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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 *)
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15 (* This file was automatically generated: do not edit *********************)
17 include "basic_1A/T/fwd.ma".
19 fact terms_props__bind_dec:
20 \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) ((eq B b1 b2) \to (\forall
23 \lambda (b1: B).(B_ind (\lambda (b: B).(\forall (b2: B).(or (eq B b b2) ((eq
24 B b b2) \to (\forall (P: Prop).P))))) (\lambda (b2: B).(B_ind (\lambda (b:
25 B).(or (eq B Abbr b) ((eq B Abbr b) \to (\forall (P: Prop).P)))) (or_introl
26 (eq B Abbr Abbr) ((eq B Abbr Abbr) \to (\forall (P: Prop).P)) (refl_equal B
27 Abbr)) (or_intror (eq B Abbr Abst) ((eq B Abbr Abst) \to (\forall (P:
28 Prop).P)) (\lambda (H: (eq B Abbr Abst)).(\lambda (P: Prop).(let H0 \def
29 (eq_ind B Abbr (\lambda (ee: B).(match ee with [Abbr \Rightarrow True | Abst
30 \Rightarrow False | Void \Rightarrow False])) I Abst H) in (False_ind P
31 H0))))) (or_intror (eq B Abbr Void) ((eq B Abbr Void) \to (\forall (P:
32 Prop).P)) (\lambda (H: (eq B Abbr Void)).(\lambda (P: Prop).(let H0 \def
33 (eq_ind B Abbr (\lambda (ee: B).(match ee with [Abbr \Rightarrow True | Abst
34 \Rightarrow False | Void \Rightarrow False])) I Void H) in (False_ind P
35 H0))))) b2)) (\lambda (b2: B).(B_ind (\lambda (b: B).(or (eq B Abst b) ((eq B
36 Abst b) \to (\forall (P: Prop).P)))) (or_intror (eq B Abst Abbr) ((eq B Abst
37 Abbr) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Abst Abbr)).(\lambda (P:
38 Prop).(let H0 \def (eq_ind B Abst (\lambda (ee: B).(match ee with [Abbr
39 \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False])) I Abbr
40 H) in (False_ind P H0))))) (or_introl (eq B Abst Abst) ((eq B Abst Abst) \to
41 (\forall (P: Prop).P)) (refl_equal B Abst)) (or_intror (eq B Abst Void) ((eq
42 B Abst Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Abst
43 Void)).(\lambda (P: Prop).(let H0 \def (eq_ind B Abst (\lambda (ee: B).(match
44 ee with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow
45 False])) I Void H) in (False_ind P H0))))) b2)) (\lambda (b2: B).(B_ind
46 (\lambda (b: B).(or (eq B Void b) ((eq B Void b) \to (\forall (P: Prop).P))))
47 (or_intror (eq B Void Abbr) ((eq B Void Abbr) \to (\forall (P: Prop).P))
48 (\lambda (H: (eq B Void Abbr)).(\lambda (P: Prop).(let H0 \def (eq_ind B Void
49 (\lambda (ee: B).(match ee with [Abbr \Rightarrow False | Abst \Rightarrow
50 False | Void \Rightarrow True])) I Abbr H) in (False_ind P H0))))) (or_intror
51 (eq B Void Abst) ((eq B Void Abst) \to (\forall (P: Prop).P)) (\lambda (H:
52 (eq B Void Abst)).(\lambda (P: Prop).(let H0 \def (eq_ind B Void (\lambda
53 (ee: B).(match ee with [Abbr \Rightarrow False | Abst \Rightarrow False |
54 Void \Rightarrow True])) I Abst H) in (False_ind P H0))))) (or_introl (eq B
55 Void Void) ((eq B Void Void) \to (\forall (P: Prop).P)) (refl_equal B Void))
59 \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) (not (eq B b1 b2))))
61 \lambda (b1: B).(\lambda (b2: B).(let H_x \def (terms_props__bind_dec b1 b2)
62 in (let H \def H_x in (or_ind (eq B b1 b2) ((eq B b1 b2) \to (\forall (P:
63 Prop).P)) (or (eq B b1 b2) ((eq B b1 b2) \to False)) (\lambda (H0: (eq B b1
64 b2)).(or_introl (eq B b1 b2) ((eq B b1 b2) \to False) H0)) (\lambda (H0:
65 (((eq B b1 b2) \to (\forall (P: Prop).P)))).(or_intror (eq B b1 b2) ((eq B b1
66 b2) \to False) (\lambda (H1: (eq B b1 b2)).(H0 H1 False)))) H)))).
68 fact terms_props__flat_dec:
69 \forall (f1: F).(\forall (f2: F).(or (eq F f1 f2) ((eq F f1 f2) \to (\forall
72 \lambda (f1: F).(F_ind (\lambda (f: F).(\forall (f2: F).(or (eq F f f2) ((eq
73 F f f2) \to (\forall (P: Prop).P))))) (\lambda (f2: F).(F_ind (\lambda (f:
74 F).(or (eq F Appl f) ((eq F Appl f) \to (\forall (P: Prop).P)))) (or_introl
75 (eq F Appl Appl) ((eq F Appl Appl) \to (\forall (P: Prop).P)) (refl_equal F
76 Appl)) (or_intror (eq F Appl Cast) ((eq F Appl Cast) \to (\forall (P:
77 Prop).P)) (\lambda (H: (eq F Appl Cast)).(\lambda (P: Prop).(let H0 \def
78 (eq_ind F Appl (\lambda (ee: F).(match ee with [Appl \Rightarrow True | Cast
79 \Rightarrow False])) I Cast H) in (False_ind P H0))))) f2)) (\lambda (f2:
80 F).(F_ind (\lambda (f: F).(or (eq F Cast f) ((eq F Cast f) \to (\forall (P:
81 Prop).P)))) (or_intror (eq F Cast Appl) ((eq F Cast Appl) \to (\forall (P:
82 Prop).P)) (\lambda (H: (eq F Cast Appl)).(\lambda (P: Prop).(let H0 \def
83 (eq_ind F Cast (\lambda (ee: F).(match ee with [Appl \Rightarrow False | Cast
84 \Rightarrow True])) I Appl H) in (False_ind P H0))))) (or_introl (eq F Cast
85 Cast) ((eq F Cast Cast) \to (\forall (P: Prop).P)) (refl_equal F Cast)) f2))
88 fact terms_props__kind_dec:
89 \forall (k1: K).(\forall (k2: K).(or (eq K k1 k2) ((eq K k1 k2) \to (\forall
92 \lambda (k1: K).(K_ind (\lambda (k: K).(\forall (k2: K).(or (eq K k k2) ((eq
93 K k k2) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (k2: K).(K_ind
94 (\lambda (k: K).(or (eq K (Bind b) k) ((eq K (Bind b) k) \to (\forall (P:
95 Prop).P)))) (\lambda (b0: B).(let H_x \def (terms_props__bind_dec b b0) in
96 (let H \def H_x in (or_ind (eq B b b0) ((eq B b b0) \to (\forall (P:
97 Prop).P)) (or (eq K (Bind b) (Bind b0)) ((eq K (Bind b) (Bind b0)) \to
98 (\forall (P: Prop).P))) (\lambda (H0: (eq B b b0)).(eq_ind B b (\lambda (b1:
99 B).(or (eq K (Bind b) (Bind b1)) ((eq K (Bind b) (Bind b1)) \to (\forall (P:
100 Prop).P)))) (or_introl (eq K (Bind b) (Bind b)) ((eq K (Bind b) (Bind b)) \to
101 (\forall (P: Prop).P)) (refl_equal K (Bind b))) b0 H0)) (\lambda (H0: (((eq B
102 b b0) \to (\forall (P: Prop).P)))).(or_intror (eq K (Bind b) (Bind b0)) ((eq
103 K (Bind b) (Bind b0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq K (Bind b)
104 (Bind b0))).(\lambda (P: Prop).(let H2 \def (f_equal K B (\lambda (e:
105 K).(match e with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])) (Bind
106 b) (Bind b0) H1) in (let H3 \def (eq_ind_r B b0 (\lambda (b1: B).((eq B b b1)
107 \to (\forall (P0: Prop).P0))) H0 b H2) in (H3 (refl_equal B b) P))))))) H))))
108 (\lambda (f: F).(or_intror (eq K (Bind b) (Flat f)) ((eq K (Bind b) (Flat f))
109 \to (\forall (P: Prop).P)) (\lambda (H: (eq K (Bind b) (Flat f))).(\lambda
110 (P: Prop).(let H0 \def (eq_ind K (Bind b) (\lambda (ee: K).(match ee with
111 [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])) I (Flat f) H) in
112 (False_ind P H0)))))) k2))) (\lambda (f: F).(\lambda (k2: K).(K_ind (\lambda
113 (k: K).(or (eq K (Flat f) k) ((eq K (Flat f) k) \to (\forall (P: Prop).P))))
114 (\lambda (b: B).(or_intror (eq K (Flat f) (Bind b)) ((eq K (Flat f) (Bind b))
115 \to (\forall (P: Prop).P)) (\lambda (H: (eq K (Flat f) (Bind b))).(\lambda
116 (P: Prop).(let H0 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee with
117 [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) I (Bind b) H) in
118 (False_ind P H0)))))) (\lambda (f0: F).(let H_x \def (terms_props__flat_dec f
119 f0) in (let H \def H_x in (or_ind (eq F f f0) ((eq F f f0) \to (\forall (P:
120 Prop).P)) (or (eq K (Flat f) (Flat f0)) ((eq K (Flat f) (Flat f0)) \to
121 (\forall (P: Prop).P))) (\lambda (H0: (eq F f f0)).(eq_ind F f (\lambda (f1:
122 F).(or (eq K (Flat f) (Flat f1)) ((eq K (Flat f) (Flat f1)) \to (\forall (P:
123 Prop).P)))) (or_introl (eq K (Flat f) (Flat f)) ((eq K (Flat f) (Flat f)) \to
124 (\forall (P: Prop).P)) (refl_equal K (Flat f))) f0 H0)) (\lambda (H0: (((eq F
125 f f0) \to (\forall (P: Prop).P)))).(or_intror (eq K (Flat f) (Flat f0)) ((eq
126 K (Flat f) (Flat f0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq K (Flat f)
127 (Flat f0))).(\lambda (P: Prop).(let H2 \def (f_equal K F (\lambda (e:
128 K).(match e with [(Bind _) \Rightarrow f | (Flat f1) \Rightarrow f1])) (Flat
129 f) (Flat f0) H1) in (let H3 \def (eq_ind_r F f0 (\lambda (f1: F).((eq F f f1)
130 \to (\forall (P0: Prop).P0))) H0 f H2) in (H3 (refl_equal F f) P))))))) H))))
134 \forall (t1: T).(\forall (t2: T).(or (eq T t1 t2) ((eq T t1 t2) \to (\forall
137 \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (t2: T).(or (eq T t t2) ((eq
138 T t t2) \to (\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (t2:
139 T).(T_ind (\lambda (t: T).(or (eq T (TSort n) t) ((eq T (TSort n) t) \to
140 (\forall (P: Prop).P)))) (\lambda (n0: nat).(let H_x \def (nat_dec n n0) in
141 (let H \def H_x in (or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P:
142 Prop).P)) (or (eq T (TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to
143 (\forall (P: Prop).P))) (\lambda (H0: (eq nat n n0)).(eq_ind nat n (\lambda
144 (n1: nat).(or (eq T (TSort n) (TSort n1)) ((eq T (TSort n) (TSort n1)) \to
145 (\forall (P: Prop).P)))) (or_introl (eq T (TSort n) (TSort n)) ((eq T (TSort
146 n) (TSort n)) \to (\forall (P: Prop).P)) (refl_equal T (TSort n))) n0 H0))
147 (\lambda (H0: (((eq nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T
148 (TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to (\forall (P: Prop).P))
149 (\lambda (H1: (eq T (TSort n) (TSort n0))).(\lambda (P: Prop).(let H2 \def
150 (f_equal T nat (\lambda (e: T).(match e with [(TSort n1) \Rightarrow n1 |
151 (TLRef _) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TSort n) (TSort n0)
152 H1) in (let H3 \def (eq_ind_r nat n0 (\lambda (n1: nat).((eq nat n n1) \to
153 (\forall (P0: Prop).P0))) H0 n H2) in (H3 (refl_equal nat n) P))))))) H))))
154 (\lambda (n0: nat).(or_intror (eq T (TSort n) (TLRef n0)) ((eq T (TSort n)
155 (TLRef n0)) \to (\forall (P: Prop).P)) (\lambda (H: (eq T (TSort n) (TLRef
156 n0))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda (ee:
157 T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
158 (THead _ _ _) \Rightarrow False])) I (TLRef n0) H) in (False_ind P H0))))))
159 (\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T (TSort n) t) ((eq T
160 (TSort n) t) \to (\forall (P: Prop).P)))).(\lambda (t0: T).(\lambda (_: (or
161 (eq T (TSort n) t0) ((eq T (TSort n) t0) \to (\forall (P:
162 Prop).P)))).(or_intror (eq T (TSort n) (THead k t t0)) ((eq T (TSort n)
163 (THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TSort n)
164 (THead k t t0))).(\lambda (P: Prop).(let H2 \def (eq_ind T (TSort n) (\lambda
165 (ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow
166 False | (THead _ _ _) \Rightarrow False])) I (THead k t t0) H1) in (False_ind
167 P H2)))))))))) t2))) (\lambda (n: nat).(\lambda (t2: T).(T_ind (\lambda (t:
168 T).(or (eq T (TLRef n) t) ((eq T (TLRef n) t) \to (\forall (P: Prop).P))))
169 (\lambda (n0: nat).(or_intror (eq T (TLRef n) (TSort n0)) ((eq T (TLRef n)
170 (TSort n0)) \to (\forall (P: Prop).P)) (\lambda (H: (eq T (TLRef n) (TSort
171 n0))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TLRef n) (\lambda (ee:
172 T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
173 (THead _ _ _) \Rightarrow False])) I (TSort n0) H) in (False_ind P H0))))))
174 (\lambda (n0: nat).(let H_x \def (nat_dec n n0) in (let H \def H_x in (or_ind
175 (eq nat n n0) ((eq nat n n0) \to (\forall (P: Prop).P)) (or (eq T (TLRef n)
176 (TLRef n0)) ((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P))) (\lambda
177 (H0: (eq nat n n0)).(eq_ind nat n (\lambda (n1: nat).(or (eq T (TLRef n)
178 (TLRef n1)) ((eq T (TLRef n) (TLRef n1)) \to (\forall (P: Prop).P))))
179 (or_introl (eq T (TLRef n) (TLRef n)) ((eq T (TLRef n) (TLRef n)) \to
180 (\forall (P: Prop).P)) (refl_equal T (TLRef n))) n0 H0)) (\lambda (H0: (((eq
181 nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T (TLRef n) (TLRef n0))
182 ((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T
183 (TLRef n) (TLRef n0))).(\lambda (P: Prop).(let H2 \def (f_equal T nat
184 (\lambda (e: T).(match e with [(TSort _) \Rightarrow n | (TLRef n1)
185 \Rightarrow n1 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef n0) H1) in
186 (let H3 \def (eq_ind_r nat n0 (\lambda (n1: nat).((eq nat n n1) \to (\forall
187 (P0: Prop).P0))) H0 n H2) in (H3 (refl_equal nat n) P))))))) H)))) (\lambda
188 (k: K).(\lambda (t: T).(\lambda (_: (or (eq T (TLRef n) t) ((eq T (TLRef n)
189 t) \to (\forall (P: Prop).P)))).(\lambda (t0: T).(\lambda (_: (or (eq T
190 (TLRef n) t0) ((eq T (TLRef n) t0) \to (\forall (P: Prop).P)))).(or_intror
191 (eq T (TLRef n) (THead k t t0)) ((eq T (TLRef n) (THead k t t0)) \to (\forall
192 (P: Prop).P)) (\lambda (H1: (eq T (TLRef n) (THead k t t0))).(\lambda (P:
193 Prop).(let H2 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee with
194 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
195 \Rightarrow False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2)))
196 (\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).(or (eq T t
197 t2) ((eq T t t2) \to (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda
198 (H0: ((\forall (t2: T).(or (eq T t0 t2) ((eq T t0 t2) \to (\forall (P:
199 Prop).P)))))).(\lambda (t2: T).(T_ind (\lambda (t3: T).(or (eq T (THead k t
200 t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (n:
201 nat).(or_intror (eq T (THead k t t0) (TSort n)) ((eq T (THead k t t0) (TSort
202 n)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TSort
203 n))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee:
204 T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
205 | (THead _ _ _) \Rightarrow True])) I (TSort n) H1) in (False_ind P H2))))))
206 (\lambda (n: nat).(or_intror (eq T (THead k t t0) (TLRef n)) ((eq T (THead k
207 t t0) (TLRef n)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (THead k t
208 t0) (TLRef n))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead k t t0)
209 (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
210 \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in
211 (False_ind P H2)))))) (\lambda (k0: K).(\lambda (t3: T).(\lambda (H1: (or (eq
212 T (THead k t t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P:
213 Prop).P)))).(\lambda (t4: T).(\lambda (H2: (or (eq T (THead k t t0) t4) ((eq
214 T (THead k t t0) t4) \to (\forall (P: Prop).P)))).(let H_x \def (H t3) in
215 (let H3 \def H_x in (or_ind (eq T t t3) ((eq T t t3) \to (\forall (P:
216 Prop).P)) (or (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k t t0)
217 (THead k0 t3 t4)) \to (\forall (P: Prop).P))) (\lambda (H4: (eq T t t3)).(let
218 H5 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T
219 (THead k t t0) t5) \to (\forall (P: Prop).P)))) H1 t H4) in (eq_ind T t
220 (\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t5 t4)) ((eq T (THead k t
221 t0) (THead k0 t5 t4)) \to (\forall (P: Prop).P)))) (let H_x0 \def (H0 t4) in
222 (let H6 \def H_x0 in (or_ind (eq T t0 t4) ((eq T t0 t4) \to (\forall (P:
223 Prop).P)) (or (eq T (THead k t t0) (THead k0 t t4)) ((eq T (THead k t t0)
224 (THead k0 t t4)) \to (\forall (P: Prop).P))) (\lambda (H7: (eq T t0 t4)).(let
225 H8 \def (eq_ind_r T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T
226 (THead k t t0) t5) \to (\forall (P: Prop).P)))) H2 t0 H7) in (eq_ind T t0
227 (\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t t5)) ((eq T (THead k t
228 t0) (THead k0 t t5)) \to (\forall (P: Prop).P)))) (let H_x1 \def
229 (terms_props__kind_dec k k0) in (let H9 \def H_x1 in (or_ind (eq K k k0) ((eq
230 K k k0) \to (\forall (P: Prop).P)) (or (eq T (THead k t t0) (THead k0 t t0))
231 ((eq T (THead k t t0) (THead k0 t t0)) \to (\forall (P: Prop).P))) (\lambda
232 (H10: (eq K k k0)).(eq_ind K k (\lambda (k1: K).(or (eq T (THead k t t0)
233 (THead k1 t t0)) ((eq T (THead k t t0) (THead k1 t t0)) \to (\forall (P:
234 Prop).P)))) (or_introl (eq T (THead k t t0) (THead k t t0)) ((eq T (THead k t
235 t0) (THead k t t0)) \to (\forall (P: Prop).P)) (refl_equal T (THead k t t0)))
236 k0 H10)) (\lambda (H10: (((eq K k k0) \to (\forall (P: Prop).P)))).(or_intror
237 (eq T (THead k t t0) (THead k0 t t0)) ((eq T (THead k t t0) (THead k0 t t0))
238 \to (\forall (P: Prop).P)) (\lambda (H11: (eq T (THead k t t0) (THead k0 t
239 t0))).(\lambda (P: Prop).(let H12 \def (f_equal T K (\lambda (e: T).(match e
240 with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _)
241 \Rightarrow k1])) (THead k t t0) (THead k0 t t0) H11) in (let H13 \def
242 (eq_ind_r K k0 (\lambda (k1: K).((eq K k k1) \to (\forall (P0: Prop).P0)))
243 H10 k H12) in (H13 (refl_equal K k) P))))))) H9))) t4 H7))) (\lambda (H7:
244 (((eq T t0 t4) \to (\forall (P: Prop).P)))).(or_intror (eq T (THead k t t0)
245 (THead k0 t t4)) ((eq T (THead k t t0) (THead k0 t t4)) \to (\forall (P:
246 Prop).P)) (\lambda (H8: (eq T (THead k t t0) (THead k0 t t4))).(\lambda (P:
247 Prop).(let H9 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _)
248 \Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1]))
249 (THead k t t0) (THead k0 t t4) H8) in ((let H10 \def (f_equal T T (\lambda
250 (e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 |
251 (THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t t4) H8) in
252 (\lambda (_: (eq K k k0)).(let H12 \def (eq_ind_r T t4 (\lambda (t5: T).((eq
253 T t0 t5) \to (\forall (P0: Prop).P0))) H7 t0 H10) in (let H13 \def (eq_ind_r
254 T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k t t0) t5)
255 \to (\forall (P0: Prop).P0)))) H2 t0 H10) in (H12 (refl_equal T t0) P)))))
256 H9)))))) H6))) t3 H4))) (\lambda (H4: (((eq T t t3) \to (\forall (P:
257 Prop).P)))).(or_intror (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k
258 t t0) (THead k0 t3 t4)) \to (\forall (P: Prop).P)) (\lambda (H5: (eq T (THead
259 k t t0) (THead k0 t3 t4))).(\lambda (P: Prop).(let H6 \def (f_equal T K
260 (\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _)
261 \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t3
262 t4) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort
263 _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t5 _) \Rightarrow t5]))
264 (THead k t t0) (THead k0 t3 t4) H5) in ((let H8 \def (f_equal T T (\lambda
265 (e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 |
266 (THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t3 t4) H5) in
267 (\lambda (H9: (eq T t t3)).(\lambda (_: (eq K k k0)).(let H11 \def (eq_ind_r
268 T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k t t0) t5)
269 \to (\forall (P0: Prop).P0)))) H2 t0 H8) in (let H12 \def (eq_ind_r T t3
270 (\lambda (t5: T).((eq T t t5) \to (\forall (P0: Prop).P0))) H4 t H9) in (let
271 H13 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T
272 (THead k t t0) t5) \to (\forall (P0: Prop).P0)))) H1 t H9) in (H12
273 (refl_equal T t) P))))))) H7)) H6)))))) H3)))))))) t2))))))) t1).
276 \forall (t: T).(or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u:
277 T).(eq T t (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall
278 (u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))
280 \lambda (t: T).(T_ind (\lambda (t0: T).(or (ex_3 B T T (\lambda (b:
281 B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u))))))
282 (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w
283 u)) \to (\forall (P: Prop).P))))))) (\lambda (n: nat).(or_intror (ex_3 B T T
284 (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (TSort n) (THead (Bind
285 b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (TSort n)
286 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda
287 (w: T).(\lambda (u: T).(\lambda (H: (eq T (TSort n) (THead (Bind b) w
288 u))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda (ee:
289 T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
290 (THead _ _ _) \Rightarrow False])) I (THead (Bind b) w u) H) in (False_ind P
291 H0))))))))) (\lambda (n: nat).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda
292 (w: T).(\lambda (u: T).(eq T (TLRef n) (THead (Bind b) w u)))))) (\forall (b:
293 B).(\forall (w: T).(\forall (u: T).((eq T (TLRef n) (THead (Bind b) w u)) \to
294 (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (w: T).(\lambda (u:
295 T).(\lambda (H: (eq T (TLRef n) (THead (Bind b) w u))).(\lambda (P:
296 Prop).(let H0 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee with
297 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
298 \Rightarrow False])) I (THead (Bind b) w u) H) in (False_ind P H0)))))))))
299 (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t0: T).((or (ex_3 B T T
300 (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w
301 u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead
302 (Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (\forall (t1: T).((or (ex_3
303 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind
304 b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead
305 (Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (or (ex_3 B T T (\lambda
306 (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead k0 t0 t1) (THead (Bind b)
307 w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead k0 t0
308 t1) (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))))))) (\lambda (b:
309 B).(\lambda (t0: T).(\lambda (_: (or (ex_3 B T T (\lambda (b0: B).(\lambda
310 (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b0) w u)))))) (\forall (b0:
311 B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b0) w u)) \to
312 (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (_: (or (ex_3 B T T
313 (\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind b0) w
314 u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead
315 (Bind b0) w u)) \to (\forall (P: Prop).P))))))).(or_introl (ex_3 B T T
316 (\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead (Bind b) t0 t1)
317 (THead (Bind b0) w u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u:
318 T).((eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u)) \to (\forall (P:
319 Prop).P))))) (ex_3_intro B T T (\lambda (b0: B).(\lambda (w: T).(\lambda (u:
320 T).(eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u))))) b t0 t1 (refl_equal
321 T (THead (Bind b) t0 t1))))))))) (\lambda (f: F).(\lambda (t0: T).(\lambda
322 (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0
323 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u:
324 T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(\lambda
325 (t1: T).(\lambda (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda
326 (u: T).(eq T t1 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w:
327 T).(\forall (u: T).((eq T t1 (THead (Bind b) w u)) \to (\forall (P:
328 Prop).P))))))).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w:
329 T).(\lambda (u: T).(eq T (THead (Flat f) t0 t1) (THead (Bind b) w u))))))
330 (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead (Flat f) t0 t1)
331 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda
332 (w: T).(\lambda (u: T).(\lambda (H1: (eq T (THead (Flat f) t0 t1) (THead
333 (Bind b) w u))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead (Flat f) t0
334 t1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
335 \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with [(Bind _)
336 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) w u) H1)
337 in (False_ind P H2))))))))))))) k)) t).
340 \forall (u: T).(\forall (v: T).(or (ex T (\lambda (t: T).(eq T u (THead
341 (Bind Abst) v t)))) (\forall (t: T).((eq T u (THead (Bind Abst) v t)) \to
342 (\forall (P: Prop).P)))))
344 \lambda (u: T).(T_ind (\lambda (t: T).(\forall (v: T).(or (ex T (\lambda
345 (t0: T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead
346 (Bind Abst) v t0)) \to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda
347 (v: T).(or_intror (ex T (\lambda (t: T).(eq T (TSort n) (THead (Bind Abst) v
348 t)))) (\forall (t: T).((eq T (TSort n) (THead (Bind Abst) v t)) \to (\forall
349 (P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TSort n) (THead (Bind
350 Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda
351 (ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow
352 False | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) v t) H) in
353 (False_ind P H0)))))))) (\lambda (n: nat).(\lambda (v: T).(or_intror (ex T
354 (\lambda (t: T).(eq T (TLRef n) (THead (Bind Abst) v t)))) (\forall (t:
355 T).((eq T (TLRef n) (THead (Bind Abst) v t)) \to (\forall (P: Prop).P)))
356 (\lambda (t: T).(\lambda (H: (eq T (TLRef n) (THead (Bind Abst) v
357 t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TLRef n) (\lambda (ee:
358 T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
359 (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) v t) H) in (False_ind
360 P H0)))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: ((\forall (v:
361 T).(or (ex T (\lambda (t0: T).(eq T t (THead (Bind Abst) v t0)))) (\forall
362 (t0: T).((eq T t (THead (Bind Abst) v t0)) \to (\forall (P:
363 Prop).P))))))).(\lambda (t0: T).(\lambda (_: ((\forall (v: T).(or (ex T
364 (\lambda (t1: T).(eq T t0 (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T
365 t0 (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))))))).(\lambda (v:
366 T).(let H_x \def (terms_props__kind_dec k (Bind Abst)) in (let H1 \def H_x in
367 (or_ind (eq K k (Bind Abst)) ((eq K k (Bind Abst)) \to (\forall (P: Prop).P))
368 (or (ex T (\lambda (t1: T).(eq T (THead k t t0) (THead (Bind Abst) v t1))))
369 (\forall (t1: T).((eq T (THead k t t0) (THead (Bind Abst) v t1)) \to (\forall
370 (P: Prop).P)))) (\lambda (H2: (eq K k (Bind Abst))).(eq_ind_r K (Bind Abst)
371 (\lambda (k0: K).(or (ex T (\lambda (t1: T).(eq T (THead k0 t t0) (THead
372 (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k0 t t0) (THead (Bind
373 Abst) v t1)) \to (\forall (P: Prop).P))))) (let H_x0 \def (term_dec t v) in
374 (let H3 \def H_x0 in (or_ind (eq T t v) ((eq T t v) \to (\forall (P:
375 Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) (THead
376 (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) (THead
377 (Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda (H4: (eq T t
378 v)).(eq_ind T t (\lambda (t1: T).(or (ex T (\lambda (t2: T).(eq T (THead
379 (Bind Abst) t t0) (THead (Bind Abst) t1 t2)))) (\forall (t2: T).((eq T (THead
380 (Bind Abst) t t0) (THead (Bind Abst) t1 t2)) \to (\forall (P: Prop).P)))))
381 (or_introl (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind
382 Abst) t t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) (THead (Bind
383 Abst) t t1)) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t1: T).(eq T
384 (THead (Bind Abst) t t0) (THead (Bind Abst) t t1))) t0 (refl_equal T (THead
385 (Bind Abst) t t0)))) v H4)) (\lambda (H4: (((eq T t v) \to (\forall (P:
386 Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0)
387 (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0)
388 (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1:
389 T).(\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) v
390 t1))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: T).(match e
391 with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t2 _)
392 \Rightarrow t2])) (THead (Bind Abst) t t0) (THead (Bind Abst) v t1) H5) in
393 ((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _)
394 \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2]))
395 (THead (Bind Abst) t t0) (THead (Bind Abst) v t1) H5) in (\lambda (H8: (eq T
396 t v)).(H4 H8 P))) H6))))))) H3))) k H2)) (\lambda (H2: (((eq K k (Bind Abst))
397 \to (\forall (P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead k
398 t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0)
399 (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1:
400 T).(\lambda (H3: (eq T (THead k t t0) (THead (Bind Abst) v t1))).(\lambda (P:
401 Prop).(let H4 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _)
402 \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0]))
403 (THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H5 \def (f_equal T T
404 (\lambda (e: T).(match e with [(TSort _) \Rightarrow t | (TLRef _)
405 \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead k t t0) (THead (Bind
406 Abst) v t1) H3) in ((let H6 \def (f_equal T T (\lambda (e: T).(match e with
407 [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2)
408 \Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in (\lambda (_:
409 (eq T t v)).(\lambda (H8: (eq K k (Bind Abst))).(H2 H8 P)))) H5)) H4)))))))