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".
19 include "Basic-1/pr3/fwd.ma".
22 \forall (c: C).(\forall (m: nat).(\forall (n: nat).((pc3 c (TSort m) (TSort
23 n)) \to (eq nat m n))))
25 \lambda (c: C).(\lambda (m: nat).(\lambda (n: nat).(\lambda (H: (pc3 c
26 (TSort m) (TSort n))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c
27 (TSort m) t)) (\lambda (t: T).(pr3 c (TSort n) t)) (eq nat m n) (\lambda (x:
28 T).(\lambda (H1: (pr3 c (TSort m) x)).(\lambda (H2: (pr3 c (TSort n) x)).(let
29 H3 \def (eq_ind T x (\lambda (t: T).(eq T t (TSort n))) (pr3_gen_sort c x n
30 H2) (TSort m) (pr3_gen_sort c x m H1)) in (let H4 \def (f_equal T nat
31 (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort n0)
32 \Rightarrow n0 | (TLRef _) \Rightarrow m | (THead _ _ _) \Rightarrow m]))
33 (TSort m) (TSort n) H3) in H4))))) H0))))).
39 \forall (c: C).(\forall (u1: T).(\forall (u2: T).(\forall (t1: T).(\forall
40 (t2: T).((pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2)) \to
41 (land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u)
44 \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t1: T).(\lambda
45 (t2: T).(\lambda (H: (pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2
46 t2))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c (THead (Bind Abst)
47 u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 t2) t)) (land (pc3 c
48 u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2))))
49 (\lambda (x: T).(\lambda (H1: (pr3 c (THead (Bind Abst) u1 t1) x)).(\lambda
50 (H2: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H3 \def (pr3_gen_abst c u2 t2
51 x H2) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead
52 (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3)))
53 (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
54 c (Bind b) u) t2 t3))))) (land (pc3 c u1 u2) (\forall (b: B).(\forall (u:
55 T).(pc3 (CHead c (Bind b) u) t1 t2)))) (\lambda (x0: T).(\lambda (x1:
56 T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr3 c u2
57 x0)).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
58 t2 x1))))).(let H7 \def (pr3_gen_abst c u1 t1 x H1) in (ex3_2_ind T T
59 (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3))))
60 (\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda
61 (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t3)))))
62 (land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u)
63 t1 t2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T x (THead
64 (Bind Abst) x2 x3))).(\lambda (H9: (pr3 c u1 x2)).(\lambda (H10: ((\forall
65 (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 x3))))).(let H11 \def
66 (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Abst) x0 x1))) H4 (THead
67 (Bind Abst) x2 x3) H8) in (let H12 \def (f_equal T T (\lambda (e: T).(match e
68 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 | (TLRef _)
69 \Rightarrow x2 | (THead _ t _) \Rightarrow t])) (THead (Bind Abst) x2 x3)
70 (THead (Bind Abst) x0 x1) H11) in ((let H13 \def (f_equal T T (\lambda (e:
71 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x3 |
72 (TLRef _) \Rightarrow x3 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abst)
73 x2 x3) (THead (Bind Abst) x0 x1) H11) in (\lambda (H14: (eq T x2 x0)).(let
74 H15 \def (eq_ind T x3 (\lambda (t: T).(\forall (b: B).(\forall (u: T).(pr3
75 (CHead c (Bind b) u) t1 t)))) H10 x1 H13) in (let H16 \def (eq_ind T x2
76 (\lambda (t: T).(pr3 c u1 t)) H9 x0 H14) in (conj (pc3 c u1 u2) (\forall (b:
77 B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2))) (pc3_pr3_t c u1 x0 H16
78 u2 H5) (\lambda (b: B).(\lambda (u: T).(pc3_pr3_t (CHead c (Bind b) u) t1 x1
79 (H15 b u) t2 (H6 b u))))))))) H12)))))))) H7))))))) H3))))) H0))))))).
84 theorem pc3_gen_abst_shift:
85 \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).((pc3 c
86 (THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (pc3 (CHead c (Bind
89 \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
90 (H: (pc3 c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2))).(let H_x \def
91 (pc3_gen_abst c u u t1 t2 H) in (let H0 \def H_x in (land_ind (pc3 c u u)
92 (\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))) (pc3
93 (CHead c (Bind Abst) u) t1 t2) (\lambda (_: (pc3 c u u)).(\lambda (H2:
94 ((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))))).(H2
100 theorem pc3_gen_lift:
101 \forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (h: nat).(\forall
102 (d: nat).((pc3 c (lift h d t1) (lift h d t2)) \to (\forall (e: C).((drop h d
103 c e) \to (pc3 e t1 t2))))))))
105 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (h: nat).(\lambda
106 (d: nat).(\lambda (H: (pc3 c (lift h d t1) (lift h d t2))).(\lambda (e:
107 C).(\lambda (H0: (drop h d c e)).(let H1 \def H in (ex2_ind T (\lambda (t:
108 T).(pr3 c (lift h d t1) t)) (\lambda (t: T).(pr3 c (lift h d t2) t)) (pc3 e
109 t1 t2) (\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t1) x)).(\lambda (H3:
110 (pr3 c (lift h d t2) x)).(let H4 \def (pr3_gen_lift c t2 x h d H3 e H0) in
111 (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3 e
112 t2 t3)) (pc3 e t1 t2) (\lambda (x0: T).(\lambda (H5: (eq T x (lift h d
113 x0))).(\lambda (H6: (pr3 e t2 x0)).(let H7 \def (pr3_gen_lift c t1 x h d H2 e
114 H0) in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3:
115 T).(pr3 e t1 t3)) (pc3 e t1 t2) (\lambda (x1: T).(\lambda (H8: (eq T x (lift
116 h d x1))).(\lambda (H9: (pr3 e t1 x1)).(let H10 \def (eq_ind T x (\lambda (t:
117 T).(eq T t (lift h d x0))) H5 (lift h d x1) H8) in (let H11 \def (eq_ind T x1
118 (\lambda (t: T).(pr3 e t1 t)) H9 x0 (lift_inj x1 x0 h d H10)) in (pc3_pr3_t e
119 t1 x0 H11 t2 H6)))))) H7))))) H4))))) H1))))))))).
124 theorem pc3_gen_not_abst:
125 \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (t1:
126 T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: T).((pc3 c (THead (Bind b)
127 u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead c (Bind b) u1) t1 (lift (S
128 O) O (THead (Bind Abst) u2 t2))))))))))
130 \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall
131 (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (u1: T).(\forall (u2:
132 T).((pc3 c (THead (Bind b0) u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead
133 c (Bind b0) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))))))))))) (\lambda
134 (_: (not (eq B Abbr Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2:
135 T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Abbr)
136 u1 t1) (THead (Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t:
137 T).(pr3 c (THead (Bind Abbr) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind
138 Abst) u2 t2) t)) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind
139 Abst) u2 t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Abbr) u1 t1)
140 x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def
141 (pr3_gen_abbr c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda
142 (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_:
143 T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr)
144 u1) t1 t3)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (pc3 (CHead
145 c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (H5:
146 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3
147 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_:
148 T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))))).(ex3_2_ind T T
149 (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3))))
150 (\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda
151 (t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))) (pc3 (CHead c (Bind Abbr) u1)
152 t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: T).(\lambda (x1:
153 T).(\lambda (H6: (eq T x (THead (Bind Abbr) x0 x1))).(\lambda (_: (pr3 c u1
154 x0)).(\lambda (_: (pr3 (CHead c (Bind Abbr) u1) t1 x1)).(let H9 \def
155 (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3:
156 T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3
157 c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u:
158 T).(pr3 (CHead c (Bind b0) u) t2 t3))))) (pc3 (CHead c (Bind Abbr) u1) t1
159 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3:
160 T).(\lambda (H10: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2
161 x2)).(\lambda (_: ((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0)
162 u) t2 x3))))).(let H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind
163 Abbr) x0 x1))) H6 (THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T
164 (THead (Bind Abst) x2 x3) (\lambda (ee: T).(match ee in T return (\lambda (_:
165 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
166 (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
167 [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr
168 \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat
169 _) \Rightarrow False])])) I (THead (Bind Abbr) x0 x1) H13) in (False_ind (pc3
170 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2)))
171 H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3 (CHead c (Bind Abbr) u1) t1
172 (lift (S O) O x))).(let H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T
173 (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3))))
174 (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda
175 (t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2
176 t3))))) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2
177 t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind
178 Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b0:
179 B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x1))))).(let H10 \def
180 (eq_ind T x (\lambda (t: T).(pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O
181 t))) H5 (THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Abbr) u1)
182 t1 (lift (S O) O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind
183 Abst) u2 t2)) (pr3_lift (CHead c (Bind Abbr) u1) c (S O) O (drop_drop (Bind
184 Abbr) O c c (drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0
185 x1) (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6)))
186 H4))))) H1))))))))) (\lambda (H: (not (eq B Abst Abst))).(\lambda (c:
187 C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (u1: T).(\lambda (u2:
188 T).(\lambda (_: (pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2
189 t2))).(let H1 \def (match (H (refl_equal B Abst)) in False return (\lambda
190 (_: False).(pc3 (CHead c (Bind Abst) u1) t1 (lift (S O) O (THead (Bind Abst)
191 u2 t2)))) with []) in H1)))))))) (\lambda (_: (not (eq B Void
192 Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (u1:
193 T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Void) u1 t1) (THead
194 (Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t: T).(pr3 c
195 (THead (Bind Void) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2
196 t2) t)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2
197 t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Void) u1 t1)
198 x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def
199 (pr3_gen_void c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda
200 (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_:
201 T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall
202 (u: T).(pr3 (CHead c (Bind b0) u) t1 t3)))))) (pr3 (CHead c (Bind Void) u1)
203 t1 (lift (S O) O x)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead
204 (Bind Abst) u2 t2))) (\lambda (H5: (ex3_2 T T (\lambda (u3: T).(\lambda (t3:
205 T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3
206 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u:
207 T).(pr3 (CHead c (Bind b0) u) t1 t3))))))).(ex3_2_ind T T (\lambda (u3:
208 T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3:
209 T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall
210 (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t3))))) (pc3 (CHead c
211 (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0:
212 T).(\lambda (x1: T).(\lambda (H6: (eq T x (THead (Bind Void) x0
213 x1))).(\lambda (_: (pr3 c u1 x0)).(\lambda (_: ((\forall (b0: B).(\forall (u:
214 T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(let H9 \def (pr3_gen_abst c u2 t2 x
215 H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind
216 Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_:
217 T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0)
218 u) t2 t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind
219 Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq T x
220 (THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2 x2)).(\lambda (_:
221 ((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x3))))).(let
222 H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Void) x0 x1))) H6
223 (THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T (THead (Bind Abst)
224 x2 x3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
225 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
226 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b0)
227 \Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr
228 \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat
229 _) \Rightarrow False])])) I (THead (Bind Void) x0 x1) H13) in (False_ind (pc3
230 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2)))
231 H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3 (CHead c (Bind Void) u1) t1
232 (lift (S O) O x))).(let H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T
233 (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3))))
234 (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda
235 (t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2
236 t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2
237 t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind
238 Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b0:
239 B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x1))))).(let H10 \def
240 (eq_ind T x (\lambda (t: T).(pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O
241 t))) H5 (THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Void) u1)
242 t1 (lift (S O) O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind
243 Abst) u2 t2)) (pr3_lift (CHead c (Bind Void) u1) c (S O) O (drop_drop (Bind
244 Void) O c c (drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0
245 x1) (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6)))
246 H4))))) H1))))))))) b).
251 theorem pc3_gen_lift_abst:
252 \forall (c: C).(\forall (t: T).(\forall (t2: T).(\forall (u2: T).(\forall
253 (h: nat).(\forall (d: nat).((pc3 c (lift h d t) (THead (Bind Abst) u2 t2))
254 \to (\forall (e: C).((drop h d c e) \to (ex3_2 T T (\lambda (u1: T).(\lambda
255 (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_:
256 T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b:
257 B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d)
260 \lambda (c: C).(\lambda (t: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda
261 (h: nat).(\lambda (d: nat).(\lambda (H: (pc3 c (lift h d t) (THead (Bind
262 Abst) u2 t2))).(\lambda (e: C).(\lambda (H0: (drop h d c e)).(let H1 \def H
263 in (ex2_ind T (\lambda (t0: T).(pr3 c (lift h d t) t0)) (\lambda (t0: T).(pr3
264 c (THead (Bind Abst) u2 t2) t0)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1:
265 T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_:
266 T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b:
267 B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1)))))))
268 (\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t) x)).(\lambda (H3: (pr3 c
269 (THead (Bind Abst) u2 t2) x)).(let H4 \def (pr3_gen_lift c t x h d H2 e H0)
270 in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3
271 e t t3)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind
272 Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1))))
273 (\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
274 c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x0: T).(\lambda (H5: (eq T
275 x (lift h d x0))).(\lambda (H6: (pr3 e t x0)).(let H7 \def (pr3_gen_abst c u2
276 t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead
277 (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3)))
278 (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
279 c (Bind b) u) t2 t3))))) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e
280 t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2
281 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall
282 (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x1:
283 T).(\lambda (x2: T).(\lambda (H8: (eq T x (THead (Bind Abst) x1
284 x2))).(\lambda (H9: (pr3 c u2 x1)).(\lambda (H10: ((\forall (b: B).(\forall
285 (u: T).(pr3 (CHead c (Bind b) u) t2 x2))))).(let H11 \def (eq_ind T x
286 (\lambda (t0: T).(eq T t0 (lift h d x0))) H5 (THead (Bind Abst) x1 x2) H8) in
287 (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y
288 z)))) (\lambda (y: T).(\lambda (_: T).(eq T x1 (lift h d y)))) (\lambda (_:
289 T).(\lambda (z: T).(eq T x2 (lift h (S d) z)))) (ex3_2 T T (\lambda (u1:
290 T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1:
291 T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1:
292 T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d)
293 t1))))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq T x0 (THead
294 (Bind Abst) x3 x4))).(\lambda (H13: (eq T x1 (lift h d x3))).(\lambda (H14:
295 (eq T x2 (lift h (S d) x4))).(let H15 \def (eq_ind T x2 (\lambda (t0:
296 T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 t0)))) H10
297 (lift h (S d) x4) H14) in (let H16 \def (eq_ind T x1 (\lambda (t0: T).(pr3 c
298 u2 t0)) H9 (lift h d x3) H13) in (let H17 \def (eq_ind T x0 (\lambda (t0:
299 T).(pr3 e t t0)) H6 (THead (Bind Abst) x3 x4) H12) in (ex3_2_intro T T
300 (\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1))))
301 (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_:
302 T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
303 t2 (lift h (S d) t1)))))) x3 x4 H17 H16 H15))))))))) (lift_gen_bind Abst x1
304 x2 x0 h d H11)))))))) H7))))) H4))))) H1)))))))))).
309 theorem pc3_gen_sort_abst:
310 \forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (n: nat).((pc3 c
311 (TSort n) (THead (Bind Abst) u t)) \to (\forall (P: Prop).P)))))
313 \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (n: nat).(\lambda
314 (H: (pc3 c (TSort n) (THead (Bind Abst) u t))).(\lambda (P: Prop).(let H0
315 \def H in (ex2_ind T (\lambda (t0: T).(pr3 c (TSort n) t0)) (\lambda (t0:
316 T).(pr3 c (THead (Bind Abst) u t) t0)) P (\lambda (x: T).(\lambda (H1: (pr3 c
317 (TSort n) x)).(\lambda (H2: (pr3 c (THead (Bind Abst) u t) x)).(let H3 \def
318 (pr3_gen_abst c u t x H2) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2:
319 T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
320 c u u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0:
321 T).(pr3 (CHead c (Bind b) u0) t t2))))) P (\lambda (x0: T).(\lambda (x1:
322 T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (_: (pr3 c u
323 x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b)
324 u0) t x1))))).(let H7 \def (eq_ind T x (\lambda (t0: T).(eq T t0 (TSort n)))
325 (pr3_gen_sort c x n H1) (THead (Bind Abst) x0 x1) H4) in (let H8 \def (eq_ind
326 T (THead (Bind Abst) x0 x1) (\lambda (ee: T).(match ee in T return (\lambda
327 (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
328 | (THead _ _ _) \Rightarrow True])) I (TSort n) H7) in (False_ind P
329 H8)))))))) H3))))) H0))))))).