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