind
1505 b0) t1) t4))))) H9 Abst H23) in (let H27 \def (eq_ind B b (\lambda (b0:
1506 B).(\forall (t4: T).((((eq T (THead (Flat Appl) (lift (S O) O x) x0) t4) \to
1507 (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) (THead (Flat Appl)
1508 (lift (S O) O x) x0) t4) \to (\forall (x5: T).(\forall (x6: T).((eq T t4
1509 (THead (Flat Appl) (lift (S O) O x5) x6)) \to (sn3 c (THead (Flat Appl) x5
1510 (THead (Bind b0) t1 x6)))))))))) H8 Abst H23) in (let H28 \def (eq_ind B b
1511 (\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P)))
1512 \to ((pr3 c t1 t4) \to (\forall (t0: T).(\forall (v0: T).((sn3 (CHead c (Bind
1513 b0) t4) (THead (Flat Appl) (lift (S O) O v0) t0)) \to (sn3 c (THead (Flat
1514 Appl) v0 (THead (Bind b0) t4 t0)))))))))) H2 Abst H23) in (let H29 \def
1515 (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H Abst H23) in (let H30
1516 \def (match (H29 (refl_equal B Abst)) in False return (\lambda (_:
1517 False).(sn3 c (THead (Bind Abbr) x3 x4))) with []) in H30)))))))))) H20))
1518 H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 B T T T T T (\lambda (b0:
1519 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1520 (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
1521 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b)
1522 t1 x0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_:
1523 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
1524 t3 (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
1525 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1526 T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
1527 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
1528 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
1529 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1
1530 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda
1531 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0
1532 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
1533 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind
1534 b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
1535 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b0) y2 (THead
1536 (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
1537 T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x
1538 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
1539 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b0:
1540 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
1541 (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2))))))) (sn3 c t3) (\lambda (x1:
1542 B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
1543 T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15: (eq T
1544 (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T t3
1545 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
1546 (H17: (pr2 c x x5)).(\lambda (H18: (pr2 c x2 x6)).(\lambda (H19: (pr2 (CHead
1547 c (Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t0: T).((eq T
1548 (THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P)))
1549 H10 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) H16) in
1550 (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))
1551 (\lambda (t0: T).(sn3 c t0)) (let H21 \def (f_equal T B (\lambda (e:
1552 T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
1553 (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
1554 (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
1555 b])])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in ((let H22 \def
1556 (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
1557 [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _)
1558 \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in
1559 ((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
1560 T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
1561 t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in
1562 (\lambda (H24: (eq T t1 x2)).(\lambda (H25: (eq B b x1)).(let H26 \def
1563 (eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H19 x0
1564 H23) in (let H27 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H18 t1
1565 H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b0: B).(pr2 (CHead c (Bind b0)
1566 x6) x0 x4)) H26 b H25) in (eq_ind B b (\lambda (b0: B).(sn3 c (THead (Bind
1567 b0) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (sn3_pr3_trans c (THead
1568 (Bind b) t1 (THead (Flat Appl) (lift (S O) O x5) x4)) (sn3_bind b c t1
1569 (sn3_sing c t1 H1) (THead (Flat Appl) (lift (S O) O x5) x4) (let H_x \def
1570 (term_dec x x5) in (let H29 \def H_x in (or_ind (eq T x x5) ((eq T x x5) \to
1571 (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S
1572 O) O x5) x4)) (\lambda (H30: (eq T x x5)).(let H31 \def (eq_ind_r T x5
1573 (\lambda (t0: T).(pr2 c x t0)) H17 x H30) in (eq_ind T x (\lambda (t0:
1574 T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x4))) (let
1575 H_x0 \def (term_dec x0 x4) in (let H32 \def H_x0 in (or_ind (eq T x0 x4) ((eq
1576 T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat
1577 Appl) (lift (S O) O x) x4)) (\lambda (H33: (eq T x0 x4)).(let H34 \def
1578 (eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0
1579 H33) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat
1580 Appl) (lift (S O) O x) t0))) (sn3_sing (CHead c (Bind b) t1) (THead (Flat
1581 Appl) (lift (S O) O x) x0) H9) x4 H33))) (\lambda (H33: (((eq T x0 x4) \to
1582 (\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x) x4) (\lambda
1583 (H34: (eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
1584 (S O) O x) x4))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e:
1585 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
1586 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
1587 Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x) x4) H34) in
1588 (let H36 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0:
1589 Prop).P0))) H33 x0 H35) in (let H37 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2
1590 (CHead c (Bind b) x6) x0 t0)) H28 x0 H35) in (H36 (refl_equal T x0) P))))))
1591 (pr3_pr3_pr3_t c t1 x6 (pr3_pr2 c t1 x6 H27) (THead (Flat Appl) (lift (S O) O
1592 x) x0) (THead (Flat Appl) (lift (S O) O x) x4) (Bind b) (pr3_pr2 (CHead c
1593 (Bind b) x6) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
1594 (S O) O x) x4) (pr2_thin_dx (CHead c (Bind b) x6) x0 x4 H28 (lift (S O) O x)
1595 Appl))))) H32))) x5 H30))) (\lambda (H30: (((eq T x x5) \to (\forall (P:
1596 Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x5) x4) (\lambda (H31: (eq T
1597 (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5)
1598 x4))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e
1599 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
1600 (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
1601 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
1602 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
1603 \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
1604 lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (TLRef _) \Rightarrow
1605 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
1606 t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
1607 (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
1608 (THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
1609 t4))]) in lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (THead _ t0 _)
1610 \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
1611 (lift (S O) O x5) x4) H31) in ((let H33 \def (f_equal T T (\lambda (e:
1612 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
1613 (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
1614 Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) x4) H31) in
1615 (\lambda (H34: (eq T (lift (S O) O x) (lift (S O) O x5))).(let H35 \def
1616 (eq_ind_r T x5 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
1617 H30 x (lift_inj x x5 (S O) O H34)) in (let H36 \def (eq_ind_r T x5 (\lambda
1618 (t0: T).(pr2 c x t0)) H17 x (lift_inj x x5 (S O) O H34)) in (let H37 \def
1619 (eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0
1620 H33) in (H35 (refl_equal T x) P)))))) H32)))) (pr3_pr3_pr3_t c t1 x6 (pr3_pr2
1621 c t1 x6 H27) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
1622 (S O) O x5) x4) (Bind b) (pr3_flat (CHead c (Bind b) x6) (lift (S O) O x)
1623 (lift (S O) O x5) (pr3_lift (CHead c (Bind b) x6) c (S O) O (drop_drop (Bind
1624 b) O c c (drop_refl c) x6) x x5 (pr3_pr2 c x x5 H17)) x0 x4 (pr3_pr2 (CHead c
1625 (Bind b) x6) x0 x4 H28) Appl)))) H29)))) (THead (Bind b) x6 (THead (Flat
1626 Appl) (lift (S O) O x5) x4)) (pr3_pr2 c (THead (Bind b) t1 (THead (Flat Appl)
1627 (lift (S O) O x5) x4)) (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O
1628 x5) x4)) (pr2_head_1 c t1 x6 H27 (Bind b) (THead (Flat Appl) (lift (S O) O
1629 x5) x4)))) x1 H25))))))) H22)) H21)) t3 H16)))))))))))))) H13))
1630 H12)))))))))))))) y H4))))) H3))))))) u H0))))).
1632 theorem sn3_appl_appl:
1633 \forall (v1: T).(\forall (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in
1634 (\forall (c: C).((sn3 c u1) \to (\forall (v2: T).((sn3 c v2) \to (((\forall
1635 (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to
1636 (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2
1639 \lambda (v1: T).(\lambda (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in
1640 (\lambda (c: C).(\lambda (H: (sn3 c (THead (Flat Appl) v1 t1))).(insert_eq T
1641 (THead (Flat Appl) v1 t1) (\lambda (t: T).(sn3 c t)) (\lambda (t: T).(\forall
1642 (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2)
1643 \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to
1644 (sn3 c (THead (Flat Appl) v2 t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c
1645 y)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Flat Appl) v1 t)) \to
1646 (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso
1647 y u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2))))))
1648 \to (sn3 c (THead (Flat Appl) v2 y))))))) (unintro T v1 (\lambda (t:
1649 T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to (\forall (v2:
1650 T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso y u2) \to
1651 (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c
1652 (THead (Flat Appl) v2 y)))))))) (sn3_ind c (\lambda (t: T).(\forall (x:
1653 T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (v2:
1654 T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to
1655 (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c
1656 (THead (Flat Appl) v2 t))))))))) (\lambda (t2: T).(\lambda (H1: ((\forall
1657 (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to
1658 (sn3 c t3)))))).(\lambda (H2: ((\forall (t3: T).((((eq T t2 t3) \to (\forall
1659 (P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x: T).(\forall (x0: T).((eq T
1660 t3 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall
1661 (u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to
1662 (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2
1663 t3))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t2
1664 (THead (Flat Appl) x x0))).(\lambda (v2: T).(\lambda (H4: (sn3 c
1665 v2)).(sn3_ind c (\lambda (t: T).(((\forall (u2: T).((pr3 c t2 u2) \to ((((iso
1666 t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t u2))))))
1667 \to (sn3 c (THead (Flat Appl) t t2)))) (\lambda (t0: T).(\lambda (H5:
1668 ((\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0
1669 t3) \to (sn3 c t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t0 t3) \to
1670 (\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to (((\forall (u2: T).((pr3 c t2
1671 u2) \to ((((iso t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat
1672 Appl) t3 u2)))))) \to (sn3 c (THead (Flat Appl) t3 t2)))))))).(\lambda (H7:
1673 ((\forall (u2: T).((pr3 c t2 u2) \to ((((iso t2 u2) \to (\forall (P:
1674 Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2))))))).(let H8 \def (eq_ind T
1675 t2 (\lambda (t: T).(\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to
1676 (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H7 (THead
1677 (Flat Appl) x x0) H3) in (let H9 \def (eq_ind T t2 (\lambda (t: T).(\forall
1678 (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to
1679 (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to (\forall (P:
1680 Prop).P))) \to (sn3 c (THead (Flat Appl) t3 u2)))))) \to (sn3 c (THead (Flat
1681 Appl) t3 t))))))) H6 (THead (Flat Appl) x x0) H3) in (let H10 \def (eq_ind T
1682 t2 (\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P:
1683 Prop).P))) \to ((pr3 c t t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3
1684 (THead (Flat Appl) x1 x2)) \to (\forall (v3: T).((sn3 c v3) \to (((\forall
1685 (u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to
1686 (sn3 c (THead (Flat Appl) v3 u2)))))) \to (sn3 c (THead (Flat Appl) v3
1687 t3)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H11 \def (eq_ind T t2
1688 (\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P: Prop).P)))
1689 \to ((pr3 c t t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in
1690 (eq_ind_r T (THead (Flat Appl) x x0) (\lambda (t: T).(sn3 c (THead (Flat
1691 Appl) t0 t))) (sn3_pr2_intro c (THead (Flat Appl) t0 (THead (Flat Appl) x
1692 x0)) (\lambda (t3: T).(\lambda (H12: (((eq T (THead (Flat Appl) t0 (THead
1693 (Flat Appl) x x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H13: (pr2 c
1694 (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t3)).(let H14 \def
1695 (pr2_gen_appl c t0 (THead (Flat Appl) x x0) t3 H13) in (or3_ind (ex3_2 T T
1696 (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
1697 (\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda
1698 (t4: T).(pr2 c (THead (Flat Appl) x x0) t4)))) (ex4_4 T T T T (\lambda (y1:
1699 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl)
1700 x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
1701 (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
1702 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2)))))
1703 (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
1704 (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T
1705 T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1706 (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
1707 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
1708 T (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
1709 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
1710 (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
1711 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1712 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_:
1713 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1714 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
1715 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
1716 y2) z1 z2)))))))) (sn3 c t3) (\lambda (H15: (ex3_2 T T (\lambda (u2:
1717 T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
1718 T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
1719 (THead (Flat Appl) x x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
1720 (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_:
1721 T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Flat Appl)
1722 x x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H16: (eq T
1723 t3 (THead (Flat Appl) x1 x2))).(\lambda (H17: (pr2 c t0 x1)).(\lambda (H18:
1724 (pr2 c (THead (Flat Appl) x x0) x2)).(let H19 \def (eq_ind T t3 (\lambda (t:
1725 T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P:
1726 Prop).P))) H12 (THead (Flat Appl) x1 x2) H16) in (eq_ind_r T (THead (Flat
1727 Appl) x1 x2) (\lambda (t: T).(sn3 c t)) (let H20 \def (pr2_gen_appl c x x0 x2
1728 H18) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead
1729 (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
1730 (\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4)))) (ex4_4 T T T T (\lambda
1731 (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead
1732 (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1733 T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
1734 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
1735 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
1736 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T
1737 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1738 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1739 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
1740 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1741 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind
1742 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
1743 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1744 (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
1745 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
1746 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
1747 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c
1748 (THead (Flat Appl) x1 x2)) (\lambda (H21: (ex3_2 T T (\lambda (u2:
1749 T).(\lambda (t4: T).(eq T x2 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
1750 T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c x0
1751 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead
1752 (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
1753 (\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4))) (sn3 c (THead (Flat Appl) x1
1754 x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H22: (eq T x2 (THead (Flat
1755 Appl) x3 x4))).(\lambda (H23: (pr2 c x x3)).(\lambda (H24: (pr2 c x0
1756 x4)).(let H25 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0
1757 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P:
1758 Prop).P))) H19 (THead (Flat Appl) x3 x4) H22) in (eq_ind_r T (THead (Flat
1759 Appl) x3 x4) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H_x \def
1760 (term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) in (let H26
1761 \def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))
1762 ((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) \to (\forall (P:
1763 Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 x4))) (\lambda
1764 (H27: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))).(let H28
1765 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
1766 with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _)
1767 \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27) in
1768 ((let H29 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
1769 T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
1770 t) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27)
1771 in (\lambda (H30: (eq T x x3)).(let H31 \def (eq_ind_r T x4 (\lambda (t:
1772 T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl)
1773 x1 (THead (Flat Appl) x3 t))) \to (\forall (P: Prop).P))) H25 x0 H29) in (let
1774 H32 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) H24 x0 H29) in (eq_ind
1775 T x0 (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 t))))
1776 (let H33 \def (eq_ind_r T x3 (\lambda (t: T).((eq T (THead (Flat Appl) t0
1777 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 (THead (Flat Appl) t x0)))
1778 \to (\forall (P: Prop).P))) H31 x H30) in (let H34 \def (eq_ind_r T x3
1779 (\lambda (t: T).(pr2 c x t)) H23 x H30) in (eq_ind T x (\lambda (t: T).(sn3 c
1780 (THead (Flat Appl) x1 (THead (Flat Appl) t x0)))) (let H_x0 \def (term_dec t0
1781 x1) in (let H35 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to (\forall
1782 (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x x0)))
1783 (\lambda (H36: (eq T t0 x1)).(let H37 \def (eq_ind_r T x1 (\lambda (t:
1784 T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl)
1785 t (THead (Flat Appl) x x0))) \to (\forall (P: Prop).P))) H33 t0 H36) in (let
1786 H38 \def (eq_ind_r T x1 (\lambda (t: T).(pr2 c t0 t)) H17 t0 H36) in (eq_ind
1787 T t0 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (THead (Flat Appl) x x0))))
1788 (H37 (refl_equal T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))) (sn3 c
1789 (THead (Flat Appl) t0 (THead (Flat Appl) x x0)))) x1 H36)))) (\lambda (H36:
1790 (((eq T t0 x1) \to (\forall (P: Prop).P)))).(H9 x1 H36 (pr3_pr2 c t0 x1 H17)
1791 (\lambda (u2: T).(\lambda (H37: (pr3 c (THead (Flat Appl) x x0) u2)).(\lambda
1792 (H38: (((iso (THead (Flat Appl) x x0) u2) \to (\forall (P:
1793 Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H8 u2 H37 H38) (THead
1794 (Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2) (THead (Flat Appl) x1
1795 u2) (pr2_head_1 c t0 x1 H17 (Flat Appl) u2)))))))) H35))) x3 H30))) x4
1796 H29))))) H28))) (\lambda (H27: (((eq T (THead (Flat Appl) x x0) (THead (Flat
1797 Appl) x3 x4)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat Appl) x3 x4) H27
1798 (pr3_flat c x x3 (pr3_pr2 c x x3 H23) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x3 x4
1799 (refl_equal T (THead (Flat Appl) x3 x4)) x1 (sn3_pr3_trans c t0 (sn3_sing c
1800 t0 H5) x1 (pr3_pr2 c t0 x1 H17)) (\lambda (u2: T).(\lambda (H28: (pr3 c
1801 (THead (Flat Appl) x3 x4) u2)).(\lambda (H29: (((iso (THead (Flat Appl) x3
1802 x4) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0
1803 u2) (H8 u2 (pr3_sing c (THead (Flat Appl) x x4) (THead (Flat Appl) x x0)
1804 (pr2_thin_dx c x0 x4 H24 x Appl) u2 (pr3_sing c (THead (Flat Appl) x3 x4)
1805 (THead (Flat Appl) x x4) (pr2_head_1 c x x3 H23 (Flat Appl) x4) u2 H28))
1806 (\lambda (H30: (iso (THead (Flat Appl) x x0) u2)).(\lambda (P: Prop).(H29
1807 (iso_trans (THead (Flat Appl) x3 x4) (THead (Flat Appl) x x0) (iso_head x3 x
1808 x4 x0 (Flat Appl)) u2 H30) P)))) (THead (Flat Appl) x1 u2) (pr3_pr2 c (THead
1809 (Flat Appl) t0 u2) (THead (Flat Appl) x1 u2) (pr2_head_1 c t0 x1 H17 (Flat
1810 Appl) u2)))))))) H26))) x2 H22))))))) H21)) (\lambda (H21: (ex4_4 T T T T
1811 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
1812 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1813 T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
1814 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
1815 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
1816 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T
1817 T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
1818 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1819 T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
1820 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
1821 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
1822 B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c (THead (Flat
1823 Appl) x1 x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda
1824 (x6: T).(\lambda (H22: (eq T x0 (THead (Bind Abst) x3 x4))).(\lambda (H23:
1825 (eq T x2 (THead (Bind Abbr) x5 x6))).(\lambda (H24: (pr2 c x x5)).(\lambda
1826 (H25: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x4
1827 x6))))).(let H26 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl)
1828 t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P:
1829 Prop).P))) H19 (THead (Bind Abbr) x5 x6) H23) in (eq_ind_r T (THead (Bind
1830 Abbr) x5 x6) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H27 \def
1831 (eq_ind T x0 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl)
1832 x t)) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6))) \to (\forall (P:
1833 Prop).P))) H26 (THead (Bind Abst) x3 x4) H22) in (let H28 \def (eq_ind T x0
1834 (\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
1835 (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c
1836 t4))))) H11 (THead (Bind Abst) x3 x4) H22) in (let H29 \def (eq_ind T x0
1837 (\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
1838 (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall
1839 (x7: T).(\forall (x8: T).((eq T t4 (THead (Flat Appl) x7 x8)) \to (\forall
1840 (v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2)
1841 \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to
1842 (sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind Abst) x3 x4)
1843 H22) in (let H30 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c
1844 (THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to
1845 (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead
1846 (Bind Abst) x3 x4) H22) in (let H31 \def (eq_ind T x0 (\lambda (t:
1847 T).(\forall (t4: T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c
1848 t0 t4) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso
1849 (THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead
1850 (Flat Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x
1851 t)))))))) H9 (THead (Bind Abst) x3 x4) H22) in (sn3_pr3_trans c (THead (Flat
1852 Appl) t0 (THead (Bind Abbr) x5 x6)) (H30 (THead (Bind Abbr) x5 x6) (pr3_sing
1853 c (THead (Bind Abbr) x x4) (THead (Flat Appl) x (THead (Bind Abst) x3 x4))
1854 (pr2_free c (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind
1855 Abbr) x x4) (pr0_beta x3 x x (pr0_refl x) x4 x4 (pr0_refl x4))) (THead (Bind
1856 Abbr) x5 x6) (pr3_head_12 c x x5 (pr3_pr2 c x x5 H24) (Bind Abbr) x4 x6
1857 (pr3_pr2 (CHead c (Bind Abbr) x5) x4 x6 (H25 Abbr x5)))) (\lambda (H32: (iso
1858 (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind Abbr) x5
1859 x6))).(\lambda (P: Prop).(let H33 \def (match H32 in iso return (\lambda (t:
1860 T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x
1861 (THead (Bind Abst) x3 x4))) \to ((eq T t4 (THead (Bind Abbr) x5 x6)) \to
1862 P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H33: (eq T (TSort n1)
1863 (THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H34: (eq T (TSort
1864 n2) (THead (Bind Abbr) x5 x6))).((let H35 \def (eq_ind T (TSort n1) (\lambda
1865 (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
1866 True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
1867 (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T
1868 (TSort n2) (THead (Bind Abbr) x5 x6)) \to P) H35)) H34))) | (iso_lref i1 i2)
1869 \Rightarrow (\lambda (H33: (eq T (TLRef i1) (THead (Flat Appl) x (THead (Bind
1870 Abst) x3 x4)))).(\lambda (H34: (eq T (TLRef i2) (THead (Bind Abbr) x5
1871 x6))).((let H35 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T
1872 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
1873 \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x
1874 (THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T (TLRef i2) (THead (Bind
1875 Abbr) x5 x6)) \to P) H35)) H34))) | (iso_head v4 v5 t4 t5 k) \Rightarrow
1876 (\lambda (H33: (eq T (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst)
1877 x3 x4)))).(\lambda (H34: (eq T (THead k v5 t5) (THead (Bind Abbr) x5
1878 x6))).((let H35 \def (f_equal T T (\lambda (e: T).(match e in T return
1879 (\lambda (_: T).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4
1880 | (THead _ _ t) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead
1881 (Bind Abst) x3 x4)) H33) in ((let H36 \def (f_equal T T (\lambda (e:
1882 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v4 |
1883 (TLRef _) \Rightarrow v4 | (THead _ t _) \Rightarrow t])) (THead k v4 t4)
1884 (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in ((let H37 \def
1885 (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
1886 [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
1887 \Rightarrow k0])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) x3
1888 x4)) H33) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x) \to ((eq T
1889 t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead k0 v5 t5) (THead (Bind Abbr)
1890 x5 x6)) \to P)))) (\lambda (H38: (eq T v4 x)).(eq_ind T x (\lambda (_:
1891 T).((eq T t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead (Flat Appl) v5 t5)
1892 (THead (Bind Abbr) x5 x6)) \to P))) (\lambda (H39: (eq T t4 (THead (Bind
1893 Abst) x3 x4))).(eq_ind T (THead (Bind Abst) x3 x4) (\lambda (_: T).((eq T
1894 (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6)) \to P)) (\lambda (H40:
1895 (eq T (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6))).(let H41 \def
1896 (eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return
1897 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
1898 \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
1899 (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
1900 True])])) I (THead (Bind Abbr) x5 x6) H40) in (False_ind P H41))) t4 (sym_eq
1901 T t4 (THead (Bind Abst) x3 x4) H39))) v4 (sym_eq T v4 x H38))) k (sym_eq K k
1902 (Flat Appl) H37))) H36)) H35)) H34)))]) in (H33 (refl_equal T (THead (Flat
1903 Appl) x (THead (Bind Abst) x3 x4))) (refl_equal T (THead (Bind Abbr) x5
1904 x6))))))) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6)) (pr3_pr2 c (THead
1905 (Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (THead (Flat Appl) x1 (THead (Bind
1906 Abbr) x5 x6)) (pr2_head_1 c t0 x1 H17 (Flat Appl) (THead (Bind Abbr) x5
1907 x6))))))))) x2 H23)))))))))) H21)) (\lambda (H21: (ex6_6 B T T T T T (\lambda
1908 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1909 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
1910 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
1911 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
1912 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind
1913 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
1914 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
1915 (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
1916 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
1917 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
1918 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind
1919 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
1920 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
1921 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
1922 (_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
1923 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
1924 x2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
1925 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
1926 T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1:
1927 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
1928 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
1929 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
1930 (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3: B).(\lambda (x4: T).(\lambda
1931 (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H22:
1932 (not (eq B x3 Abst))).(\lambda (H23: (eq T x0 (THead (Bind x3) x4
1933 x5))).(\lambda (H24: (eq T x2 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S
1934 O) O x7) x6)))).(\lambda (H25: (pr2 c x x7)).(\lambda (H26: (pr2 c x4
1935 x8)).(\lambda (H27: (pr2 (CHead c (Bind x3) x8) x5 x6)).(let H28 \def (eq_ind
1936 T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))
1937 (THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H19 (THead (Bind x3) x8
1938 (THead (Flat Appl) (lift (S O) O x7) x6)) H24) in (eq_ind_r T (THead (Bind
1939 x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (\lambda (t: T).(sn3 c
1940 (THead (Flat Appl) x1 t))) (let H29 \def (eq_ind T x0 (\lambda (t: T).((eq T
1941 (THead (Flat Appl) t0 (THead (Flat Appl) x t)) (THead (Flat Appl) x1 (THead
1942 (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))) \to (\forall (P:
1943 Prop).P))) H28 (THead (Bind x3) x4 x5) H23) in (let H30 \def (eq_ind T x0
1944 (\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
1945 (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c
1946 t4))))) H11 (THead (Bind x3) x4 x5) H23) in (let H31 \def (eq_ind T x0
1947 (\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to
1948 (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall
1949 (x9: T).(\forall (x10: T).((eq T t4 (THead (Flat Appl) x9 x10)) \to (\forall
1950 (v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2)
1951 \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to
1952 (sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind x3) x4 x5) H23)
1953 in (let H32 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c (THead
1954 (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P:
1955 Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead (Bind x3) x4
1956 x5) H23) in (let H33 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4:
1957 T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t4) \to
1958 (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead
1959 (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat
1960 Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x
1961 t)))))))) H9 (THead (Bind x3) x4 x5) H23) in (sn3_pr3_trans c (THead (Flat
1962 Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (H32
1963 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (pr3_sing c
1964 (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) x5)) (THead (Flat
1965 Appl) x (THead (Bind x3) x4 x5)) (pr2_free c (THead (Flat Appl) x (THead
1966 (Bind x3) x4 x5)) (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x)
1967 x5)) (pr0_upsilon x3 H22 x x (pr0_refl x) x4 x4 (pr0_refl x4) x5 x5 (pr0_refl
1968 x5))) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))
1969 (pr3_head_12 c x4 x8 (pr3_pr2 c x4 x8 H26) (Bind x3) (THead (Flat Appl) (lift
1970 (S O) O x) x5) (THead (Flat Appl) (lift (S O) O x7) x6) (pr3_head_12 (CHead c
1971 (Bind x3) x8) (lift (S O) O x) (lift (S O) O x7) (pr3_lift (CHead c (Bind x3)
1972 x8) c (S O) O (drop_drop (Bind x3) O c c (drop_refl c) x8) x x7 (pr3_pr2 c x
1973 x7 H25)) (Flat Appl) x5 x6 (pr3_pr2 (CHead (CHead c (Bind x3) x8) (Flat Appl)
1974 (lift (S O) O x7)) x5 x6 (pr2_cflat (CHead c (Bind x3) x8) x5 x6 H27 Appl
1975 (lift (S O) O x7)))))) (\lambda (H34: (iso (THead (Flat Appl) x (THead (Bind
1976 x3) x4 x5)) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
1977 x6)))).(\lambda (P: Prop).(let H35 \def (match H34 in iso return (\lambda (t:
1978 T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x
1979 (THead (Bind x3) x4 x5))) \to ((eq T t4 (THead (Bind x3) x8 (THead (Flat
1980 Appl) (lift (S O) O x7) x6))) \to P))))) with [(iso_sort n1 n2) \Rightarrow
1981 (\lambda (H35: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind x3) x4
1982 x5)))).(\lambda (H36: (eq T (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl)
1983 (lift (S O) O x7) x6)))).((let H37 \def (eq_ind T (TSort n1) (\lambda (e:
1984 T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
1985 True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
1986 (THead (Flat Appl) x (THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T
1987 (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to
1988 P) H37)) H36))) | (iso_lref i1 i2) \Rightarrow (\lambda (H35: (eq T (TLRef
1989 i1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T
1990 (TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
1991 x6)))).((let H37 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T
1992 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
1993 \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x
1994 (THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T (TLRef i2) (THead (Bind
1995 x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H37)) H36))) |
1996 (iso_head v4 v5 t4 t5 k) \Rightarrow (\lambda (H35: (eq T (THead k v4 t4)
1997 (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T (THead k
1998 v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let
1999 H37 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
2000 with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t)
2001 \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4
2002 x5)) H35) in ((let H38 \def (f_equal T T (\lambda (e: T).(match e in T return
2003 (\lambda (_: T).T) with [(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4
2004 | (THead _ t _) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead
2005 (Bind x3) x4 x5)) H35) in ((let H39 \def (f_equal T K (\lambda (e: T).(match
2006 e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
2007 \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k v4 t4) (THead (Flat
2008 Appl) x (THead (Bind x3) x4 x5)) H35) in (eq_ind K (Flat Appl) (\lambda (k0:
2009 K).((eq T v4 x) \to ((eq T t4 (THead (Bind x3) x4 x5)) \to ((eq T (THead k0
2010 v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to
2011 P)))) (\lambda (H40: (eq T v4 x)).(eq_ind T x (\lambda (_: T).((eq T t4
2012 (THead (Bind x3) x4 x5)) \to ((eq T (THead (Flat Appl) v5 t5) (THead (Bind
2013 x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P))) (\lambda (H41: (eq
2014 T t4 (THead (Bind x3) x4 x5))).(eq_ind T (THead (Bind x3) x4 x5) (\lambda (_:
2015 T).((eq T (THead (Flat Appl) v5 t5) (THead (Bind x3) x8 (THead (Flat Appl)
2016 (lift (S O) O x7) x6))) \to P)) (\lambda (H42: (eq T (THead (Flat Appl) v5
2017 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(let H43
2018 \def (eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return
2019 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
2020 \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
2021 (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
2022 True])])) I (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))
2023 H42) in (False_ind P H43))) t4 (sym_eq T t4 (THead (Bind x3) x4 x5) H41))) v4
2024 (sym_eq T v4 x H40))) k (sym_eq K k (Flat Appl) H39))) H38)) H37)) H36)))])
2025 in (H35 (refl_equal T (THead (Flat Appl) x (THead (Bind x3) x4 x5)))
2026 (refl_equal T (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
2027 x6)))))))) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift
2028 (S O) O x7) x6))) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x3) x8 (THead
2029 (Flat Appl) (lift (S O) O x7) x6))) (THead (Flat Appl) x1 (THead (Bind x3) x8
2030 (THead (Flat Appl) (lift (S O) O x7) x6))) (pr2_head_1 c t0 x1 H17 (Flat
2031 Appl) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))))))))))
2032 x2 H24)))))))))))))) H21)) H20)) t3 H16))))))) H15)) (\lambda (H15: (ex4_4 T
2033 T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
2034 (THead (Flat Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
2035 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
2036 Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2037 (_: T).(pr2 c t0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
2038 T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
2039 z1 t4))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
2040 (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind Abst) y1
2041 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
2042 T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
2043 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) (\lambda (_:
2044 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
2045 (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c t3) (\lambda (x1:
2046 T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H16: (eq T
2047 (THead (Flat Appl) x x0) (THead (Bind Abst) x1 x2))).(\lambda (H17: (eq T t3
2048 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c t0 x3)).(\lambda (_:
2049 ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let
2050 H20 \def (eq_ind T t3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead
2051 (Flat Appl) x x0)) t) \to (\forall (P: Prop).P))) H12 (THead (Bind Abbr) x3
2052 x4) H17) in (eq_ind_r T (THead (Bind Abbr) x3 x4) (\lambda (t: T).(sn3 c t))
2053 (let H21 \def (eq_ind T (THead (Flat Appl) x x0) (\lambda (ee: T).(match ee
2054 in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
2055 _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
2056 (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
2057 True])])) I (THead (Bind Abst) x1 x2) H16) in (False_ind (sn3 c (THead (Bind
2058 Abbr) x3 x4)) H21)) t3 H17)))))))))) H15)) (\lambda (H15: (ex6_6 B T T T T T
2059 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2060 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
2061 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
2062 (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
2063 B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
2064 (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
2065 z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
2066 T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_:
2067 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
2068 (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
2069 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
2070 y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_:
2071 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
2072 b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
2073 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead
2074 (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
2075 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
2076 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
2077 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
2078 (_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
2079 T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
2080 (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
2081 (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3)
2082 (\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda
2083 (x5: T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H17:
2084 (eq T (THead (Flat Appl) x x0) (THead (Bind x1) x2 x3))).(\lambda (H18: (eq T
2085 t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
2086 (_: (pr2 c t0 x5)).(\lambda (_: (pr2 c x2 x6)).(\lambda (_: (pr2 (CHead c
2087 (Bind x1) x6) x3 x4)).(let H22 \def (eq_ind T t3 (\lambda (t: T).((eq T
2088 (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P:
2089 Prop).P))) H12 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))
2090 H18) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5)
2091 x4)) (\lambda (t: T).(sn3 c t)) (let H23 \def (eq_ind T (THead (Flat Appl) x
2092 x0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
2093 _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
2094 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
2095 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x1) x2 x3)
2096 H17) in (False_ind (sn3 c (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O)
2097 O x5) x4))) H23)) t3 H18)))))))))))))) H15)) H14)))))) t2 H3))))))))) v2
2098 H4))))))))) y H0))))) H))))).
2100 theorem sn3_appl_beta:
2101 \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((sn3 c
2102 (THead (Flat Appl) u (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w)
2103 \to (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind Abst) w
2106 \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
2107 (sn3 c (THead (Flat Appl) u (THead (Bind Abbr) v t)))).(\lambda (w:
2108 T).(\lambda (H0: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THead (Bind
2109 Abbr) v t) H) in (let H1 \def H_x in (and_ind (sn3 c u) (sn3 c (THead (Bind
2110 Abbr) v t)) (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind
2111 Abst) w t)))) (\lambda (H2: (sn3 c u)).(\lambda (H3: (sn3 c (THead (Bind
2112 Abbr) v t))).(sn3_appl_appl v (THead (Bind Abst) w t) c (sn3_beta c v t H3 w
2113 H0) u H2 (\lambda (u2: T).(\lambda (H4: (pr3 c (THead (Flat Appl) v (THead
2114 (Bind Abst) w t)) u2)).(\lambda (H5: (((iso (THead (Flat Appl) v (THead (Bind
2115 Abst) w t)) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat
2116 Appl) u (THead (Bind Abbr) v t)) H (THead (Flat Appl) u u2) (pr3_thin_dx c
2117 (THead (Bind Abbr) v t) u2 (pr3_iso_beta v w t c u2 H4 H5) u Appl))))))))
2120 theorem sn3_appl_appls:
2121 \forall (v1: T).(\forall (t1: T).(\forall (vs: TList).(let u1 \def (THeads
2122 (Flat Appl) (TCons v1 vs) t1) in (\forall (c: C).((sn3 c u1) \to (\forall
2123 (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2)
2124 \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to
2125 (sn3 c (THead (Flat Appl) v2 u1))))))))))
2127 \lambda (v1: T).(\lambda (t1: T).(\lambda (vs: TList).(let u1 \def (THeads
2128 (Flat Appl) (TCons v1 vs) t1) in (\lambda (c: C).(\lambda (H: (sn3 c (THead
2129 (Flat Appl) v1 (THeads (Flat Appl) vs t1)))).(\lambda (v2: T).(\lambda (H0:
2130 (sn3 c v2)).(\lambda (H1: ((\forall (u2: T).((pr3 c (THead (Flat Appl) v1
2131 (THeads (Flat Appl) vs t1)) u2) \to ((((iso (THead (Flat Appl) v1 (THeads
2132 (Flat Appl) vs t1)) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat
2133 Appl) v2 u2))))))).(sn3_appl_appl v1 (THeads (Flat Appl) vs t1) c H v2 H0
2136 theorem sn3_appls_lref:
2137 \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (us:
2138 TList).((sns3 c us) \to (sn3 c (THeads (Flat Appl) us (TLRef i)))))))
2140 \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda
2141 (us: TList).(TList_ind (\lambda (t: TList).((sns3 c t) \to (sn3 c (THeads
2142 (Flat Appl) t (TLRef i))))) (\lambda (_: True).(sn3_nf2 c (TLRef i) H))
2143 (\lambda (t: T).(\lambda (t0: TList).(TList_ind (\lambda (t1: TList).((((sns3
2144 c t1) \to (sn3 c (THeads (Flat Appl) t1 (TLRef i))))) \to ((land (sn3 c t)
2145 (sns3 c t1)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (TLRef
2146 i))))))) (\lambda (_: (((sns3 c TNil) \to (sn3 c (THeads (Flat Appl) TNil
2147 (TLRef i)))))).(\lambda (H1: (land (sn3 c t) (sns3 c TNil))).(let H2 \def H1
2148 in (and_ind (sn3 c t) True (sn3 c (THead (Flat Appl) t (THeads (Flat Appl)
2149 TNil (TLRef i)))) (\lambda (H3: (sn3 c t)).(\lambda (_: True).(sn3_appl_lref
2150 c i H t H3))) H2)))) (\lambda (t1: T).(\lambda (t2: TList).(\lambda (_:
2151 (((((sns3 c t2) \to (sn3 c (THeads (Flat Appl) t2 (TLRef i))))) \to ((land
2152 (sn3 c t) (sns3 c t2)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2
2153 (TLRef i)))))))).(\lambda (H1: (((sns3 c (TCons t1 t2)) \to (sn3 c (THeads
2154 (Flat Appl) (TCons t1 t2) (TLRef i)))))).(\lambda (H2: (land (sn3 c t) (sns3
2155 c (TCons t1 t2)))).(let H3 \def H2 in (and_ind (sn3 c t) (land (sn3 c t1)
2156 (sns3 c t2)) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2)
2157 (TLRef i)))) (\lambda (H4: (sn3 c t)).(\lambda (H5: (land (sn3 c t1) (sns3 c
2158 t2))).(and_ind (sn3 c t1) (sns3 c t2) (sn3 c (THead (Flat Appl) t (THeads
2159 (Flat Appl) (TCons t1 t2) (TLRef i)))) (\lambda (H6: (sn3 c t1)).(\lambda
2160 (H7: (sns3 c t2)).(sn3_appl_appls t1 (TLRef i) t2 c (H1 (conj (sn3 c t1)
2161 (sns3 c t2) H6 H7)) t H4 (\lambda (u2: T).(\lambda (H8: (pr3 c (THeads (Flat
2162 Appl) (TCons t1 t2) (TLRef i)) u2)).(\lambda (H9: (((iso (THeads (Flat Appl)
2163 (TCons t1 t2) (TLRef i)) u2) \to (\forall (P: Prop).P)))).(H9
2164 (nf2_iso_appls_lref c i H (TCons t1 t2) u2 H8) (sn3 c (THead (Flat Appl) t
2165 u2))))))))) H5))) H3))))))) t0))) us)))).
2167 theorem sn3_appls_cast:
2168 \forall (c: C).(\forall (vs: TList).(\forall (u: T).((sn3 c (THeads (Flat
2169 Appl) vs u)) \to (\forall (t: T).((sn3 c (THeads (Flat Appl) vs t)) \to (sn3
2170 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))
2172 \lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: TList).(\forall
2173 (u: T).((sn3 c (THeads (Flat Appl) t u)) \to (\forall (t0: T).((sn3 c (THeads
2174 (Flat Appl) t t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Flat Cast) u
2175 t0)))))))) (\lambda (u: T).(\lambda (H: (sn3 c u)).(\lambda (t: T).(\lambda
2176 (H0: (sn3 c t)).(sn3_cast c u H t H0))))) (\lambda (t: T).(\lambda (t0:
2177 TList).(TList_ind (\lambda (t1: TList).(((\forall (u: T).((sn3 c (THeads
2178 (Flat Appl) t1 u)) \to (\forall (t2: T).((sn3 c (THeads (Flat Appl) t1 t2))
2179 \to (sn3 c (THeads (Flat Appl) t1 (THead (Flat Cast) u t2)))))))) \to
2180 (\forall (u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 u))) \to
2181 (\forall (t2: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 t2)))
2182 \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (THead (Flat Cast) u
2183 t2)))))))))) (\lambda (_: ((\forall (u: T).((sn3 c (THeads (Flat Appl) TNil
2184 u)) \to (\forall (t1: T).((sn3 c (THeads (Flat Appl) TNil t1)) \to (sn3 c
2185 (THeads (Flat Appl) TNil (THead (Flat Cast) u t1))))))))).(\lambda (u:
2186 T).(\lambda (H0: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) TNil
2187 u)))).(\lambda (t1: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads
2188 (Flat Appl) TNil t1)))).(sn3_appl_cast c t u H0 t1 H1)))))) (\lambda (t1:
2189 T).(\lambda (t2: TList).(\lambda (_: ((((\forall (u: T).((sn3 c (THeads (Flat
2190 Appl) t2 u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) t2 t3)) \to
2191 (sn3 c (THeads (Flat Appl) t2 (THead (Flat Cast) u t3)))))))) \to (\forall
2192 (u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 u))) \to (\forall
2193 (t3: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 t3))) \to (sn3 c
2194 (THead (Flat Appl) t (THeads (Flat Appl) t2 (THead (Flat Cast) u
2195 t3))))))))))).(\lambda (H0: ((\forall (u: T).((sn3 c (THeads (Flat Appl)
2196 (TCons t1 t2) u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) (TCons t1
2197 t2) t3)) \to (sn3 c (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u
2198 t3))))))))).(\lambda (u: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads
2199 (Flat Appl) (TCons t1 t2) u)))).(\lambda (t3: T).(\lambda (H2: (sn3 c (THead
2200 (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) t3)))).(let H_x \def
2201 (sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) t3) H2) in (let H3
2202 \def H_x in (and_ind (sn3 c t) (sn3 c (THeads (Flat Appl) (TCons t1 t2) t3))
2203 (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) (THead (Flat
2204 Cast) u t3)))) (\lambda (_: (sn3 c t)).(\lambda (H5: (sn3 c (THeads (Flat
2205 Appl) (TCons t1 t2) t3))).(let H6 \def H5 in (let H_x0 \def (sn3_gen_flat
2206 Appl c t (THeads (Flat Appl) (TCons t1 t2) u) H1) in (let H7 \def H_x0 in
2207 (and_ind (sn3 c t) (sn3 c (THeads (Flat Appl) (TCons t1 t2) u)) (sn3 c (THead
2208 (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u t3))))
2209 (\lambda (H8: (sn3 c t)).(\lambda (H9: (sn3 c (THeads (Flat Appl) (TCons t1
2210 t2) u))).(let H10 \def H9 in (sn3_appl_appls t1 (THead (Flat Cast) u t3) t2 c
2211 (H0 u H10 t3 H6) t H8 (\lambda (u2: T).(\lambda (H11: (pr3 c (THeads (Flat
2212 Appl) (TCons t1 t2) (THead (Flat Cast) u t3)) u2)).(\lambda (H12: (((iso
2213 (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u t3)) u2) \to (\forall
2214 (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t (THeads (Flat Appl)
2215 (TCons t1 t2) t3)) H2 (THead (Flat Appl) t u2) (pr3_thin_dx c (THeads (Flat
2216 Appl) (TCons t1 t2) t3) u2 (pr3_iso_appls_cast c u t3 (TCons t1 t2) u2 H11
2217 H12) t Appl))))))))) H7)))))) H3))))))))))) t0))) vs)).
2219 theorem sn3_appls_bind:
2220 \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u:
2221 T).((sn3 c u) \to (\forall (vs: TList).(\forall (t: T).((sn3 (CHead c (Bind
2222 b) u) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to (sn3 c (THeads (Flat
2223 Appl) vs (THead (Bind b) u t))))))))))
2225 \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda
2226 (u: T).(\lambda (H0: (sn3 c u)).(\lambda (vs: TList).(TList_ind (\lambda (t:
2227 TList).(\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) (lifts
2228 (S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u t0))))))
2229 (\lambda (t: T).(\lambda (H1: (sn3 (CHead c (Bind b) u) t)).(sn3_bind b c u
2230 H0 t H1))) (\lambda (v: T).(\lambda (vs0: TList).(TList_ind (\lambda (t:
2231 TList).(((\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl)
2232 (lifts (S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u
2233 t0)))))) \to (\forall (t0: T).((sn3 (CHead c (Bind b) u) (THead (Flat Appl)
2234 (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t) t0))) \to (sn3 c
2235 (THead (Flat Appl) v (THeads (Flat Appl) t (THead (Bind b) u t0))))))))
2236 (\lambda (_: ((\forall (t: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl)
2237 (lifts (S O) O TNil) t)) \to (sn3 c (THeads (Flat Appl) TNil (THead (Bind b)
2238 u t))))))).(\lambda (t: T).(\lambda (H2: (sn3 (CHead c (Bind b) u) (THead
2239 (Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O TNil)
2240 t)))).(sn3_appl_bind b H c u H0 t v H2)))) (\lambda (t: T).(\lambda (t0:
2241 TList).(\lambda (_: ((((\forall (t1: T).((sn3 (CHead c (Bind b) u) (THeads
2242 (Flat Appl) (lifts (S O) O t0) t1)) \to (sn3 c (THeads (Flat Appl) t0 (THead
2243 (Bind b) u t1)))))) \to (\forall (t1: T).((sn3 (CHead c (Bind b) u) (THead
2244 (Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t0) t1))) \to
2245 (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (THead (Bind b) u
2246 t1))))))))).(\lambda (H2: ((\forall (t1: T).((sn3 (CHead c (Bind b) u)
2247 (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) \to (sn3 c (THeads
2248 (Flat Appl) (TCons t t0) (THead (Bind b) u t1))))))).(\lambda (t1:
2249 T).(\lambda (H3: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O
2250 v) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))).(let H_x \def
2251 (sn3_gen_flat Appl (CHead c (Bind b) u) (lift (S O) O v) (THeads (Flat Appl)
2252 (lifts (S O) O (TCons t t0)) t1) H3) in (let H4 \def H_x in (and_ind (sn3
2253 (CHead c (Bind b) u) (lift (S O) O v)) (sn3 (CHead c (Bind b) u) (THeads
2254 (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) (sn3 c (THead (Flat Appl) v
2255 (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u t1)))) (\lambda (H5: (sn3
2256 (CHead c (Bind b) u) (lift (S O) O v))).(\lambda (H6: (sn3 (CHead c (Bind b)
2257 u) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1))).(let H_y \def
2258 (sn3_gen_lift (CHead c (Bind b) u) v (S O) O H5 c) in (sn3_appl_appls t
2259 (THead (Bind b) u t1) t0 c (H2 t1 H6) v (H_y (drop_drop (Bind b) O c c
2260 (drop_refl c) u)) (\lambda (u2: T).(\lambda (H7: (pr3 c (THeads (Flat Appl)
2261 (TCons t t0) (THead (Bind b) u t1)) u2)).(\lambda (H8: (((iso (THeads (Flat
2262 Appl) (TCons t t0) (THead (Bind b) u t1)) u2) \to (\forall (P:
2263 Prop).P)))).(let H9 \def (pr3_iso_appls_bind b H (TCons t t0) u t1 c u2 H7
2264 H8) in (sn3_pr3_trans c (THead (Flat Appl) v (THead (Bind b) u (THeads (Flat
2265 Appl) (lifts (S O) O (TCons t t0)) t1))) (sn3_appl_bind b H c u H0 (THeads
2266 (Flat Appl) (lifts (S O) O (TCons t t0)) t1) v H3) (THead (Flat Appl) v u2)
2267 (pr3_flat c v v (pr3_refl c v) (THead (Bind b) u (THeads (Flat Appl) (lifts
2268 (S O) O (TCons t t0)) t1)) u2 H9 Appl)))))))))) H4))))))))) vs0))) vs)))))).
2270 theorem sn3_appls_beta:
2271 \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (us: TList).((sn3 c
2272 (THeads (Flat Appl) us (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c
2273 w) \to (sn3 c (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst)
2276 \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (us:
2277 TList).(TList_ind (\lambda (t0: TList).((sn3 c (THeads (Flat Appl) t0 (THead
2278 (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat
2279 Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) (\lambda (H:
2280 (sn3 c (THead (Bind Abbr) v t))).(\lambda (w: T).(\lambda (H0: (sn3 c
2281 w)).(sn3_beta c v t H w H0)))) (\lambda (u: T).(\lambda (us0:
2282 TList).(TList_ind (\lambda (t0: TList).((((sn3 c (THeads (Flat Appl) t0
2283 (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads
2284 (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3
2285 c (THead (Flat Appl) u (THeads (Flat Appl) t0 (THead (Bind Abbr) v t)))) \to
2286 (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat
2287 Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))) (\lambda (_:
2288 (((sn3 c (THeads (Flat Appl) TNil (THead (Bind Abbr) v t))) \to (\forall (w:
2289 T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) TNil (THead (Flat Appl) v (THead
2290 (Bind Abst) w t))))))))).(\lambda (H0: (sn3 c (THead (Flat Appl) u (THeads
2291 (Flat Appl) TNil (THead (Bind Abbr) v t))))).(\lambda (w: T).(\lambda (H1:
2292 (sn3 c w)).(sn3_appl_beta c u v t H0 w H1))))) (\lambda (t0: T).(\lambda (t1:
2293 TList).(\lambda (_: (((((sn3 c (THeads (Flat Appl) t1 (THead (Bind Abbr) v
2294 t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) t1 (THead
2295 (Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 c (THead (Flat Appl) u
2296 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)))) \to (\forall (w: T).((sn3 c
2297 w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) t1 (THead (Flat Appl)
2298 v (THead (Bind Abst) w t))))))))))).(\lambda (H0: (((sn3 c (THeads (Flat
2299 Appl) (TCons t0 t1) (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w)
2300 \to (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead
2301 (Bind Abst) w t))))))))).(\lambda (H1: (sn3 c (THead (Flat Appl) u (THeads
2302 (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t))))).(\lambda (w:
2303 T).(\lambda (H2: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THeads
2304 (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t)) H1) in (let H3 \def H_x in
2305 (and_ind (sn3 c u) (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Bind
2306 Abbr) v t))) (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1)
2307 (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H4: (sn3 c
2308 u)).(\lambda (H5: (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr)
2309 v t)))).(sn3_appl_appls t0 (THead (Flat Appl) v (THead (Bind Abst) w t)) t1 c
2310 (H0 H5 w H2) u H4 (\lambda (u2: T).(\lambda (H6: (pr3 c (THeads (Flat Appl)
2311 (TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w t))) u2)).(\lambda
2312 (H7: (((iso (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead
2313 (Bind Abst) w t))) u2) \to (\forall (P: Prop).P)))).(let H8 \def
2314 (pr3_iso_appls_beta (TCons t0 t1) v w t c u2 H6 H7) in (sn3_pr3_trans c
2315 (THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v
2316 t))) H1 (THead (Flat Appl) u u2) (pr3_thin_dx c (THeads (Flat Appl) (TCons t0
2317 t1) (THead (Bind Abbr) v t)) u2 H8 u Appl))))))))) H3)))))))))) us0))) us)))).
2320 \forall (d: C).(\forall (t: T).((sn3 d t) \to (\forall (c: C).(\forall (h:
2321 nat).(\forall (i: nat).((drop h i c d) \to (sn3 c (lift h i t))))))))
2323 \lambda (d: C).(\lambda (t: T).(\lambda (H: (sn3 d t)).(sn3_ind d (\lambda
2324 (t0: T).(\forall (c: C).(\forall (h: nat).(\forall (i: nat).((drop h i c d)
2325 \to (sn3 c (lift h i t0))))))) (\lambda (t1: T).(\lambda (_: ((\forall (t2:
2326 T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 d t1 t2) \to (sn3 d
2327 t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
2328 Prop).P))) \to ((pr3 d t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall
2329 (i: nat).((drop h i c d) \to (sn3 c (lift h i t2))))))))))).(\lambda (c:
2330 C).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H2: (drop h i c
2331 d)).(sn3_pr2_intro c (lift h i t1) (\lambda (t2: T).(\lambda (H3: (((eq T
2332 (lift h i t1) t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr2 c (lift h i
2333 t1) t2)).(let H5 \def (pr2_gen_lift c t1 t2 h i H4 d H2) in (ex2_ind T
2334 (\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t1 t3))
2335 (sn3 c t2) (\lambda (x: T).(\lambda (H6: (eq T t2 (lift h i x))).(\lambda
2336 (H7: (pr2 d t1 x)).(let H8 \def (eq_ind T t2 (\lambda (t0: T).((eq T (lift h
2337 i t1) t0) \to (\forall (P: Prop).P))) H3 (lift h i x) H6) in (eq_ind_r T
2338 (lift h i x) (\lambda (t0: T).(sn3 c t0)) (H1 x (\lambda (H9: (eq T t1
2339 x)).(\lambda (P: Prop).(let H10 \def (eq_ind_r T x (\lambda (t0: T).((eq T
2340 (lift h i t1) (lift h i t0)) \to (\forall (P0: Prop).P0))) H8 t1 H9) in (let
2341 H11 \def (eq_ind_r T x (\lambda (t0: T).(pr2 d t1 t0)) H7 t1 H9) in (H10
2342 (refl_equal T (lift h i t1)) P))))) (pr3_pr2 d t1 x H7) c h i H2) t2 H6)))))
2343 H5))))))))))))) t H))).
2346 \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
2347 (CHead d (Bind Abbr) v)) \to ((sn3 d v) \to (sn3 c (TLRef i)))))))
2349 \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
2350 (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 d
2351 v)).(sn3_pr2_intro c (TLRef i) (\lambda (t2: T).(\lambda (H1: (((eq T (TLRef
2352 i) t2) \to (\forall (P: Prop).P)))).(\lambda (H2: (pr2 c (TLRef i) t2)).(let
2353 H3 \def (pr2_gen_lref c t2 i H2) in (or_ind (eq T t2 (TLRef i)) (ex2_2 C T
2354 (\lambda (d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u))))
2355 (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S i) O u))))) (sn3 c t2)
2356 (\lambda (H4: (eq T t2 (TLRef i))).(let H5 \def (eq_ind T t2 (\lambda (t:
2357 T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H4) in
2358 (eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c t)) (H5 (refl_equal T (TLRef i))
2359 (sn3 c (TLRef i))) t2 H4))) (\lambda (H4: (ex2_2 C T (\lambda (d0:
2360 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)))))).(ex2_2_ind C T (\lambda
2362 (d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_:
2363 C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))) (sn3 c t2) (\lambda (x0:
2364 C).(\lambda (x1: T).(\lambda (H5: (getl i c (CHead x0 (Bind Abbr)
2365 x1))).(\lambda (H6: (eq T t2 (lift (S i) O x1))).(let H7 \def (eq_ind T t2
2366 (\lambda (t: T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (lift (S
2367 i) O x1) H6) in (eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(sn3 c t)) (let
2368 H8 \def (eq_ind C (CHead d (Bind Abbr) v) (\lambda (c0: C).(getl i c c0)) H
2369 (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0
2370 (Bind Abbr) x1) H5)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in
2371 C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
2372 \Rightarrow c0])) (CHead d (Bind Abbr) v) (CHead x0 (Bind Abbr) x1)
2373 (getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 (Bind Abbr) x1) H5)) in
2374 ((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
2375 C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead d
2376 (Bind Abbr) v) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v)
2377 i H (CHead x0 (Bind Abbr) x1) H5)) in (\lambda (H11: (eq C d x0)).(let H12
2378 \def (eq_ind_r T x1 (\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 v
2379 H10) in (eq_ind T v (\lambda (t: T).(sn3 c (lift (S i) O t))) (let H13 \def
2380 (eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) v))) H12 d
2381 H11) in (sn3_lift d v H0 c (S i) O (getl_drop Abbr c d v i H13))) x1 H10))))
2382 H9))) t2 H6)))))) H4)) H3))))))))))).
2384 theorem sn3_appls_abbr:
2385 \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c
2386 (CHead d (Bind Abbr) w)) \to (\forall (vs: TList).((sn3 c (THeads (Flat Appl)
2387 vs (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))
2389 \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda
2390 (H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind
2391 (\lambda (t: TList).((sn3 c (THeads (Flat Appl) t (lift (S i) O w))) \to (sn3
2392 c (THeads (Flat Appl) t (TLRef i))))) (\lambda (H0: (sn3 c (lift (S i) O
2393 w))).(let H_y \def (sn3_gen_lift c w (S i) O H0 d (getl_drop Abbr c d w i H))
2394 in (sn3_abbr c d w i H H_y))) (\lambda (v: T).(\lambda (vs0:
2395 TList).(TList_ind (\lambda (t: TList).((((sn3 c (THeads (Flat Appl) t (lift
2396 (S i) O w))) \to (sn3 c (THeads (Flat Appl) t (TLRef i))))) \to ((sn3 c
2397 (THead (Flat Appl) v (THeads (Flat Appl) t (lift (S i) O w)))) \to (sn3 c
2398 (THead (Flat Appl) v (THeads (Flat Appl) t (TLRef i))))))) (\lambda (_:
2399 (((sn3 c (THeads (Flat Appl) TNil (lift (S i) O w))) \to (sn3 c (THeads (Flat
2400 Appl) TNil (TLRef i)))))).(\lambda (H1: (sn3 c (THead (Flat Appl) v (THeads
2401 (Flat Appl) TNil (lift (S i) O w))))).(sn3_appl_abbr c d w i H v H1)))
2402 (\lambda (t: T).(\lambda (t0: TList).(\lambda (_: (((((sn3 c (THeads (Flat
2403 Appl) t0 (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) t0 (TLRef i)))))
2404 \to ((sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (lift (S i) O w))))
2405 \to (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (TLRef
2406 i)))))))).(\lambda (H1: (((sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i)
2407 O w))) \to (sn3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)))))).(\lambda
2408 (H2: (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (lift (S i)
2409 O w))))).(let H_x \def (sn3_gen_flat Appl c v (THeads (Flat Appl) (TCons t
2410 t0) (lift (S i) O w)) H2) in (let H3 \def H_x in (and_ind (sn3 c v) (sn3 c
2411 (THeads (Flat Appl) (TCons t t0) (lift (S i) O w))) (sn3 c (THead (Flat Appl)
2412 v (THeads (Flat Appl) (TCons t t0) (TLRef i)))) (\lambda (H4: (sn3 c
2413 v)).(\lambda (H5: (sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i) O
2414 w)))).(sn3_appl_appls t (TLRef i) t0 c (H1 H5) v H4 (\lambda (u2: T).(\lambda
2415 (H6: (pr3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2)).(\lambda (H7:
2416 (((iso (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2) \to (\forall (P:
2417 Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) v (THeads (Flat Appl) (TCons
2418 t t0) (lift (S i) O w))) H2 (THead (Flat Appl) v u2) (pr3_thin_dx c (THeads
2419 (Flat Appl) (TCons t t0) (lift (S i) O w)) u2 (pr3_iso_appls_abbr c d w i H
2420 (TCons t t0) u2 H6 H7) v Appl)))))))) H3)))))))) vs0))) vs)))))).
2423 \forall (c: C).(\forall (d: C).(\forall (h: nat).(\forall (i: nat).((drop h
2424 i c d) \to (\forall (ts: TList).((sns3 d ts) \to (sns3 c (lifts h i ts))))))))
2426 \lambda (c: C).(\lambda (d: C).(\lambda (h: nat).(\lambda (i: nat).(\lambda
2427 (H: (drop h i c d)).(\lambda (ts: TList).(TList_ind (\lambda (t:
2428 TList).((sns3 d t) \to (sns3 c (lifts h i t)))) (\lambda (H0: True).H0)
2429 (\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: (((sns3 d t0) \to (sns3 c
2430 (lifts h i t0))))).(\lambda (H1: (land (sn3 d t) (sns3 d t0))).(let H2 \def
2431 H1 in (and_ind (sn3 d t) (sns3 d t0) (land (sn3 c (lift h i t)) (sns3 c
2432 (lifts h i t0))) (\lambda (H3: (sn3 d t)).(\lambda (H4: (sns3 d t0)).(conj
2433 (sn3 c (lift h i t)) (sns3 c (lifts h i t0)) (sn3_lift d t H3 c h i H) (H0
2434 H4)))) H2)))))) ts)))))).
2436 theorem sn3_gen_def:
2437 \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
2438 (CHead d (Bind Abbr) v)) \to ((sn3 c (TLRef i)) \to (sn3 d v))))))
2440 \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
2441 (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 c (TLRef
2442 i))).(sn3_gen_lift c v (S i) O (sn3_pr3_trans c (TLRef i) H0 (lift (S i) O v)
2443 (pr3_pr2 c (TLRef i) (lift (S i) O v) (pr2_delta c d v i H (TLRef i) (TLRef
2444 i) (pr0_refl (TLRef i)) (lift (S i) O v) (subst0_lref v i)))) d (getl_drop
2445 Abbr c d v i H))))))).
2448 \forall (v: T).(\forall (t: T).(\forall (i: nat).(((\forall (w: T).(ex T
2449 (\lambda (u: T).(subst0 i w t u))))) \to (\forall (c: C).(\forall (d:
2450 C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to (sn3 d v))))))))
2452 \lambda (v: T).(\lambda (t: T).(\lambda (i: nat).(\lambda (H: ((\forall (w:
2453 T).(ex T (\lambda (u: T).(subst0 i w t u)))))).(let H_x \def (H v) in (let H0
2454 \def H_x in (ex_ind T (\lambda (u: T).(subst0 i v t u)) (\forall (c:
2455 C).(\forall (d: C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to
2456 (sn3 d v))))) (\lambda (x: T).(\lambda (H1: (subst0 i v t x)).(subst0_ind
2457 (\lambda (n: nat).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_: T).(\forall
2458 (c: C).(\forall (d: C).((getl n c (CHead d (Bind Abbr) t0)) \to ((sn3 c t1)
2459 \to (sn3 d t0))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (c:
2460 C).(\lambda (d: C).(\lambda (H2: (getl i0 c (CHead d (Bind Abbr)
2461 v0))).(\lambda (H3: (sn3 c (TLRef i0))).(sn3_gen_def c d v0 i0 H2 H3)))))))
2462 (\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0:
2463 nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c:
2464 C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to
2465 (sn3 d v0))))))).(\lambda (t0: T).(\lambda (k: K).(\lambda (c: C).(\lambda
2466 (d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) v0))).(\lambda (H5: (sn3
2467 c (THead k u1 t0))).(let H_y \def (sn3_gen_head k c u1 t0 H5) in (H3 c d H4
2468 H_y)))))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: T).(\lambda
2469 (t1: T).(\lambda (i0: nat).(\lambda (H2: (subst0 (s k i0) v0 t1 t2)).(\lambda
2470 (H3: ((\forall (c: C).(\forall (d: C).((getl (s k i0) c (CHead d (Bind Abbr)
2471 v0)) \to ((sn3 c t1) \to (sn3 d v0))))))).(\lambda (u: T).(\lambda (c:
2472 C).(\lambda (d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr)
2473 v0))).(\lambda (H5: (sn3 c (THead k u t1))).(K_ind (\lambda (k0: K).((subst0
2474 (s k0 i0) v0 t1 t2) \to (((\forall (c0: C).(\forall (d0: C).((getl (s k0 i0)
2475 c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0)))))) \to ((sn3
2476 c (THead k0 u t1)) \to (sn3 d v0))))) (\lambda (b: B).(\lambda (_: (subst0 (s
2477 (Bind b) i0) v0 t1 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0:
2478 C).((getl (s (Bind b) i0) c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to
2479 (sn3 d0 v0))))))).(\lambda (H8: (sn3 c (THead (Bind b) u t1))).(let H_x0 \def
2480 (sn3_gen_bind b c u t1 H8) in (let H9 \def H_x0 in (and_ind (sn3 c u) (sn3
2481 (CHead c (Bind b) u) t1) (sn3 d v0) (\lambda (_: (sn3 c u)).(\lambda (H11:
2482 (sn3 (CHead c (Bind b) u) t1)).(H7 (CHead c (Bind b) u) d (getl_clear_bind b
2483 (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d (Bind Abbr) v0) i0 H4)
2484 H11))) H9))))))) (\lambda (f: F).(\lambda (_: (subst0 (s (Flat f) i0) v0 t1
2485 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: C).((getl (s (Flat f) i0)
2486 c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0))))))).(\lambda
2487 (H8: (sn3 c (THead (Flat f) u t1))).(let H_x0 \def (sn3_gen_flat f c u t1 H8)
2488 in (let H9 \def H_x0 in (and_ind (sn3 c u) (sn3 c t1) (sn3 d v0) (\lambda (_:
2489 (sn3 c u)).(\lambda (H11: (sn3 c t1)).(H7 c d H4 H11))) H9))))))) k H2 H3
2490 H5))))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda
2491 (i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c:
2492 C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to
2493 (sn3 d v0))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda
2494 (_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: ((\forall (c: C).(\forall (d:
2495 C).((getl (s k i0) c (CHead d (Bind Abbr) v0)) \to ((sn3 c t1) \to (sn3 d
2496 v0))))))).(\lambda (c: C).(\lambda (d: C).(\lambda (H6: (getl i0 c (CHead d
2497 (Bind Abbr) v0))).(\lambda (H7: (sn3 c (THead k u1 t1))).(let H_y \def
2498 (sn3_gen_head k c u1 t1 H7) in (H3 c d H6 H_y))))))))))))))))) i v t x H1)))
projects / helm.git / blob