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7 (* ||T|| The HELM team. *)
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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/T/dec".
21 theorem terms_props__bind_dec:
22 \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) ((eq B b1 b2) \to (\forall
25 \lambda (b1: B).(B_ind (\lambda (b: B).(\forall (b2: B).(or (eq B b b2) ((eq
26 B b b2) \to (\forall (P: Prop).P))))) (\lambda (b2: B).(B_ind (\lambda (b:
27 B).(or (eq B Abbr b) ((eq B Abbr b) \to (\forall (P: Prop).P)))) (or_introl
28 (eq B Abbr Abbr) ((eq B Abbr Abbr) \to (\forall (P: Prop).P)) (refl_equal B
29 Abbr)) (or_intror (eq B Abbr Abst) ((eq B Abbr Abst) \to (\forall (P:
30 Prop).P)) (\lambda (H: (eq B Abbr Abst)).(\lambda (P: Prop).(let H0 \def
31 (eq_ind B Abbr (\lambda (ee: B).(match ee in B return (\lambda (_: B).Prop)
32 with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow
33 False])) I Abst H) in (False_ind P H0))))) (or_intror (eq B Abbr Void) ((eq B
34 Abbr Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Abbr Void)).(\lambda
35 (P: Prop).(let H0 \def (eq_ind B Abbr (\lambda (ee: B).(match ee in B return
36 (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False |
37 Void \Rightarrow False])) I Void H) in (False_ind P H0))))) b2)) (\lambda
38 (b2: B).(B_ind (\lambda (b: B).(or (eq B Abst b) ((eq B Abst b) \to (\forall
39 (P: Prop).P)))) (or_intror (eq B Abst Abbr) ((eq B Abst Abbr) \to (\forall
40 (P: Prop).P)) (\lambda (H: (eq B Abst Abbr)).(\lambda (P: Prop).(let H0 \def
41 (eq_ind B Abst (\lambda (ee: B).(match ee in B return (\lambda (_: B).Prop)
42 with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow
43 False])) I Abbr H) in (False_ind P H0))))) (or_introl (eq B Abst Abst) ((eq B
44 Abst Abst) \to (\forall (P: Prop).P)) (refl_equal B Abst)) (or_intror (eq B
45 Abst Void) ((eq B Abst Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B
46 Abst Void)).(\lambda (P: Prop).(let H0 \def (eq_ind B Abst (\lambda (ee:
47 B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False |
48 Abst \Rightarrow True | Void \Rightarrow False])) I Void H) in (False_ind P
49 H0))))) b2)) (\lambda (b2: B).(B_ind (\lambda (b: B).(or (eq B Void b) ((eq B
50 Void b) \to (\forall (P: Prop).P)))) (or_intror (eq B Void Abbr) ((eq B Void
51 Abbr) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Void Abbr)).(\lambda (P:
52 Prop).(let H0 \def (eq_ind B Void (\lambda (ee: B).(match ee in B return
53 (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False |
54 Void \Rightarrow True])) I Abbr H) in (False_ind P H0))))) (or_intror (eq B
55 Void Abst) ((eq B Void Abst) \to (\forall (P: Prop).P)) (\lambda (H: (eq B
56 Void Abst)).(\lambda (P: Prop).(let H0 \def (eq_ind B Void (\lambda (ee:
57 B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False |
58 Abst \Rightarrow False | Void \Rightarrow True])) I Abst H) in (False_ind P
59 H0))))) (or_introl (eq B Void Void) ((eq B Void Void) \to (\forall (P:
60 Prop).P)) (refl_equal B Void)) b2)) b1).
63 \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) (not (eq B b1 b2))))
65 \lambda (b1: B).(\lambda (b2: B).(let H_x \def (terms_props__bind_dec b1 b2)
66 in (let H \def H_x in (or_ind (eq B b1 b2) ((eq B b1 b2) \to (\forall (P:
67 Prop).P)) (or (eq B b1 b2) ((eq B b1 b2) \to False)) (\lambda (H0: (eq B b1
68 b2)).(or_introl (eq B b1 b2) ((eq B b1 b2) \to False) H0)) (\lambda (H0:
69 (((eq B b1 b2) \to (\forall (P: Prop).P)))).(or_intror (eq B b1 b2) ((eq B b1
70 b2) \to False) (\lambda (H1: (eq B b1 b2)).(H0 H1 False)))) H)))).
72 theorem terms_props__flat_dec:
73 \forall (f1: F).(\forall (f2: F).(or (eq F f1 f2) ((eq F f1 f2) \to (\forall
76 \lambda (f1: F).(F_ind (\lambda (f: F).(\forall (f2: F).(or (eq F f f2) ((eq
77 F f f2) \to (\forall (P: Prop).P))))) (\lambda (f2: F).(F_ind (\lambda (f:
78 F).(or (eq F Appl f) ((eq F Appl f) \to (\forall (P: Prop).P)))) (or_introl
79 (eq F Appl Appl) ((eq F Appl Appl) \to (\forall (P: Prop).P)) (refl_equal F
80 Appl)) (or_intror (eq F Appl Cast) ((eq F Appl Cast) \to (\forall (P:
81 Prop).P)) (\lambda (H: (eq F Appl Cast)).(\lambda (P: Prop).(let H0 \def
82 (eq_ind F Appl (\lambda (ee: F).(match ee in F return (\lambda (_: F).Prop)
83 with [Appl \Rightarrow True | Cast \Rightarrow False])) I Cast H) in
84 (False_ind P H0))))) f2)) (\lambda (f2: F).(F_ind (\lambda (f: F).(or (eq F
85 Cast f) ((eq F Cast f) \to (\forall (P: Prop).P)))) (or_intror (eq F Cast
86 Appl) ((eq F Cast Appl) \to (\forall (P: Prop).P)) (\lambda (H: (eq F Cast
87 Appl)).(\lambda (P: Prop).(let H0 \def (eq_ind F Cast (\lambda (ee: F).(match
88 ee in F return (\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast
89 \Rightarrow True])) I Appl H) in (False_ind P H0))))) (or_introl (eq F Cast
90 Cast) ((eq F Cast Cast) \to (\forall (P: Prop).P)) (refl_equal F Cast)) f2))
93 theorem terms_props__kind_dec:
94 \forall (k1: K).(\forall (k2: K).(or (eq K k1 k2) ((eq K k1 k2) \to (\forall
97 \lambda (k1: K).(K_ind (\lambda (k: K).(\forall (k2: K).(or (eq K k k2) ((eq
98 K k k2) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (k2: K).(K_ind
99 (\lambda (k: K).(or (eq K (Bind b) k) ((eq K (Bind b) k) \to (\forall (P:
100 Prop).P)))) (\lambda (b0: B).(let H_x \def (terms_props__bind_dec b b0) in
101 (let H \def H_x in (or_ind (eq B b b0) ((eq B b b0) \to (\forall (P:
102 Prop).P)) (or (eq K (Bind b) (Bind b0)) ((eq K (Bind b) (Bind b0)) \to
103 (\forall (P: Prop).P))) (\lambda (H0: (eq B b b0)).(eq_ind B b (\lambda (b1:
104 B).(or (eq K (Bind b) (Bind b1)) ((eq K (Bind b) (Bind b1)) \to (\forall (P:
105 Prop).P)))) (or_introl (eq K (Bind b) (Bind b)) ((eq K (Bind b) (Bind b)) \to
106 (\forall (P: Prop).P)) (refl_equal K (Bind b))) b0 H0)) (\lambda (H0: (((eq B
107 b b0) \to (\forall (P: Prop).P)))).(or_intror (eq K (Bind b) (Bind b0)) ((eq
108 K (Bind b) (Bind b0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq K (Bind b)
109 (Bind b0))).(\lambda (P: Prop).(let H2 \def (f_equal K B (\lambda (e:
110 K).(match e in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 |
111 (Flat _) \Rightarrow b])) (Bind b) (Bind b0) H1) in (let H3 \def (eq_ind_r B
112 b0 (\lambda (b1: B).((eq B b b1) \to (\forall (P0: Prop).P0))) H0 b H2) in
113 (H3 (refl_equal B b) P))))))) H)))) (\lambda (f: F).(or_intror (eq K (Bind b)
114 (Flat f)) ((eq K (Bind b) (Flat f)) \to (\forall (P: Prop).P)) (\lambda (H:
115 (eq K (Bind b) (Flat f))).(\lambda (P: Prop).(let H0 \def (eq_ind K (Bind b)
116 (\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with [(Bind _)
117 \Rightarrow True | (Flat _) \Rightarrow False])) I (Flat f) H) in (False_ind
118 P H0)))))) k2))) (\lambda (f: F).(\lambda (k2: K).(K_ind (\lambda (k: K).(or
119 (eq K (Flat f) k) ((eq K (Flat f) k) \to (\forall (P: Prop).P)))) (\lambda
120 (b: B).(or_intror (eq K (Flat f) (Bind b)) ((eq K (Flat f) (Bind b)) \to
121 (\forall (P: Prop).P)) (\lambda (H: (eq K (Flat f) (Bind b))).(\lambda (P:
122 Prop).(let H0 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return
123 (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
124 True])) I (Bind b) H) in (False_ind P H0)))))) (\lambda (f0: F).(let H_x \def
125 (terms_props__flat_dec f f0) in (let H \def H_x in (or_ind (eq F f f0) ((eq F
126 f f0) \to (\forall (P: Prop).P)) (or (eq K (Flat f) (Flat f0)) ((eq K (Flat
127 f) (Flat f0)) \to (\forall (P: Prop).P))) (\lambda (H0: (eq F f f0)).(eq_ind
128 F f (\lambda (f1: F).(or (eq K (Flat f) (Flat f1)) ((eq K (Flat f) (Flat f1))
129 \to (\forall (P: Prop).P)))) (or_introl (eq K (Flat f) (Flat f)) ((eq K (Flat
130 f) (Flat f)) \to (\forall (P: Prop).P)) (refl_equal K (Flat f))) f0 H0))
131 (\lambda (H0: (((eq F f f0) \to (\forall (P: Prop).P)))).(or_intror (eq K
132 (Flat f) (Flat f0)) ((eq K (Flat f) (Flat f0)) \to (\forall (P: Prop).P))
133 (\lambda (H1: (eq K (Flat f) (Flat f0))).(\lambda (P: Prop).(let H2 \def
134 (f_equal K F (\lambda (e: K).(match e in K return (\lambda (_: K).F) with
135 [(Bind _) \Rightarrow f | (Flat f1) \Rightarrow f1])) (Flat f) (Flat f0) H1)
136 in (let H3 \def (eq_ind_r F f0 (\lambda (f1: F).((eq F f f1) \to (\forall
137 (P0: Prop).P0))) H0 f H2) in (H3 (refl_equal F f) P))))))) H)))) k2))) k1).
140 \forall (t1: T).(\forall (t2: T).(or (eq T t1 t2) ((eq T t1 t2) \to (\forall
143 \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (t2: T).(or (eq T t t2) ((eq
144 T t t2) \to (\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (t2:
145 T).(T_ind (\lambda (t: T).(or (eq T (TSort n) t) ((eq T (TSort n) t) \to
146 (\forall (P: Prop).P)))) (\lambda (n0: nat).(let H_x \def (nat_dec n n0) in
147 (let H \def H_x in (or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P:
148 Prop).P)) (or (eq T (TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to
149 (\forall (P: Prop).P))) (\lambda (H0: (eq nat n n0)).(eq_ind nat n (\lambda
150 (n1: nat).(or (eq T (TSort n) (TSort n1)) ((eq T (TSort n) (TSort n1)) \to
151 (\forall (P: Prop).P)))) (or_introl (eq T (TSort n) (TSort n)) ((eq T (TSort
152 n) (TSort n)) \to (\forall (P: Prop).P)) (refl_equal T (TSort n))) n0 H0))
153 (\lambda (H0: (((eq nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T
154 (TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to (\forall (P: Prop).P))
155 (\lambda (H1: (eq T (TSort n) (TSort n0))).(\lambda (P: Prop).(let H2 \def
156 (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
157 [(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n | (THead _ _ _)
158 \Rightarrow n])) (TSort n) (TSort n0) H1) in (let H3 \def (eq_ind_r nat n0
159 (\lambda (n1: nat).((eq nat n n1) \to (\forall (P0: Prop).P0))) H0 n H2) in
160 (H3 (refl_equal nat n) P))))))) H)))) (\lambda (n0: nat).(or_intror (eq T
161 (TSort n) (TLRef n0)) ((eq T (TSort n) (TLRef n0)) \to (\forall (P: Prop).P))
162 (\lambda (H: (eq T (TSort n) (TLRef n0))).(\lambda (P: Prop).(let H0 \def
163 (eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_:
164 T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
165 (THead _ _ _) \Rightarrow False])) I (TLRef n0) H) in (False_ind P H0))))))
166 (\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T (TSort n) t) ((eq T
167 (TSort n) t) \to (\forall (P: Prop).P)))).(\lambda (t0: T).(\lambda (_: (or
168 (eq T (TSort n) t0) ((eq T (TSort n) t0) \to (\forall (P:
169 Prop).P)))).(or_intror (eq T (TSort n) (THead k t t0)) ((eq T (TSort n)
170 (THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TSort n)
171 (THead k t t0))).(\lambda (P: Prop).(let H2 \def (eq_ind T (TSort n) (\lambda
172 (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
173 \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
174 False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2))) (\lambda (n:
175 nat).(\lambda (t2: T).(T_ind (\lambda (t: T).(or (eq T (TLRef n) t) ((eq T
176 (TLRef n) t) \to (\forall (P: Prop).P)))) (\lambda (n0: nat).(or_intror (eq T
177 (TLRef n) (TSort n0)) ((eq T (TLRef n) (TSort n0)) \to (\forall (P: Prop).P))
178 (\lambda (H: (eq T (TLRef n) (TSort n0))).(\lambda (P: Prop).(let H0 \def
179 (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_:
180 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
181 (THead _ _ _) \Rightarrow False])) I (TSort n0) H) in (False_ind P H0))))))
182 (\lambda (n0: nat).(let H_x \def (nat_dec n n0) in (let H \def H_x in (or_ind
183 (eq nat n n0) ((eq nat n n0) \to (\forall (P: Prop).P)) (or (eq T (TLRef n)
184 (TLRef n0)) ((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P))) (\lambda
185 (H0: (eq nat n n0)).(eq_ind nat n (\lambda (n1: nat).(or (eq T (TLRef n)
186 (TLRef n1)) ((eq T (TLRef n) (TLRef n1)) \to (\forall (P: Prop).P))))
187 (or_introl (eq T (TLRef n) (TLRef n)) ((eq T (TLRef n) (TLRef n)) \to
188 (\forall (P: Prop).P)) (refl_equal T (TLRef n))) n0 H0)) (\lambda (H0: (((eq
189 nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T (TLRef n) (TLRef n0))
190 ((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T
191 (TLRef n) (TLRef n0))).(\lambda (P: Prop).(let H2 \def (f_equal T nat
192 (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _)
193 \Rightarrow n | (TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n]))
194 (TLRef n) (TLRef n0) H1) in (let H3 \def (eq_ind_r nat n0 (\lambda (n1:
195 nat).((eq nat n n1) \to (\forall (P0: Prop).P0))) H0 n H2) in (H3 (refl_equal
196 nat n) P))))))) H)))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T
197 (TLRef n) t) ((eq T (TLRef n) t) \to (\forall (P: Prop).P)))).(\lambda (t0:
198 T).(\lambda (_: (or (eq T (TLRef n) t0) ((eq T (TLRef n) t0) \to (\forall (P:
199 Prop).P)))).(or_intror (eq T (TLRef n) (THead k t t0)) ((eq T (TLRef n)
200 (THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TLRef n)
201 (THead k t t0))).(\lambda (P: Prop).(let H2 \def (eq_ind T (TLRef n) (\lambda
202 (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
203 \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
204 False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2))) (\lambda (k:
205 K).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).(or (eq T t t2) ((eq T t
206 t2) \to (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall
207 (t2: T).(or (eq T t0 t2) ((eq T t0 t2) \to (\forall (P:
208 Prop).P)))))).(\lambda (t2: T).(T_ind (\lambda (t3: T).(or (eq T (THead k t
209 t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (n:
210 nat).(or_intror (eq T (THead k t t0) (TSort n)) ((eq T (THead k t t0) (TSort
211 n)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TSort
212 n))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee:
213 T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
214 False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
215 (TSort n) H1) in (False_ind P H2)))))) (\lambda (n: nat).(or_intror (eq T
216 (THead k t t0) (TLRef n)) ((eq T (THead k t t0) (TLRef n)) \to (\forall (P:
217 Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TLRef n))).(\lambda (P:
218 Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee: T).(match ee in T
219 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
220 \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in
221 (False_ind P H2)))))) (\lambda (k0: K).(\lambda (t3: T).(\lambda (H1: (or (eq
222 T (THead k t t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P:
223 Prop).P)))).(\lambda (t4: T).(\lambda (H2: (or (eq T (THead k t t0) t4) ((eq
224 T (THead k t t0) t4) \to (\forall (P: Prop).P)))).(let H_x \def (H t3) in
225 (let H3 \def H_x in (or_ind (eq T t t3) ((eq T t t3) \to (\forall (P:
226 Prop).P)) (or (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k t t0)
227 (THead k0 t3 t4)) \to (\forall (P: Prop).P))) (\lambda (H4: (eq T t t3)).(let
228 H5 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T
229 (THead k t t0) t5) \to (\forall (P: Prop).P)))) H1 t H4) in (eq_ind T t
230 (\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t5 t4)) ((eq T (THead k t
231 t0) (THead k0 t5 t4)) \to (\forall (P: Prop).P)))) (let H_x0 \def (H0 t4) in
232 (let H6 \def H_x0 in (or_ind (eq T t0 t4) ((eq T t0 t4) \to (\forall (P:
233 Prop).P)) (or (eq T (THead k t t0) (THead k0 t t4)) ((eq T (THead k t t0)
234 (THead k0 t t4)) \to (\forall (P: Prop).P))) (\lambda (H7: (eq T t0 t4)).(let
235 H8 \def (eq_ind_r T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T
236 (THead k t t0) t5) \to (\forall (P: Prop).P)))) H2 t0 H7) in (eq_ind T t0
237 (\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t t5)) ((eq T (THead k t
238 t0) (THead k0 t t5)) \to (\forall (P: Prop).P)))) (let H_x1 \def
239 (terms_props__kind_dec k k0) in (let H9 \def H_x1 in (or_ind (eq K k k0) ((eq
240 K k k0) \to (\forall (P: Prop).P)) (or (eq T (THead k t t0) (THead k0 t t0))
241 ((eq T (THead k t t0) (THead k0 t t0)) \to (\forall (P: Prop).P))) (\lambda
242 (H10: (eq K k k0)).(eq_ind K k (\lambda (k1: K).(or (eq T (THead k t t0)
243 (THead k1 t t0)) ((eq T (THead k t t0) (THead k1 t t0)) \to (\forall (P:
244 Prop).P)))) (or_introl (eq T (THead k t t0) (THead k t t0)) ((eq T (THead k t
245 t0) (THead k t t0)) \to (\forall (P: Prop).P)) (refl_equal T (THead k t t0)))
246 k0 H10)) (\lambda (H10: (((eq K k k0) \to (\forall (P: Prop).P)))).(or_intror
247 (eq T (THead k t t0) (THead k0 t t0)) ((eq T (THead k t t0) (THead k0 t t0))
248 \to (\forall (P: Prop).P)) (\lambda (H11: (eq T (THead k t t0) (THead k0 t
249 t0))).(\lambda (P: Prop).(let H12 \def (f_equal T K (\lambda (e: T).(match e
250 in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
251 \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t
252 t0) H11) in (let H13 \def (eq_ind_r K k0 (\lambda (k1: K).((eq K k k1) \to
253 (\forall (P0: Prop).P0))) H10 k H12) in (H13 (refl_equal K k) P))))))) H9)))
254 t4 H7))) (\lambda (H7: (((eq T t0 t4) \to (\forall (P: Prop).P)))).(or_intror
255 (eq T (THead k t t0) (THead k0 t t4)) ((eq T (THead k t t0) (THead k0 t t4))
256 \to (\forall (P: Prop).P)) (\lambda (H8: (eq T (THead k t t0) (THead k0 t
257 t4))).(\lambda (P: Prop).(let H9 \def (f_equal T K (\lambda (e: T).(match e
258 in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
259 \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t
260 t4) H8) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e in T return
261 (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0
262 | (THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t t4) H8) in
263 (\lambda (_: (eq K k k0)).(let H12 \def (eq_ind_r T t4 (\lambda (t5: T).((eq
264 T t0 t5) \to (\forall (P0: Prop).P0))) H7 t0 H10) in (let H13 \def (eq_ind_r
265 T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k t t0) t5)
266 \to (\forall (P0: Prop).P0)))) H2 t0 H10) in (H12 (refl_equal T t0) P)))))
267 H9)))))) H6))) t3 H4))) (\lambda (H4: (((eq T t t3) \to (\forall (P:
268 Prop).P)))).(or_intror (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k
269 t t0) (THead k0 t3 t4)) \to (\forall (P: Prop).P)) (\lambda (H5: (eq T (THead
270 k t t0) (THead k0 t3 t4))).(\lambda (P: Prop).(let H6 \def (f_equal T K
271 (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
272 \Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1]))
273 (THead k t t0) (THead k0 t3 t4) H5) in ((let H7 \def (f_equal T T (\lambda
274 (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t
275 | (TLRef _) \Rightarrow t | (THead _ t5 _) \Rightarrow t5])) (THead k t t0)
276 (THead k0 t3 t4) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e
277 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
278 \Rightarrow t0 | (THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t3
279 t4) H5) in (\lambda (H9: (eq T t t3)).(\lambda (_: (eq K k k0)).(let H11 \def
280 (eq_ind_r T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k
281 t t0) t5) \to (\forall (P0: Prop).P0)))) H2 t0 H8) in (let H12 \def (eq_ind_r
282 T t3 (\lambda (t5: T).((eq T t t5) \to (\forall (P0: Prop).P0))) H4 t H9) in
283 (let H13 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5)
284 ((eq T (THead k t t0) t5) \to (\forall (P0: Prop).P0)))) H1 t H9) in (H12
285 (refl_equal T t) P))))))) H7)) H6)))))) H3)))))))) t2))))))) t1).
288 \forall (t: T).(or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u:
289 T).(eq T t (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall
290 (u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))
292 \lambda (t: T).(T_ind (\lambda (t0: T).(or (ex_3 B T T (\lambda (b:
293 B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u))))))
294 (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w
295 u)) \to (\forall (P: Prop).P))))))) (\lambda (n: nat).(or_intror (ex_3 B T T
296 (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (TSort n) (THead (Bind
297 b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (TSort n)
298 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda
299 (w: T).(\lambda (u: T).(\lambda (H: (eq T (TSort n) (THead (Bind b) w
300 u))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda (ee:
301 T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
302 True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
303 (THead (Bind b) w u) H) in (False_ind P H0))))))))) (\lambda (n:
304 nat).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u:
305 T).(eq T (TLRef n) (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w:
306 T).(\forall (u: T).((eq T (TLRef n) (THead (Bind b) w u)) \to (\forall (P:
307 Prop).P))))) (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(\lambda (H: (eq
308 T (TLRef n) (THead (Bind b) w u))).(\lambda (P: Prop).(let H0 \def (eq_ind T
309 (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
310 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
311 \Rightarrow False])) I (THead (Bind b) w u) H) in (False_ind P H0)))))))))
312 (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t0: T).((or (ex_3 B T T
313 (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w
314 u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead
315 (Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (\forall (t1: T).((or (ex_3
316 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind
317 b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead
318 (Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (or (ex_3 B T T (\lambda
319 (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead k0 t0 t1) (THead (Bind b)
320 w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead k0 t0
321 t1) (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))))))) (\lambda (b:
322 B).(\lambda (t0: T).(\lambda (_: (or (ex_3 B T T (\lambda (b0: B).(\lambda
323 (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b0) w u)))))) (\forall (b0:
324 B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b0) w u)) \to
325 (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (_: (or (ex_3 B T T
326 (\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind b0) w
327 u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead
328 (Bind b0) w u)) \to (\forall (P: Prop).P))))))).(or_introl (ex_3 B T T
329 (\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead (Bind b) t0 t1)
330 (THead (Bind b0) w u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u:
331 T).((eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u)) \to (\forall (P:
332 Prop).P))))) (ex_3_intro B T T (\lambda (b0: B).(\lambda (w: T).(\lambda (u:
333 T).(eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u))))) b t0 t1 (refl_equal
334 T (THead (Bind b) t0 t1))))))))) (\lambda (f: F).(\lambda (t0: T).(\lambda
335 (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0
336 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u:
337 T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(\lambda
338 (t1: T).(\lambda (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda
339 (u: T).(eq T t1 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w:
340 T).(\forall (u: T).((eq T t1 (THead (Bind b) w u)) \to (\forall (P:
341 Prop).P))))))).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w:
342 T).(\lambda (u: T).(eq T (THead (Flat f) t0 t1) (THead (Bind b) w u))))))
343 (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead (Flat f) t0 t1)
344 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda
345 (w: T).(\lambda (u: T).(\lambda (H1: (eq T (THead (Flat f) t0 t1) (THead
346 (Bind b) w u))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead (Flat f) t0
347 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
348 _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _)
349 \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _)
350 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) w u) H1)
351 in (False_ind P H2))))))))))))) k)) t).
354 \forall (u: T).(\forall (v: T).(or (ex T (\lambda (t: T).(eq T u (THead
355 (Bind Abst) v t)))) (\forall (t: T).((eq T u (THead (Bind Abst) v t)) \to
356 (\forall (P: Prop).P)))))
358 \lambda (u: T).(T_ind (\lambda (t: T).(\forall (v: T).(or (ex T (\lambda
359 (t0: T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead
360 (Bind Abst) v t0)) \to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda
361 (v: T).(or_intror (ex T (\lambda (t: T).(eq T (TSort n) (THead (Bind Abst) v
362 t)))) (\forall (t: T).((eq T (TSort n) (THead (Bind Abst) v t)) \to (\forall
363 (P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TSort n) (THead (Bind
364 Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda
365 (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
366 \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
367 False])) I (THead (Bind Abst) v t) H) in (False_ind P H0)))))))) (\lambda (n:
368 nat).(\lambda (v: T).(or_intror (ex T (\lambda (t: T).(eq T (TLRef n) (THead
369 (Bind Abst) v t)))) (\forall (t: T).((eq T (TLRef n) (THead (Bind Abst) v t))
370 \to (\forall (P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TLRef n)
371 (THead (Bind Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TLRef n)
372 (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
373 \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
374 False])) I (THead (Bind Abst) v t) H) in (False_ind P H0)))))))) (\lambda (k:
375 K).(\lambda (t: T).(\lambda (_: ((\forall (v: T).(or (ex T (\lambda (t0:
376 T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead (Bind
377 Abst) v t0)) \to (\forall (P: Prop).P))))))).(\lambda (t0: T).(\lambda (_:
378 ((\forall (v: T).(or (ex T (\lambda (t1: T).(eq T t0 (THead (Bind Abst) v
379 t1)))) (\forall (t1: T).((eq T t0 (THead (Bind Abst) v t1)) \to (\forall (P:
380 Prop).P))))))).(\lambda (v: T).(let H_x \def (terms_props__kind_dec k (Bind
381 Abst)) in (let H1 \def H_x in (or_ind (eq K k (Bind Abst)) ((eq K k (Bind
382 Abst)) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead k t
383 t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead
384 (Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda (H2: (eq K k (Bind
385 Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex T (\lambda (t1:
386 T).(eq T (THead k0 t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T
387 (THead k0 t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))))) (let
388 H_x0 \def (term_dec t v) in (let H3 \def H_x0 in (or_ind (eq T t v) ((eq T t
389 v) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead (Bind
390 Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind
391 Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda
392 (H4: (eq T t v)).(eq_ind T t (\lambda (t1: T).(or (ex T (\lambda (t2: T).(eq
393 T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)))) (\forall (t2: T).((eq
394 T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)) \to (\forall (P:
395 Prop).P))))) (or_introl (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0)
396 (THead (Bind Abst) t t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0)
397 (THead (Bind Abst) t t1)) \to (\forall (P: Prop).P))) (ex_intro T (\lambda
398 (t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t t1))) t0
399 (refl_equal T (THead (Bind Abst) t t0)))) v H4)) (\lambda (H4: (((eq T t v)
400 \to (\forall (P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead
401 (Bind Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead
402 (Bind Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P)))
403 (\lambda (t1: T).(\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind
404 Abst) v t1))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e:
405 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t |
406 (TLRef _) \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst)
407 t t0) (THead (Bind Abst) v t1) H5) in ((let H7 \def (f_equal T T (\lambda (e:
408 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
409 (TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind
410 Abst) t t0) (THead (Bind Abst) v t1) H5) in (\lambda (H8: (eq T t v)).(H4 H8
411 P))) H6))))))) H3))) k H2)) (\lambda (H2: (((eq K k (Bind Abst)) \to (\forall
412 (P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead k t t0) (THead
413 (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead (Bind
414 Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1: T).(\lambda (H3: (eq T
415 (THead k t t0) (THead (Bind Abst) v t1))).(\lambda (P: Prop).(let H4 \def
416 (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
417 [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
418 \Rightarrow k0])) (THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H5
419 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
420 with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t2 _)
421 \Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H6
422 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
423 with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2)
424 \Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in (\lambda (_:
425 (eq T t v)).(\lambda (H8: (eq K k (Bind Abst))).(H2 H8 P)))) H5)) H4)))))))