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 "basic_1/pc3/props.ma".
20 \forall (c: C).(\forall (m: nat).(\forall (n: nat).((pc3 c (TSort m) (TSort
21 n)) \to (eq nat m n))))
23 \lambda (c: C).(\lambda (m: nat).(\lambda (n: nat).(\lambda (H: (pc3 c
24 (TSort m) (TSort n))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c
25 (TSort m) t)) (\lambda (t: T).(pr3 c (TSort n) t)) (eq nat m n) (\lambda (x:
26 T).(\lambda (H1: (pr3 c (TSort m) x)).(\lambda (H2: (pr3 c (TSort n) x)).(let
27 H3 \def (eq_ind T x (\lambda (t: T).(eq T t (TSort n))) (pr3_gen_sort c x n
28 H2) (TSort m) (pr3_gen_sort c x m H1)) in (let H4 \def (f_equal T nat
29 (\lambda (e: T).(match e with [(TSort n0) \Rightarrow n0 | (TLRef _)
30 \Rightarrow m | (THead _ _ _) \Rightarrow m])) (TSort m) (TSort n) H3) in
34 \forall (c: C).(\forall (u1: T).(\forall (u2: T).(\forall (t1: T).(\forall
35 (t2: T).((pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2)) \to
36 (land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u)
39 \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t1: T).(\lambda
40 (t2: T).(\lambda (H: (pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2
41 t2))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c (THead (Bind Abst)
42 u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 t2) t)) (land (pc3 c
43 u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2))))
44 (\lambda (x: T).(\lambda (H1: (pr3 c (THead (Bind Abst) u1 t1) x)).(\lambda
45 (H2: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H3 \def (pr3_gen_abst c u2 t2
46 x H2) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead
47 (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3)))
48 (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
49 c (Bind b) u) t2 t3))))) (land (pc3 c u1 u2) (\forall (b: B).(\forall (u:
50 T).(pc3 (CHead c (Bind b) u) t1 t2)))) (\lambda (x0: T).(\lambda (x1:
51 T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr3 c u2
52 x0)).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
53 t2 x1))))).(let H7 \def (pr3_gen_abst c u1 t1 x H1) in (ex3_2_ind T T
54 (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3))))
55 (\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda
56 (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t3)))))
57 (land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u)
58 t1 t2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T x (THead
59 (Bind Abst) x2 x3))).(\lambda (H9: (pr3 c u1 x2)).(\lambda (H10: ((\forall
60 (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 x3))))).(let H11 \def
61 (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Abst) x0 x1))) H4 (THead
62 (Bind Abst) x2 x3) H8) in (let H12 \def (f_equal T T (\lambda (e: T).(match e
63 with [(TSort _) \Rightarrow x2 | (TLRef _) \Rightarrow x2 | (THead _ t _)
64 \Rightarrow t])) (THead (Bind Abst) x2 x3) (THead (Bind Abst) x0 x1) H11) in
65 ((let H13 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _)
66 \Rightarrow x3 | (TLRef _) \Rightarrow x3 | (THead _ _ t) \Rightarrow t]))
67 (THead (Bind Abst) x2 x3) (THead (Bind Abst) x0 x1) H11) in (\lambda (H14:
68 (eq T x2 x0)).(let H15 \def (eq_ind T x3 (\lambda (t: T).(\forall (b:
69 B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t)))) H10 x1 H13) in (let H16
70 \def (eq_ind T x2 (\lambda (t: T).(pr3 c u1 t)) H9 x0 H14) in (conj (pc3 c u1
71 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2)))
72 (pc3_pr3_t c u1 x0 H16 u2 H5) (\lambda (b: B).(\lambda (u: T).(pc3_pr3_t
73 (CHead c (Bind b) u) t1 x1 (H15 b u) t2 (H6 b u))))))))) H12))))))))
74 H7))))))) H3))))) H0))))))).
76 lemma pc3_gen_abst_shift:
77 \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).((pc3 c
78 (THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (pc3 (CHead c (Bind
81 \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
82 (H: (pc3 c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2))).(let H_x \def
83 (pc3_gen_abst c u u t1 t2 H) in (let H0 \def H_x in (land_ind (pc3 c u u)
84 (\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))) (pc3
85 (CHead c (Bind Abst) u) t1 t2) (\lambda (_: (pc3 c u u)).(\lambda (H2:
86 ((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))))).(H2
90 \forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (h: nat).(\forall
91 (d: nat).((pc3 c (lift h d t1) (lift h d t2)) \to (\forall (e: C).((drop h d
92 c e) \to (pc3 e t1 t2))))))))
94 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (h: nat).(\lambda
95 (d: nat).(\lambda (H: (pc3 c (lift h d t1) (lift h d t2))).(\lambda (e:
96 C).(\lambda (H0: (drop h d c e)).(let H1 \def H in (ex2_ind T (\lambda (t:
97 T).(pr3 c (lift h d t1) t)) (\lambda (t: T).(pr3 c (lift h d t2) t)) (pc3 e
98 t1 t2) (\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t1) x)).(\lambda (H3:
99 (pr3 c (lift h d t2) x)).(let H4 \def (pr3_gen_lift c t2 x h d H3 e H0) in
100 (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3 e
101 t2 t3)) (pc3 e t1 t2) (\lambda (x0: T).(\lambda (H5: (eq T x (lift h d
102 x0))).(\lambda (H6: (pr3 e t2 x0)).(let H7 \def (pr3_gen_lift c t1 x h d H2 e
103 H0) in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3:
104 T).(pr3 e t1 t3)) (pc3 e t1 t2) (\lambda (x1: T).(\lambda (H8: (eq T x (lift
105 h d x1))).(\lambda (H9: (pr3 e t1 x1)).(let H10 \def (eq_ind T x (\lambda (t:
106 T).(eq T t (lift h d x0))) H5 (lift h d x1) H8) in (let H11 \def (eq_ind T x1
107 (\lambda (t: T).(pr3 e t1 t)) H9 x0 (lift_inj x1 x0 h d H10)) in (pc3_pr3_t e
108 t1 x0 H11 t2 H6)))))) H7))))) H4))))) H1))))))))).
110 lemma pc3_gen_not_abst:
111 \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (t1:
112 T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: T).((pc3 c (THead (Bind b)
113 u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead c (Bind b) u1) t1 (lift (S
114 O) O (THead (Bind Abst) u2 t2))))))))))
116 \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall
117 (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (u1: T).(\forall (u2:
118 T).((pc3 c (THead (Bind b0) u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead
119 c (Bind b0) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))))))))))) (\lambda
120 (_: (not (eq B Abbr Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2:
121 T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Abbr)
122 u1 t1) (THead (Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t:
123 T).(pr3 c (THead (Bind Abbr) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind
124 Abst) u2 t2) t)) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind
125 Abst) u2 t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Abbr) u1 t1)
126 x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def
127 (pr3_gen_abbr c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda
128 (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_:
129 T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr)
130 u1) t1 t3)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (pc3 (CHead
131 c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (H5:
132 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3
133 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_:
134 T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))))).(ex3_2_ind T T
135 (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3))))
136 (\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda
137 (t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))) (pc3 (CHead c (Bind Abbr) u1)
138 t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: T).(\lambda (x1:
139 T).(\lambda (H6: (eq T x (THead (Bind Abbr) x0 x1))).(\lambda (_: (pr3 c u1
140 x0)).(\lambda (_: (pr3 (CHead c (Bind Abbr) u1) t1 x1)).(let H9 \def
141 (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3:
142 T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3
143 c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u:
144 T).(pr3 (CHead c (Bind b0) u) t2 t3))))) (pc3 (CHead c (Bind Abbr) u1) t1
145 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3:
146 T).(\lambda (H10: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2
147 x2)).(\lambda (_: ((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0)
148 u) t2 x3))))).(let H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind
149 Abbr) x0 x1))) H6 (THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T
150 (THead (Bind Abst) x2 x3) (\lambda (ee: T).(match ee with [(TSort _)
151 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
152 (match k with [(Bind b0) \Rightarrow (match b0 with [Abbr \Rightarrow False |
153 Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow
154 False])])) I (THead (Bind Abbr) x0 x1) H13) in (False_ind (pc3 (CHead c (Bind
155 Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) H14)))))))) H9)))))))
156 H5)) (\lambda (H5: (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x))).(let
157 H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T (\lambda (u3:
158 T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3:
159 T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall
160 (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 t3))))) (pc3 (CHead c
161 (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0:
162 T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind Abst) x0
163 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b0: B).(\forall
164 (u: T).(pr3 (CHead c (Bind b0) u) t2 x1))))).(let H10 \def (eq_ind T x
165 (\lambda (t: T).(pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O t))) H5 (THead
166 (Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Abbr) u1) t1 (lift (S O)
167 O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind Abst) u2 t2))
168 (pr3_lift (CHead c (Bind Abbr) u1) c (S O) O (drop_drop (Bind Abbr) O c c
169 (drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0 x1)
170 (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6))) H4)))))
171 H1))))))))) (\lambda (H: (not (eq B Abst Abst))).(\lambda (c: C).(\lambda
172 (t1: T).(\lambda (t2: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pc3
173 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2))).(let H1 \def (match
174 (H (refl_equal B Abst)) in False with []) in H1)))))))) (\lambda (_: (not (eq
175 B Void Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
176 (u1: T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Void) u1 t1)
177 (THead (Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t:
178 T).(pr3 c (THead (Bind Void) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind
179 Abst) u2 t2) t)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind
180 Abst) u2 t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Void) u1 t1)
181 x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def
182 (pr3_gen_void c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda
183 (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_:
184 T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall
185 (u: T).(pr3 (CHead c (Bind b0) u) t1 t3)))))) (pr3 (CHead c (Bind Void) u1)
186 t1 (lift (S O) O x)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead
187 (Bind Abst) u2 t2))) (\lambda (H5: (ex3_2 T T (\lambda (u3: T).(\lambda (t3:
188 T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3
189 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u:
190 T).(pr3 (CHead c (Bind b0) u) t1 t3))))))).(ex3_2_ind T T (\lambda (u3:
191 T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3:
192 T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall
193 (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t3))))) (pc3 (CHead c
194 (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0:
195 T).(\lambda (x1: T).(\lambda (H6: (eq T x (THead (Bind Void) x0
196 x1))).(\lambda (_: (pr3 c u1 x0)).(\lambda (_: ((\forall (b0: B).(\forall (u:
197 T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(let H9 \def (pr3_gen_abst c u2 t2 x
198 H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind
199 Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_:
200 T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0)
201 u) t2 t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind
202 Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq T x
203 (THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2 x2)).(\lambda (_:
204 ((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x3))))).(let
205 H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Void) x0 x1))) H6
206 (THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T (THead (Bind Abst)
207 x2 x3) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef
208 _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind b0)
209 \Rightarrow (match b0 with [Abbr \Rightarrow False | Abst \Rightarrow True |
210 Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind
211 Void) x0 x1) H13) in (False_ind (pc3 (CHead c (Bind Void) u1) t1 (lift (S O)
212 O (THead (Bind Abst) u2 t2))) H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3
213 (CHead c (Bind Void) u1) t1 (lift (S O) O x))).(let H6 \def (pr3_gen_abst c
214 u2 t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x
215 (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3)))
216 (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead
217 c (Bind b0) u) t2 t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O
218 (THead (Bind Abst) u2 t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7:
219 (eq T x (THead (Bind Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda
220 (H9: ((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2
221 x1))))).(let H10 \def (eq_ind T x (\lambda (t: T).(pr3 (CHead c (Bind Void)
222 u1) t1 (lift (S O) O t))) H5 (THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t
223 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) x0 x1)) H10
224 (lift (S O) O (THead (Bind Abst) u2 t2)) (pr3_lift (CHead c (Bind Void) u1) c
225 (S O) O (drop_drop (Bind Void) O c c (drop_refl c) u1) (THead (Bind Abst) u2
226 t2) (THead (Bind Abst) x0 x1) (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9
227 Abst x0)))))))))) H6))) H4))))) H1))))))))) b).
229 lemma pc3_gen_lift_abst:
230 \forall (c: C).(\forall (t: T).(\forall (t2: T).(\forall (u2: T).(\forall
231 (h: nat).(\forall (d: nat).((pc3 c (lift h d t) (THead (Bind Abst) u2 t2))
232 \to (\forall (e: C).((drop h d c e) \to (ex3_2 T T (\lambda (u1: T).(\lambda
233 (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_:
234 T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b:
235 B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d)
238 \lambda (c: C).(\lambda (t: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda
239 (h: nat).(\lambda (d: nat).(\lambda (H: (pc3 c (lift h d t) (THead (Bind
240 Abst) u2 t2))).(\lambda (e: C).(\lambda (H0: (drop h d c e)).(let H1 \def H
241 in (ex2_ind T (\lambda (t0: T).(pr3 c (lift h d t) t0)) (\lambda (t0: T).(pr3
242 c (THead (Bind Abst) u2 t2) t0)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1:
243 T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_:
244 T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b:
245 B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1)))))))
246 (\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t) x)).(\lambda (H3: (pr3 c
247 (THead (Bind Abst) u2 t2) x)).(let H4 \def (pr3_gen_lift c t x h d H2 e H0)
248 in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3
249 e t t3)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind
250 Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1))))
251 (\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
252 c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x0: T).(\lambda (H5: (eq T
253 x (lift h d x0))).(\lambda (H6: (pr3 e t x0)).(let H7 \def (pr3_gen_abst c u2
254 t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead
255 (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3)))
256 (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
257 c (Bind b) u) t2 t3))))) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e
258 t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2
259 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall
260 (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x1:
261 T).(\lambda (x2: T).(\lambda (H8: (eq T x (THead (Bind Abst) x1
262 x2))).(\lambda (H9: (pr3 c u2 x1)).(\lambda (H10: ((\forall (b: B).(\forall
263 (u: T).(pr3 (CHead c (Bind b) u) t2 x2))))).(let H11 \def (eq_ind T x
264 (\lambda (t0: T).(eq T t0 (lift h d x0))) H5 (THead (Bind Abst) x1 x2) H8) in
265 (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y
266 z)))) (\lambda (y: T).(\lambda (_: T).(eq T x1 (lift h d y)))) (\lambda (_:
267 T).(\lambda (z: T).(eq T x2 (lift h (S d) z)))) (ex3_2 T T (\lambda (u1:
268 T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1:
269 T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1:
270 T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d)
271 t1))))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq T x0 (THead
272 (Bind Abst) x3 x4))).(\lambda (H13: (eq T x1 (lift h d x3))).(\lambda (H14:
273 (eq T x2 (lift h (S d) x4))).(let H15 \def (eq_ind T x2 (\lambda (t0:
274 T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 t0)))) H10
275 (lift h (S d) x4) H14) in (let H16 \def (eq_ind T x1 (\lambda (t0: T).(pr3 c
276 u2 t0)) H9 (lift h d x3) H13) in (let H17 \def (eq_ind T x0 (\lambda (t0:
277 T).(pr3 e t t0)) H6 (THead (Bind Abst) x3 x4) H12) in (ex3_2_intro T T
278 (\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1))))
279 (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_:
280 T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
281 t2 (lift h (S d) t1)))))) x3 x4 H17 H16 H15))))))))) (lift_gen_bind Abst x1
282 x2 x0 h d H11)))))))) H7))))) H4))))) H1)))))))))).
284 lemma pc3_gen_sort_abst:
285 \forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (n: nat).((pc3 c
286 (TSort n) (THead (Bind Abst) u t)) \to (\forall (P: Prop).P)))))
288 \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (n: nat).(\lambda
289 (H: (pc3 c (TSort n) (THead (Bind Abst) u t))).(\lambda (P: Prop).(let H0
290 \def H in (ex2_ind T (\lambda (t0: T).(pr3 c (TSort n) t0)) (\lambda (t0:
291 T).(pr3 c (THead (Bind Abst) u t) t0)) P (\lambda (x: T).(\lambda (H1: (pr3 c
292 (TSort n) x)).(\lambda (H2: (pr3 c (THead (Bind Abst) u t) x)).(let H3 \def
293 (pr3_gen_abst c u t x H2) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2:
294 T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
295 c u u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0:
296 T).(pr3 (CHead c (Bind b) u0) t t2))))) P (\lambda (x0: T).(\lambda (x1:
297 T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (_: (pr3 c u
298 x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b)
299 u0) t x1))))).(let H7 \def (eq_ind T x (\lambda (t0: T).(eq T t0 (TSort n)))
300 (pr3_gen_sort c x n H1) (THead (Bind Abst) x0 x1) H4) in (let H8 \def (eq_ind
301 T (THead (Bind Abst) x0 x1) (\lambda (ee: T).(match ee with [(TSort _)
302 \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
303 True])) I (TSort n) H7) in (False_ind P H8)))))))) H3))))) H0))))))).