1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 include "LambdaDelta-1/pr0/fwd.ma".
19 include "LambdaDelta-1/subst0/dec.ma".
21 include "LambdaDelta-1/T/dec.ma".
23 include "LambdaDelta-1/T/props.ma".
26 \forall (t1: T).(or (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T
27 (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
30 \lambda (t1: T).(T_ind (\lambda (t: T).(or (\forall (t2: T).((pr0 t t2) \to
31 (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P:
32 Prop).P))) (\lambda (t2: T).(pr0 t t2))))) (\lambda (n: nat).(or_introl
33 (\forall (t2: T).((pr0 (TSort n) t2) \to (eq T (TSort n) t2))) (ex2 T
34 (\lambda (t2: T).((eq T (TSort n) t2) \to (\forall (P: Prop).P))) (\lambda
35 (t2: T).(pr0 (TSort n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TSort n)
36 t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T
37 (TSort n)) t2 (pr0_gen_sort t2 n H)))))) (\lambda (n: nat).(or_introl
38 (\forall (t2: T).((pr0 (TLRef n) t2) \to (eq T (TLRef n) t2))) (ex2 T
39 (\lambda (t2: T).((eq T (TLRef n) t2) \to (\forall (P: Prop).P))) (\lambda
40 (t2: T).(pr0 (TLRef n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TLRef n)
41 t2)).(eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t)) (refl_equal T
42 (TLRef n)) t2 (pr0_gen_lref t2 n H)))))) (\lambda (k: K).(\lambda (t:
43 T).(\lambda (H: (or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T
44 (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2:
45 T).(pr0 t t2))))).(\lambda (t0: T).(\lambda (H0: (or (\forall (t2: T).((pr0
46 t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall
47 (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))))).(K_ind (\lambda (k0: K).(or
48 (\forall (t2: T).((pr0 (THead k0 t t0) t2) \to (eq T (THead k0 t t0) t2)))
49 (ex2 T (\lambda (t2: T).((eq T (THead k0 t t0) t2) \to (\forall (P:
50 Prop).P))) (\lambda (t2: T).(pr0 (THead k0 t t0) t2))))) (\lambda (b:
51 B).(B_ind (\lambda (b0: B).(or (\forall (t2: T).((pr0 (THead (Bind b0) t t0)
52 t2) \to (eq T (THead (Bind b0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T
53 (THead (Bind b0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
54 (THead (Bind b0) t t0) t2))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind
55 Abbr) t t0) t2) \to (eq T (THead (Bind Abbr) t t0) t2))) (ex2 T (\lambda (t2:
56 T).((eq T (THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda
57 (t2: T).(pr0 (THead (Bind Abbr) t t0) t2))) (let H_x \def (dnf_dec t t0 O) in
58 (let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 O t t0 (lift (S O)
59 O v)) (eq T t0 (lift (S O) O v)))) (ex2 T (\lambda (t2: T).((eq T (THead
60 (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
61 (THead (Bind Abbr) t t0) t2))) (\lambda (x: T).(\lambda (H2: (or (subst0 O t
62 t0 (lift (S O) O x)) (eq T t0 (lift (S O) O x)))).(or_ind (subst0 O t t0
63 (lift (S O) O x)) (eq T t0 (lift (S O) O x)) (ex2 T (\lambda (t2: T).((eq T
64 (THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
65 T).(pr0 (THead (Bind Abbr) t t0) t2))) (\lambda (H3: (subst0 O t t0 (lift (S
66 O) O x))).(ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t t0) t2)
67 \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0)
68 t2)) (THead (Bind Abbr) t (lift (S O) O x)) (\lambda (H4: (eq T (THead (Bind
69 Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O x)))).(\lambda (P: Prop).(let
70 H5 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
71 with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2)
72 \Rightarrow t2])) (THead (Bind Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O
73 x)) H4) in (let H6 \def (eq_ind T t0 (\lambda (t2: T).(subst0 O t t2 (lift (S
74 O) O x))) H3 (lift (S O) O x) H5) in (subst0_refl t (lift (S O) O x) O H6
75 P))))) (pr0_delta t t (pr0_refl t) t0 t0 (pr0_refl t0) (lift (S O) O x) H3)))
76 (\lambda (H3: (eq T t0 (lift (S O) O x))).(eq_ind_r T (lift (S O) O x)
77 (\lambda (t2: T).(ex2 T (\lambda (t3: T).((eq T (THead (Bind Abbr) t t2) t3)
78 \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Abbr) t t2)
79 t3)))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t (lift (S O)
80 O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind
81 Abbr) t (lift (S O) O x)) t2)) x (\lambda (H4: (eq T (THead (Bind Abbr) t
82 (lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y (Bind Abbr) x t (S
83 O) O H4 P))) (pr0_zeta Abbr not_abbr_abst x x (pr0_refl x) t)) t0 H3)) H2)))
84 H1)))) (let H1 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t
85 t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P)))
86 (\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t
87 t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq
88 T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
89 T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (H2: ((\forall (t2: T).((pr0
90 t t2) \to (eq T t t2))))).(let H3 \def H0 in (or_ind (\forall (t2: T).((pr0
91 t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall
92 (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0
93 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T
94 (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P:
95 Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda
96 (H4: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall
97 (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0)
98 t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to
99 (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))
100 (\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) t t0) t2)).(ex3_2_ind
101 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3))))
102 (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3:
103 T).(pr0 t0 t3))) (eq T (THead (Bind Abst) t t0) t2) (\lambda (x0: T).(\lambda
104 (x1: T).(\lambda (H6: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H7: (pr0
105 t x0)).(\lambda (H8: (pr0 t0 x1)).(let H_y \def (H4 x1 H8) in (let H_y0 \def
106 (H2 x0 H7) in (let H9 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H8 t0
107 H_y) in (let H10 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead (Bind
108 Abst) x0 t3))) H6 t0 H_y) in (let H11 \def (eq_ind_r T x0 (\lambda (t3:
109 T).(pr0 t t3)) H7 t H_y0) in (let H12 \def (eq_ind_r T x0 (\lambda (t3:
110 T).(eq T t2 (THead (Bind Abst) t3 t0))) H10 t H_y0) in (eq_ind_r T (THead
111 (Bind Abst) t t0) (\lambda (t3: T).(eq T (THead (Bind Abst) t t0) t3))
112 (refl_equal T (THead (Bind Abst) t t0)) t2 H12)))))))))))) (pr0_gen_abst t t0
113 t2 H5)))))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall
114 (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2:
115 T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))
116 (or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead
117 (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t
118 t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst)
119 t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t0 x) \to (\forall (P:
120 Prop).P)))).(\lambda (H6: (pr0 t0 x)).(or_intror (\forall (t2: T).((pr0
121 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T
122 (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P:
123 Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T
124 (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P:
125 Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind
126 Abst) t x) (\lambda (H7: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t
127 x))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e in
128 T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
129 \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind Abst) t t0)
130 (THead (Bind Abst) t x) H7) in (let H9 \def (eq_ind_r T x (\lambda (t2:
131 T).(pr0 t0 t2)) H6 t0 H8) in (let H10 \def (eq_ind_r T x (\lambda (t2:
132 T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H5 t0 H8) in (H10 (refl_equal
133 T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H6 (Bind Abst))))))) H4)) H3)))
134 (\lambda (H2: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P:
135 Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T
136 t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall
137 (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0)
138 t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to
139 (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))))
140 (\lambda (x: T).(\lambda (H3: (((eq T t x) \to (\forall (P:
141 Prop).P)))).(\lambda (H4: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead
142 (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T
143 (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P:
144 Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T
145 (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P:
146 Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind
147 Abst) x t0) (\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) x
148 t0))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: T).(match e
149 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _)
150 \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst) t t0)
151 (THead (Bind Abst) x t0) H5) in (let H7 \def (eq_ind_r T x (\lambda (t2:
152 T).(pr0 t t2)) H4 t H6) in (let H8 \def (eq_ind_r T x (\lambda (t2: T).((eq T
153 t t2) \to (\forall (P0: Prop).P0))) H3 t H6) in (H8 (refl_equal T t) P))))))
154 (pr0_comp t x H4 t0 t0 (pr0_refl t0) (Bind Abst))))))) H2)) H1)) (let H_x
155 \def (dnf_dec t t0 O) in (let H1 \def H_x in (ex_ind T (\lambda (v: T).(or
156 (subst0 O t t0 (lift (S O) O v)) (eq T t0 (lift (S O) O v)))) (or (\forall
157 (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0)
158 t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to
159 (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))))
160 (\lambda (x: T).(\lambda (H2: (or (subst0 O t t0 (lift (S O) O x)) (eq T t0
161 (lift (S O) O x)))).(or_ind (subst0 O t t0 (lift (S O) O x)) (eq T t0 (lift
162 (S O) O x)) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T
163 (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind
164 Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
165 (Bind Void) t t0) t2)))) (\lambda (H3: (subst0 O t t0 (lift (S O) O x))).(let
166 H4 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T
167 (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2:
168 T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to
169 (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead
170 (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
171 (THead (Bind Void) t t0) t2)))) (\lambda (H5: ((\forall (t2: T).((pr0 t t2)
172 \to (eq T t t2))))).(let H6 \def H0 in (or_ind (\forall (t2: T).((pr0 t0 t2)
173 \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P:
174 Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead
175 (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T
176 (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P:
177 Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda
178 (H7: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall
179 (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0)
180 t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to
181 (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))
182 (\lambda (t2: T).(\lambda (H8: (pr0 (THead (Bind Void) t t0) t2)).(or_ind
183 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2
184 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda
185 (t3: T).(pr0 t0 t3)))) (pr0 t0 (lift (S O) O t2)) (eq T (THead (Bind Void) t
186 t0) t2) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
187 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2)))
188 (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2:
189 T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2:
190 T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0
191 t3))) (eq T (THead (Bind Void) t t0) t2) (\lambda (x0: T).(\lambda (x1:
192 T).(\lambda (H10: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H11: (pr0 t
193 x0)).(\lambda (H12: (pr0 t0 x1)).(let H_y \def (H7 x1 H12) in (let H_y0 \def
194 (H5 x0 H11) in (let H13 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H12
195 t0 H_y) in (let H14 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead
196 (Bind Void) x0 t3))) H10 t0 H_y) in (let H15 \def (eq_ind_r T x0 (\lambda
197 (t3: T).(pr0 t t3)) H11 t H_y0) in (let H16 \def (eq_ind_r T x0 (\lambda (t3:
198 T).(eq T t2 (THead (Bind Void) t3 t0))) H14 t H_y0) in (eq_ind_r T (THead
199 (Bind Void) t t0) (\lambda (t3: T).(eq T (THead (Bind Void) t t0) t3))
200 (refl_equal T (THead (Bind Void) t t0)) t2 H16)))))))))))) H9)) (\lambda (H9:
201 (pr0 t0 (lift (S O) O t2))).(let H_y \def (H7 (lift (S O) O t2) H9) in (let
202 H10 \def (eq_ind T t0 (\lambda (t3: T).(subst0 O t t3 (lift (S O) O x))) H3
203 (lift (S O) O t2) H_y) in (eq_ind_r T (lift (S O) O t2) (\lambda (t3: T).(eq
204 T (THead (Bind Void) t t3) t2)) (subst0_gen_lift_false t2 t (lift (S O) O x)
205 (S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n))
206 (le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H10 (eq T (THead
207 (Bind Void) t (lift (S O) O t2)) t2)) t0 H_y)))) (pr0_gen_void t t0 t2
208 H8)))))) (\lambda (H7: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P:
209 Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T
210 t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall
211 (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0)
212 t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to
213 (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))))
214 (\lambda (x0: T).(\lambda (H8: (((eq T t0 x0) \to (\forall (P:
215 Prop).P)))).(\lambda (H9: (pr0 t0 x0)).(or_intror (\forall (t2: T).((pr0
216 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T
217 (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P:
218 Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T
219 (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P:
220 Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind
221 Void) t x0) (\lambda (H10: (eq T (THead (Bind Void) t t0) (THead (Bind Void)
222 t x0))).(\lambda (P: Prop).(let H11 \def (f_equal T T (\lambda (e: T).(match
223 e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
224 \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind Void) t t0)
225 (THead (Bind Void) t x0) H10) in (let H12 \def (eq_ind_r T x0 (\lambda (t2:
226 T).(pr0 t0 t2)) H9 t0 H11) in (let H13 \def (eq_ind_r T x0 (\lambda (t2:
227 T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H8 t0 H11) in (H13 (refl_equal
228 T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x0 H9 (Bind Void))))))) H7))
229 H6))) (\lambda (H5: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P:
230 Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T
231 t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall
232 (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0)
233 t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to
234 (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))))
235 (\lambda (x0: T).(\lambda (H6: (((eq T t x0) \to (\forall (P:
236 Prop).P)))).(\lambda (H7: (pr0 t x0)).(or_intror (\forall (t2: T).((pr0
237 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T
238 (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P:
239 Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T
240 (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P:
241 Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind
242 Void) x0 t0) (\lambda (H8: (eq T (THead (Bind Void) t t0) (THead (Bind Void)
243 x0 t0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match
244 e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _)
245 \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Void) t t0)
246 (THead (Bind Void) x0 t0) H8) in (let H10 \def (eq_ind_r T x0 (\lambda (t2:
247 T).(pr0 t t2)) H7 t H9) in (let H11 \def (eq_ind_r T x0 (\lambda (t2: T).((eq
248 T t t2) \to (\forall (P0: Prop).P0))) H6 t H9) in (H11 (refl_equal T t)
249 P)))))) (pr0_comp t x0 H7 t0 t0 (pr0_refl t0) (Bind Void))))))) H5)) H4)))
250 (\lambda (H3: (eq T t0 (lift (S O) O x))).(let H4 \def (eq_ind T t0 (\lambda
251 (t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq T t2 t3))) (ex2 T (\lambda
252 (t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 t2
253 t3))))) H0 (lift (S O) O x) H3) in (eq_ind_r T (lift (S O) O x) (\lambda (t2:
254 T).(or (\forall (t3: T).((pr0 (THead (Bind Void) t t2) t3) \to (eq T (THead
255 (Bind Void) t t2) t3))) (ex2 T (\lambda (t3: T).((eq T (THead (Bind Void) t
256 t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Void)
257 t t2) t3))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind Void) t (lift (S
258 O) O x)) t2) \to (eq T (THead (Bind Void) t (lift (S O) O x)) t2))) (ex2 T
259 (\lambda (t2: T).((eq T (THead (Bind Void) t (lift (S O) O x)) t2) \to
260 (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S
261 O) O x)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Void) t
262 (lift (S O) O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
263 (THead (Bind Void) t (lift (S O) O x)) t2)) x (\lambda (H5: (eq T (THead
264 (Bind Void) t (lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y
265 (Bind Void) x t (S O) O H5 P))) (pr0_zeta Void not_void_abst x x (pr0_refl x)
266 t))) t0 H3))) H2))) H1))) b)) (\lambda (f: F).(F_ind (\lambda (f0: F).(or
267 (\forall (t2: T).((pr0 (THead (Flat f0) t t0) t2) \to (eq T (THead (Flat f0)
268 t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat f0) t t0) t2) \to
269 (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat f0) t t0) t2)))))
270 (let H_x \def (binder_dec t0) in (let H1 \def H_x in (or_ind (ex_3 B T T
271 (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w
272 u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead
273 (Bind b) w u)) \to (\forall (P: Prop).P))))) (or (\forall (t2: T).((pr0
274 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T
275 (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P:
276 Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda
277 (H2: (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0
278 (THead (Bind b) w u))))))).(ex_3_ind B T T (\lambda (b: B).(\lambda (w:
279 T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u))))) (or (\forall (t2:
280 T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0)
281 t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to
282 (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))))
283 (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H3: (eq T t0
284 (THead (Bind x0) x1 x2))).(let H4 \def (eq_ind T t0 (\lambda (t2: T).(or
285 (\forall (t3: T).((pr0 t2 t3) \to (eq T t2 t3))) (ex2 T (\lambda (t3: T).((eq
286 T t2 t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 t2 t3))))) H0
287 (THead (Bind x0) x1 x2) H3) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda
288 (t2: T).(or (\forall (t3: T).((pr0 (THead (Flat Appl) t t2) t3) \to (eq T
289 (THead (Flat Appl) t t2) t3))) (ex2 T (\lambda (t3: T).((eq T (THead (Flat
290 Appl) t t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead
291 (Flat Appl) t t2) t3))))) (B_ind (\lambda (b: B).((or (\forall (t2: T).((pr0
292 (THead (Bind b) x1 x2) t2) \to (eq T (THead (Bind b) x1 x2) t2))) (ex2 T
293 (\lambda (t2: T).((eq T (THead (Bind b) x1 x2) t2) \to (\forall (P:
294 Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind b) x1 x2) t2)))) \to (or
295 (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to
296 (eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2))) (ex2 T (\lambda (t2:
297 T).((eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to (\forall (P:
298 Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2))
299 t2)))))) (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abbr) x1 x2)
300 t2) \to (eq T (THead (Bind Abbr) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T
301 (THead (Bind Abbr) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
302 T).(pr0 (THead (Bind Abbr) x1 x2) t2))))).(or_intror (\forall (t2: T).((pr0
303 (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (eq T (THead (Flat
304 Appl) t (THead (Bind Abbr) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T
305 (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (\forall (P:
306 Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abbr) x1
307 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead
308 (Bind Abbr) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
309 (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2)) (THead (Bind Abbr) x1
310 (THead (Flat Appl) (lift (S O) O t) x2)) (\lambda (H6: (eq T (THead (Flat
311 Appl) t (THead (Bind Abbr) x1 x2)) (THead (Bind Abbr) x1 (THead (Flat Appl)
312 (lift (S O) O t) x2)))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead
313 (Flat Appl) t (THead (Bind Abbr) x1 x2)) (\lambda (ee: T).(match ee in T
314 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
315 \Rightarrow False | (THead _ _ t2) \Rightarrow (match t2 in T return (\lambda
316 (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
317 | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with
318 [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])])) I (THead (Bind
319 Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)) H6) in (False_ind P H7))))
320 (pr0_upsilon Abbr not_abbr_abst t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2
321 (pr0_refl x2))))) (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abst)
322 x1 x2) t2) \to (eq T (THead (Bind Abst) x1 x2) t2))) (ex2 T (\lambda (t2:
323 T).((eq T (THead (Bind Abst) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda
324 (t2: T).(pr0 (THead (Bind Abst) x1 x2) t2))))).(or_intror (\forall (t2:
325 T).((pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (eq T (THead
326 (Flat Appl) t (THead (Bind Abst) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T
327 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (\forall (P:
328 Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abst) x1
329 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead
330 (Bind Abst) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
331 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2)) (THead (Bind Abbr) t x2)
332 (\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) (THead
333 (Bind Abbr) t x2))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead (Flat
334 Appl) t (THead (Bind Abst) x1 x2)) (\lambda (ee: T).(match ee in T return
335 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
336 \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
337 (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
338 True])])) I (THead (Bind Abbr) t x2) H6) in (False_ind P H7)))) (pr0_beta x1
339 t t (pr0_refl t) x2 x2 (pr0_refl x2))))) (\lambda (_: (or (\forall (t2:
340 T).((pr0 (THead (Bind Void) x1 x2) t2) \to (eq T (THead (Bind Void) x1 x2)
341 t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) x1 x2) t2) \to
342 (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) x1 x2)
343 t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind
344 Void) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2))
345 t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Void)
346 x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat
347 Appl) t (THead (Bind Void) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T
348 (THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) \to (\forall (P:
349 Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Void) x1
350 x2)) t2)) (THead (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2))
351 (\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) (THead
352 (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)))).(\lambda (P:
353 Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Void) x1 x2))
354 (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
355 \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ t2) \Rightarrow
356 (match t2 in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False
357 | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K
358 return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _)
359 \Rightarrow False])])])) I (THead (Bind Void) x1 (THead (Flat Appl) (lift (S
360 O) O t) x2)) H6) in (False_ind P H7)))) (pr0_upsilon Void not_void_abst t t
361 (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2))))) x0 H4) t0 H3))))))
362 H2)) (\lambda (H2: ((\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0
363 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(let H3 \def H in
364 (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2:
365 T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))) (or
366 (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat
367 Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2)
368 \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0)
369 t2)))) (\lambda (H4: ((\forall (t2: T).((pr0 t t2) \to (eq T t t2))))).(let
370 H5 \def H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))) (ex2 T
371 (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
372 T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to
373 (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead
374 (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
375 (THead (Flat Appl) t t0) t2)))) (\lambda (H6: ((\forall (t2: T).((pr0 t0 t2)
376 \to (eq T t0 t2))))).(or_introl (\forall (t2: T).((pr0 (THead (Flat Appl) t
377 t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq
378 T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
379 T).(pr0 (THead (Flat Appl) t t0) t2))) (\lambda (t2: T).(\lambda (H7: (pr0
380 (THead (Flat Appl) t t0) t2)).(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda
381 (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
382 T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3)))) (ex4_4 T T T
383 T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0
384 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
385 T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
386 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda
387 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))
388 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
389 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
390 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
391 (_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
392 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T
393 t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3)))))))))
394 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
395 T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda
396 (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
397 (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
398 T).(\lambda (t3: T).(pr0 z1 t3)))))))) (eq T (THead (Flat Appl) t t0) t2)
399 (\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
400 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda
401 (_: T).(\lambda (t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2:
402 T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
403 T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0
404 t3))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: T).(\lambda (x1:
405 T).(\lambda (H9: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H10: (pr0 t
406 x0)).(\lambda (H11: (pr0 t0 x1)).(let H_y \def (H6 x1 H11) in (let H_y0 \def
407 (H4 x0 H10) in (let H12 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H11
408 t0 H_y) in (let H13 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead
409 (Flat Appl) x0 t3))) H9 t0 H_y) in (let H14 \def (eq_ind_r T x0 (\lambda (t3:
410 T).(pr0 t t3)) H10 t H_y0) in (let H15 \def (eq_ind_r T x0 (\lambda (t3:
411 T).(eq T t2 (THead (Flat Appl) t3 t0))) H13 t H_y0) in (eq_ind_r T (THead
412 (Flat Appl) t t0) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3))
413 (refl_equal T (THead (Flat Appl) t t0)) t2 H15)))))))))))) H8)) (\lambda (H8:
414 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
415 T).(eq T t0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
416 T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))))
417 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t
418 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
419 T).(pr0 z1 t3))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1:
420 T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) y1 z1))))))
421 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2
422 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
423 T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
424 (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (eq T (THead (Flat Appl) t t0) t2)
425 (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda
426 (H9: (eq T t0 (THead (Bind Abst) x0 x1))).(\lambda (H10: (eq T t2 (THead
427 (Bind Abbr) x2 x3))).(\lambda (_: (pr0 t x2)).(\lambda (_: (pr0 x1
428 x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t3: T).(eq T (THead
429 (Flat Appl) t t0) t3)) (let H13 \def (eq_ind T t0 (\lambda (t3: T).(\forall
430 (t4: T).((pr0 t3 t4) \to (eq T t3 t4)))) H6 (THead (Bind Abst) x0 x1) H9) in
431 (let H14 \def (eq_ind T t0 (\lambda (t3: T).(\forall (b: B).(\forall (w:
432 T).(\forall (u: T).((eq T t3 (THead (Bind b) w u)) \to (\forall (P:
433 Prop).P)))))) H2 (THead (Bind Abst) x0 x1) H9) in (eq_ind_r T (THead (Bind
434 Abst) x0 x1) (\lambda (t3: T).(eq T (THead (Flat Appl) t t3) (THead (Bind
435 Abbr) x2 x3))) (H14 Abst x0 x1 (H13 (THead (Bind Abst) x0 x1) (pr0_refl
436 (THead (Bind Abst) x0 x1))) (eq T (THead (Flat Appl) t (THead (Bind Abst) x0
437 x1)) (THead (Bind Abbr) x2 x3))) t0 H9))) t2 H10))))))))) H8)) (\lambda (H8:
438 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
439 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
440 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
441 (_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
442 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T
443 t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3)))))))))
444 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
445 T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda
446 (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
447 (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
448 T).(\lambda (t3: T).(pr0 z1 t3))))))))).(ex6_6_ind B T T T T T (\lambda (b:
449 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
450 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
451 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind
452 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
453 (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead
454 (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
455 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t
456 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
457 T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
458 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
459 (t3: T).(pr0 z1 t3))))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (x0:
460 B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
461 T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H10: (eq T
462 t0 (THead (Bind x0) x1 x2))).(\lambda (H11: (eq T t2 (THead (Bind x0) x4
463 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda (_: (pr0 t x3)).(\lambda
464 (_: (pr0 x1 x4)).(\lambda (_: (pr0 x2 x5)).(eq_ind_r T (THead (Bind x0) x4
465 (THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(eq T (THead (Flat
466 Appl) t t0) t3)) (let H15 \def (eq_ind T t0 (\lambda (t3: T).(\forall (t4:
467 T).((pr0 t3 t4) \to (eq T t3 t4)))) H6 (THead (Bind x0) x1 x2) H10) in (let
468 H16 \def (eq_ind T t0 (\lambda (t3: T).(\forall (b: B).(\forall (w:
469 T).(\forall (u: T).((eq T t3 (THead (Bind b) w u)) \to (\forall (P:
470 Prop).P)))))) H2 (THead (Bind x0) x1 x2) H10) in (eq_ind_r T (THead (Bind x0)
471 x1 x2) (\lambda (t3: T).(eq T (THead (Flat Appl) t t3) (THead (Bind x0) x4
472 (THead (Flat Appl) (lift (S O) O x3) x5)))) (H16 x0 x1 x2 (H15 (THead (Bind
473 x0) x1 x2) (pr0_refl (THead (Bind x0) x1 x2))) (eq T (THead (Flat Appl) t
474 (THead (Bind x0) x1 x2)) (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O
475 x3) x5)))) t0 H10))) t2 H11))))))))))))) H8)) (pr0_gen_appl t t0 t2 H7))))))
476 (\lambda (H6: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P:
477 Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T
478 t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall
479 (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0)
480 t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to
481 (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))))
482 (\lambda (x: T).(\lambda (H7: (((eq T t0 x) \to (\forall (P:
483 Prop).P)))).(\lambda (H8: (pr0 t0 x)).(or_intror (\forall (t2: T).((pr0
484 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T
485 (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P:
486 Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (ex_intro2 T
487 (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P:
488 Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) (THead (Flat
489 Appl) t x) (\lambda (H9: (eq T (THead (Flat Appl) t t0) (THead (Flat Appl) t
490 x))).(\lambda (P: Prop).(let H10 \def (f_equal T T (\lambda (e: T).(match e
491 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
492 \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Flat Appl) t t0)
493 (THead (Flat Appl) t x) H9) in (let H11 \def (eq_ind_r T x (\lambda (t2:
494 T).(pr0 t0 t2)) H8 t0 H10) in (let H12 \def (eq_ind_r T x (\lambda (t2:
495 T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H7 t0 H10) in (H12 (refl_equal
496 T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H8 (Flat Appl))))))) H6)) H5)))
497 (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P:
498 Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T
499 t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall
500 (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0)
501 t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to
502 (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))))
503 (\lambda (x: T).(\lambda (H5: (((eq T t x) \to (\forall (P:
504 Prop).P)))).(\lambda (H6: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead
505 (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T
506 (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P:
507 Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (ex_intro2 T
508 (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P:
509 Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) (THead (Flat
510 Appl) x t0) (\lambda (H7: (eq T (THead (Flat Appl) t t0) (THead (Flat Appl) x
511 t0))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e
512 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _)
513 \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Flat Appl) t t0)
514 (THead (Flat Appl) x t0) H7) in (let H9 \def (eq_ind_r T x (\lambda (t2:
515 T).(pr0 t t2)) H6 t H8) in (let H10 \def (eq_ind_r T x (\lambda (t2: T).((eq
516 T t t2) \to (\forall (P0: Prop).P0))) H5 t H8) in (H10 (refl_equal T t)
517 P)))))) (pr0_comp t x H6 t0 t0 (pr0_refl t0) (Flat Appl))))))) H4)) H3)))
518 H1))) (or_intror (\forall (t2: T).((pr0 (THead (Flat Cast) t t0) t2) \to (eq
519 T (THead (Flat Cast) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat
520 Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
521 (Flat Cast) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat
522 Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
523 (Flat Cast) t t0) t2)) t0 (\lambda (H1: (eq T (THead (Flat Cast) t t0)
524 t0)).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) t t0 H1 P))) (pr0_tau t0 t0
525 (pr0_refl t0) t))) f)) k)))))) t1).
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