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 *********************)
21 include "pr2/props.ma".
23 include "pr1/props.ma".
25 theorem clear_pr3_trans:
26 \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr3 c2 t1 t2) \to
27 (\forall (c1: C).((clear c1 c2) \to (pr3 c1 t1 t2))))))
29 \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c2 t1
30 t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(pr3_ind c2 (\lambda (t:
31 T).(\lambda (t0: T).(pr3 c1 t t0))) (\lambda (t: T).(pr3_refl c1 t)) (\lambda
32 (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c2 t4 t3)).(\lambda (t5:
33 T).(\lambda (_: (pr3 c2 t3 t5)).(\lambda (H3: (pr3 c1 t3 t5)).(pr3_sing c1 t3
34 t4 (clear_pr2_trans c2 t4 t3 H1 c1 H0) t5 H3))))))) t1 t2 H)))))).
37 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pr3 c
40 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
41 t2)).(pr3_sing c t2 t1 H t2 (pr3_refl c t2))))).
44 \forall (t2: T).(\forall (t1: T).(\forall (c: C).((pr3 c t1 t2) \to (\forall
45 (t3: T).((pr3 c t2 t3) \to (pr3 c t1 t3))))))
47 \lambda (t2: T).(\lambda (t1: T).(\lambda (c: C).(\lambda (H: (pr3 c t1
48 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3: T).((pr3 c t0
49 t3) \to (pr3 c t t3))))) (\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr3
50 c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3
51 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall
52 (t5: T).((pr3 c t4 t5) \to (pr3 c t0 t5))))).(\lambda (t5: T).(\lambda (H3:
53 (pr3 c t4 t5)).(pr3_sing c t0 t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
56 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
57 (u: T).(\forall (f: F).(pr3 c (THead (Flat f) u t1) (THead (Flat f) u
60 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
61 t2)).(\lambda (u: T).(\lambda (f: F).(pr3_ind c (\lambda (t: T).(\lambda (t0:
62 T).(pr3 c (THead (Flat f) u t) (THead (Flat f) u t0)))) (\lambda (t:
63 T).(pr3_refl c (THead (Flat f) u t))) (\lambda (t0: T).(\lambda (t3:
64 T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0
65 t4)).(\lambda (H2: (pr3 c (THead (Flat f) u t0) (THead (Flat f) u
66 t4))).(pr3_sing c (THead (Flat f) u t0) (THead (Flat f) u t3) (pr2_thin_dx c
67 t3 t0 H0 u f) (THead (Flat f) u t4) H2))))))) t1 t2 H)))))).
70 \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
71 (k: K).(\forall (t: T).(pr3 c (THead k u1 t) (THead k u2 t)))))))
73 \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
74 u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (k: K).(\forall
75 (t1: T).(pr3 c (THead k t t1) (THead k t0 t1)))))) (\lambda (t: T).(\lambda
76 (k: K).(\lambda (t0: T).(pr3_refl c (THead k t t0))))) (\lambda (t2:
77 T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda
78 (_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (k: K).(\forall (t: T).(pr3 c
79 (THead k t2 t) (THead k t3 t)))))).(\lambda (k: K).(\lambda (t: T).(pr3_sing
80 c (THead k t2 t) (THead k t1 t) (pr2_head_1 c t1 t2 H0 k t) (THead k t3 t)
81 (H2 k t)))))))))) u1 u2 H)))).
84 \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall
85 (k: K).((pr3 (CHead c k u) t1 t2) \to (pr3 c (THead k u t1) (THead k u
88 \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
89 (k: K).(\lambda (H: (pr3 (CHead c k u) t1 t2)).(pr3_ind (CHead c k u)
90 (\lambda (t: T).(\lambda (t0: T).(pr3 c (THead k u t) (THead k u t0))))
91 (\lambda (t: T).(pr3_refl c (THead k u t))) (\lambda (t0: T).(\lambda (t3:
92 T).(\lambda (H0: (pr2 (CHead c k u) t3 t0)).(\lambda (t4: T).(\lambda (_:
93 (pr3 (CHead c k u) t0 t4)).(\lambda (H2: (pr3 c (THead k u t0) (THead k u
94 t4))).(pr3_sing c (THead k u t0) (THead k u t3) (pr2_head_2 c u t3 t0 k H0)
95 (THead k u t4) H2))))))) t1 t2 H)))))).
98 \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
99 (k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u1) t1 t2) \to (pr3
100 c (THead k u1 t1) (THead k u2 t2)))))))))
102 \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
103 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3
104 (CHead c k u1) t1 t2)).(pr3_t (THead k u1 t2) (THead k u1 t1) c (pr3_head_2 c
105 u1 t1 t2 k H0) (THead k u2 t2) (pr3_head_1 c u1 u2 H k t2))))))))).
108 \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
109 (k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u2) t1 t2) \to (pr3
110 c (THead k u1 t1) (THead k u2 t2)))))))))
112 \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
113 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3
114 (CHead c k u2) t1 t2)).(pr3_t (THead k u2 t1) (THead k u1 t1) c (pr3_head_1 c
115 u1 u2 H k t1) (THead k u2 t2) (pr3_head_2 c u2 t1 t2 k H0))))))))).
118 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
119 (f: F).(\forall (v: T).(pr3 (CHead c (Flat f) v) t1 t2))))))
121 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
122 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (f: F).(\forall (v:
123 T).(pr3 (CHead c (Flat f) v) t t0))))) (\lambda (t: T).(\lambda (f:
124 F).(\lambda (v: T).(pr3_refl (CHead c (Flat f) v) t)))) (\lambda (t3:
125 T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda
126 (_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (f: F).(\forall (v: T).(pr3 (CHead
127 c (Flat f) v) t3 t5))))).(\lambda (f: F).(\lambda (v: T).(pr3_sing (CHead c
128 (Flat f) v) t3 t4 (pr2_cflat c t4 t3 H0 f v) t5 (H2 f v)))))))))) t1 t2 H)))).
131 \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
132 (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (f: F).(pr3 c (THead
133 (Flat f) u1 t1) (THead (Flat f) u2 t2)))))))))
135 \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
136 u2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda
137 (f: F).(pr3_head_12 c u1 u2 H (Flat f) t1 t2 (pr3_cflat c t1 t2 H0 f
140 theorem pr3_pr0_pr2_t:
141 \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (c: C).(\forall
142 (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3
143 (CHead c k u1) t1 t2))))))))
145 \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (c:
146 C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2
147 (CHead c k u2) t1 t2)).(let H1 \def (match H0 in pr2 return (\lambda (c0:
148 C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0
149 (CHead c k u2)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr3 (CHead c k u1) t1
150 t2)))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0
151 (CHead c k u2))).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3
152 t2)).(eq_ind C (CHead c k u2) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2)
153 \to ((pr0 t0 t3) \to (pr3 (CHead c k u1) t1 t2))))) (\lambda (H5: (eq T t0
154 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr3
155 (CHead c k u1) t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda
156 (t: T).((pr0 t1 t) \to (pr3 (CHead c k u1) t1 t2))) (\lambda (H7: (pr0 t1
157 t2)).(pr3_pr2 (CHead c k u1) t1 t2 (pr2_free (CHead c k u1) t1 t2 H7))) t3
158 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 (CHead c k u2)
159 H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t0 t3 H2 t H3) \Rightarrow (\lambda
160 (H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq
161 T t t2)).(eq_ind C (CHead c k u2) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t
162 t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i
163 u t3 t) \to (pr3 (CHead c k u1) t1 t2))))))) (\lambda (H7: (eq T t0
164 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c k u2)
165 (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr3
166 (CHead c k u1) t1 t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda
167 (t4: T).((getl i (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to
168 ((subst0 i u t3 t4) \to (pr3 (CHead c k u1) t1 t2))))) (\lambda (H9: (getl i
169 (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda
170 (H11: (subst0 i u t3 t2)).(nat_ind (\lambda (n: nat).((getl n (CHead c k u2)
171 (CHead d (Bind Abbr) u)) \to ((subst0 n u t3 t2) \to (pr3 (CHead c k u1) t1
172 t2)))) (\lambda (H12: (getl O (CHead c k u2) (CHead d (Bind Abbr)
173 u))).(\lambda (H13: (subst0 O u t3 t2)).(K_ind (\lambda (k0: K).((getl O
174 (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to (pr3 (CHead c k0 u1) t1 t2)))
175 (\lambda (b: B).(\lambda (H14: (getl O (CHead c (Bind b) u2) (CHead d (Bind
176 Abbr) u))).(let H15 \def (f_equal C C (\lambda (e: C).(match e in C return
177 (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow
178 c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c
179 (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind
180 Abbr) u) H14))) in ((let H16 \def (f_equal C B (\lambda (e: C).(match e in C
181 return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _)
182 \Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0)
183 \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u)
184 (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2
185 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H14))) in ((let H17
186 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
187 with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d
188 (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr)
189 u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H14))) in
190 (\lambda (H18: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H20 \def (eq_ind
191 T u (\lambda (t4: T).(subst0 O t4 t3 t2)) H13 u2 H17) in (eq_ind B Abbr
192 (\lambda (b0: B).(pr3 (CHead c (Bind b0) u1) t1 t2)) (ex2_ind T (\lambda (t4:
193 T).(subst0 O u1 t3 t4)) (\lambda (t4: T).(pr0 t4 t2)) (pr3 (CHead c (Bind
194 Abbr) u1) t1 t2) (\lambda (x: T).(\lambda (H21: (subst0 O u1 t3 x)).(\lambda
195 (H22: (pr0 x t2)).(pr3_sing (CHead c (Bind Abbr) u1) x t1 (pr2_delta (CHead c
196 (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 t3 H10 x H21) t2 (pr3_pr2
197 (CHead c (Bind Abbr) u1) x t2 (pr2_free (CHead c (Bind Abbr) u1) x t2
198 H22)))))) (pr0_subst0_back u2 t3 t2 O H20 u1 H)) b H18))))) H16)) H15))))
199 (\lambda (f: F).(\lambda (H14: (getl O (CHead c (Flat f) u2) (CHead d (Bind
200 Abbr) u))).(pr3_pr2 (CHead c (Flat f) u1) t1 t2 (pr2_cflat c t1 t2 (pr2_delta
201 c d u O (getl_intro O c (CHead d (Bind Abbr) u) c (drop_refl c)
202 (clear_gen_flat f c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Flat f)
203 u2) (CHead d (Bind Abbr) u) H14))) t1 t3 H10 t2 H13) f u1)))) k H12)))
204 (\lambda (i0: nat).(\lambda (IHi: (((getl i0 (CHead c k u2) (CHead d (Bind
205 Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c k u1) t1
206 t2))))).(\lambda (H12: (getl (S i0) (CHead c k u2) (CHead d (Bind Abbr)
207 u))).(\lambda (H13: (subst0 (S i0) u t3 t2)).(K_ind (\lambda (k0: K).((getl
208 (S i0) (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((((getl i0 (CHead c k0
209 u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c k0
210 u1) t1 t2)))) \to (pr3 (CHead c k0 u1) t1 t2)))) (\lambda (b: B).(\lambda
211 (H14: (getl (S i0) (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(\lambda
212 (_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0
213 u t3 t2) \to (pr3 (CHead c (Bind b) u1) t1 t2))))).(pr3_pr2 (CHead c (Bind b)
214 u1) t1 t2 (pr2_delta (CHead c (Bind b) u1) d u (S i0) (getl_head (Bind b) i0
215 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c (CHead d (Bind Abbr) u) u2
216 i0 H14) u1) t1 t3 H10 t2 H13))))) (\lambda (f: F).(\lambda (H14: (getl (S i0)
217 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(\lambda (_: (((getl i0
218 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to
219 (pr3 (CHead c (Flat f) u1) t1 t2))))).(pr3_pr2 (CHead c (Flat f) u1) t1 t2
220 (pr2_cflat c t1 t2 (pr2_delta c d u (r (Flat f) i0) (getl_gen_S (Flat f) c
221 (CHead d (Bind Abbr) u) u2 i0 H14) t1 t3 H10 t2 H13) f u1))))) k H12 IHi)))))
222 i H9 H11)))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0
223 (CHead c k u2) H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C (CHead c k u2))
224 (refl_equal T t1) (refl_equal T t2)))))))))).
226 theorem pr3_pr2_pr2_t:
227 \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall
228 (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3
229 (CHead c k u1) t1 t2))))))))
231 \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1
232 u2)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda (t:
233 T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t u1)
234 \to ((eq T t0 u2) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2
235 (CHead c k u2) t1 t2) \to (pr3 (CHead c k u1) t1 t2)))))))))))) with
236 [(pr2_free c0 t1 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2:
237 (eq T t1 u1)).(\lambda (H3: (eq T t2 u2)).(eq_ind C c (\lambda (_: C).((eq T
238 t1 u1) \to ((eq T t2 u2) \to ((pr0 t1 t2) \to (\forall (t3: T).(\forall (t4:
239 T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3
240 t4))))))))) (\lambda (H4: (eq T t1 u1)).(eq_ind T u1 (\lambda (t: T).((eq T
241 t2 u2) \to ((pr0 t t2) \to (\forall (t3: T).(\forall (t4: T).(\forall (k:
242 K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4)))))))) (\lambda
243 (H5: (eq T t2 u2)).(eq_ind T u2 (\lambda (t: T).((pr0 u1 t) \to (\forall (t3:
244 T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3
245 (CHead c k u1) t3 t4))))))) (\lambda (H6: (pr0 u1 u2)).(\lambda (t3:
246 T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H7: (pr2 (CHead c k u2) t3
247 t4)).(pr3_pr0_pr2_t u1 u2 H6 c t3 t4 k H7)))))) t2 (sym_eq T t2 u2 H5))) t1
248 (sym_eq T t1 u1 H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u
249 i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq
250 T t1 u1)).(\lambda (H5: (eq T t u2)).(eq_ind C c (\lambda (c1: C).((eq T t1
251 u1) \to ((eq T t u2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1
252 t2) \to ((subst0 i u t2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k:
253 K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4)))))))))))
254 (\lambda (H6: (eq T t1 u1)).(eq_ind T u1 (\lambda (t0: T).((eq T t u2) \to
255 ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t)
256 \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3
257 t4) \to (pr3 (CHead c k u1) t3 t4)))))))))) (\lambda (H7: (eq T t
258 u2)).(eq_ind T u2 (\lambda (t0: T).((getl i c (CHead d (Bind Abbr) u)) \to
259 ((pr0 u1 t2) \to ((subst0 i u t2 t0) \to (\forall (t3: T).(\forall (t4:
260 T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3
261 t4))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9:
262 (pr0 u1 t2)).(\lambda (H10: (subst0 i u t2 u2)).(\lambda (t3: T).(\lambda
263 (t0: T).(\lambda (k: K).(\lambda (H11: (pr2 (CHead c k u2) t3 t0)).(let H12
264 \def (match H11 in pr2 return (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5:
265 T).(\lambda (_: (pr2 c1 t4 t5)).((eq C c1 (CHead c k u2)) \to ((eq T t4 t3)
266 \to ((eq T t5 t0) \to (pr3 (CHead c k u1) t3 t0)))))))) with [(pr2_free c1 t4
267 t5 H12) \Rightarrow (\lambda (H13: (eq C c1 (CHead c k u2))).(\lambda (H14:
268 (eq T t4 t3)).(\lambda (H15: (eq T t5 t0)).(eq_ind C (CHead c k u2) (\lambda
269 (_: C).((eq T t4 t3) \to ((eq T t5 t0) \to ((pr0 t4 t5) \to (pr3 (CHead c k
270 u1) t3 t0))))) (\lambda (H16: (eq T t4 t3)).(eq_ind T t3 (\lambda (t6:
271 T).((eq T t5 t0) \to ((pr0 t6 t5) \to (pr3 (CHead c k u1) t3 t0)))) (\lambda
272 (H17: (eq T t5 t0)).(eq_ind T t0 (\lambda (t6: T).((pr0 t3 t6) \to (pr3
273 (CHead c k u1) t3 t0))) (\lambda (H18: (pr0 t3 t0)).(pr3_pr2 (CHead c k u1)
274 t3 t0 (pr2_free (CHead c k u1) t3 t0 H18))) t5 (sym_eq T t5 t0 H17))) t4
275 (sym_eq T t4 t3 H16))) c1 (sym_eq C c1 (CHead c k u2) H13) H14 H15 H12)))) |
276 (pr2_delta c1 d0 u0 i0 H12 t4 t5 H13 t6 H14) \Rightarrow (\lambda (H15: (eq C
277 c1 (CHead c k u2))).(\lambda (H16: (eq T t4 t3)).(\lambda (H17: (eq T t6
278 t0)).(eq_ind C (CHead c k u2) (\lambda (c2: C).((eq T t4 t3) \to ((eq T t6
279 t0) \to ((getl i0 c2 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t4 t5) \to ((subst0
280 i0 u0 t5 t6) \to (pr3 (CHead c k u1) t3 t0))))))) (\lambda (H18: (eq T t4
281 t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t6 t0) \to ((getl i0 (CHead c k u2)
282 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t7 t5) \to ((subst0 i0 u0 t5 t6) \to
283 (pr3 (CHead c k u1) t3 t0)))))) (\lambda (H19: (eq T t6 t0)).(eq_ind T t0
284 (\lambda (t7: T).((getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to
285 ((pr0 t3 t5) \to ((subst0 i0 u0 t5 t7) \to (pr3 (CHead c k u1) t3 t0)))))
286 (\lambda (H20: (getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda
287 (H21: (pr0 t3 t5)).(\lambda (H22: (subst0 i0 u0 t5 t0)).(nat_ind (\lambda (n:
288 nat).((getl n (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5
289 t0) \to (pr3 (CHead c k u1) t3 t0)))) (\lambda (H23: (getl O (CHead c k u2)
290 (CHead d0 (Bind Abbr) u0))).(\lambda (H24: (subst0 O u0 t5 t0)).(K_ind
291 (\lambda (k0: K).((clear (CHead c k0 u2) (CHead d0 (Bind Abbr) u0)) \to (pr3
292 (CHead c k0 u1) t3 t0))) (\lambda (b: B).(\lambda (H25: (clear (CHead c (Bind
293 b) u2) (CHead d0 (Bind Abbr) u0))).(let H26 \def (f_equal C C (\lambda (e:
294 C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 |
295 (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b)
296 u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H25)) in ((let H27 \def
297 (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
298 [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K
299 return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
300 \Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2)
301 (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H25)) in ((let H28 \def
302 (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
303 [(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind
304 Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0)
305 u2 H25)) in (\lambda (H29: (eq B Abbr b)).(\lambda (_: (eq C d0 c)).(let H31
306 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t0)) H24 u2 H28) in
307 (eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c (Bind b0) u1) t3 t0)) (ex2_ind
308 T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(subst0 (S (plus i
309 O)) u t7 t0)) (pr3 (CHead c (Bind Abbr) u1) t3 t0) (\lambda (x: T).(\lambda
310 (H32: (subst0 O t2 t5 x)).(\lambda (H33: (subst0 (S (plus i O)) u x t0)).(let
311 H34 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O
312 i))) in (let H35 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n
313 u x t0)) H33 (S i) H34) in (ex2_ind T (\lambda (t7: T).(subst0 O u1 t5 t7))
314 (\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c (Bind Abbr) u1) t3 t0) (\lambda
315 (x0: T).(\lambda (H36: (subst0 O u1 t5 x0)).(\lambda (H37: (pr0 x0
316 x)).(pr3_sing (CHead c (Bind Abbr) u1) x0 t3 (pr2_delta (CHead c (Bind Abbr)
317 u1) c u1 O (getl_refl Abbr c u1) t3 t5 H21 x0 H36) t0 (pr3_pr2 (CHead c (Bind
318 Abbr) u1) x0 t0 (pr2_delta (CHead c (Bind Abbr) u1) d u (S i)
319 (getl_clear_bind Abbr (CHead c (Bind Abbr) u1) c u1 (clear_bind Abbr c u1)
320 (CHead d (Bind Abbr) u) i H8) x0 x H37 t0 H35)))))) (pr0_subst0_back t2 t5 x
321 O H32 u1 H9))))))) (subst0_subst0 t5 t0 u2 O H31 t2 u i H10)) b H29)))))
322 H27)) H26)))) (\lambda (f: F).(\lambda (H25: (clear (CHead c (Flat f) u2)
323 (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c (Flat f) u1) t3 t0 (pr2_cflat c
324 t3 t0 (pr2_delta c d0 u0 O (getl_intro O c (CHead d0 (Bind Abbr) u0) c
325 (drop_refl c) (clear_gen_flat f c (CHead d0 (Bind Abbr) u0) u2 H25)) t3 t5
326 H21 t0 H24) f u1)))) k (getl_gen_O (CHead c k u2) (CHead d0 (Bind Abbr) u0)
327 H23)))) (\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c k u2) (CHead d0
328 (Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t0) \to (pr3 (CHead c k u1) t3
329 t0))))).(\lambda (H23: (getl (S i1) (CHead c k u2) (CHead d0 (Bind Abbr)
330 u0))).(\lambda (H24: (subst0 (S i1) u0 t5 t0)).(K_ind (\lambda (k0: K).((getl
331 (S i1) (CHead c k0 u2) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c k0 u1) t3
332 t0))) (\lambda (b: B).(\lambda (H25: (getl (S i1) (CHead c (Bind b) u2)
333 (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c (Bind b) u1) t3 t0 (pr2_delta
334 (CHead c (Bind b) u1) d0 u0 (S i1) (getl_head (Bind b) i1 c (CHead d0 (Bind
335 Abbr) u0) (getl_gen_S (Bind b) c (CHead d0 (Bind Abbr) u0) u2 i1 H25) u1) t3
336 t5 H21 t0 H24)))) (\lambda (f: F).(\lambda (H25: (getl (S i1) (CHead c (Flat
337 f) u2) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c (Flat f) u1) t3 t0
338 (pr2_cflat c t3 t0 (pr2_delta c d0 u0 (r (Flat f) i1) (getl_gen_S (Flat f) c
339 (CHead d0 (Bind Abbr) u0) u2 i1 H25) t3 t5 H21 t0 H24) f u1)))) k H23))))) i0
340 H20 H22)))) t6 (sym_eq T t6 t0 H19))) t4 (sym_eq T t4 t3 H18))) c1 (sym_eq C
341 c1 (CHead c k u2) H15) H16 H17 H12 H13 H14))))]) in (H12 (refl_equal C (CHead
342 c k u2)) (refl_equal T t3) (refl_equal T t0)))))))))) t (sym_eq T t u2 H7)))
343 t1 (sym_eq T t1 u1 H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0
344 (refl_equal C c) (refl_equal T u1) (refl_equal T u2)))))).
346 theorem pr3_pr2_pr3_t:
347 \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall
348 (k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u1 u2) \to
349 (pr3 (CHead c k u1) t1 t2))))))))
351 \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
352 (k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2)
353 (\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u1 u2) \to (pr3
354 (CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c
355 u1 u2)).(pr3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3:
356 T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_:
357 (pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u1 u2)
358 \to (pr3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u1
359 u2)).(pr3_t t0 t3 (CHead c k u1) (pr3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2
360 u1 H3)))))))))) t1 t2 H)))))).
362 theorem pr3_pr3_pr3_t:
363 \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
364 (t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u2) t1 t2) \to (pr3
365 (CHead c k u1) t1 t2))))))))
367 \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
368 u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall
369 (t2: T).(\forall (k: K).((pr3 (CHead c k t0) t1 t2) \to (pr3 (CHead c k t) t1
370 t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k:
371 K).(\lambda (H0: (pr3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda
372 (t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2
373 t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pr3
374 (CHead c k t3) t4 t5) \to (pr3 (CHead c k t2) t4 t5))))))).(\lambda (t0:
375 T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pr3 (CHead c k t3) t0
376 t4)).(pr3_pr2_pr3_t c t2 t0 t4 k (H2 t0 t4 k H3) t1 H0))))))))))) u1 u2 H)))).
379 \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h
380 d c e) \to (\forall (t1: T).(\forall (t2: T).((pr3 e t1 t2) \to (pr3 c (lift
381 h d t1) (lift h d t2)))))))))
383 \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
384 (H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 e t1
385 t2)).(pr3_ind e (\lambda (t: T).(\lambda (t0: T).(pr3 c (lift h d t) (lift h
386 d t0)))) (\lambda (t: T).(pr3_refl c (lift h d t))) (\lambda (t0: T).(\lambda
387 (t3: T).(\lambda (H1: (pr2 e t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 e t0
388 t4)).(\lambda (H3: (pr3 c (lift h d t0) (lift h d t4))).(pr3_sing c (lift h d
389 t0) (lift h d t3) (pr2_lift c e h d H t3 t0 H1) (lift h d t4) H3))))))) t1 t2
393 \forall (c: C).(\forall (w: T).(\forall (u: T).(let t \def (THead (Bind
394 Abst) w u) in (\forall (v: T).((pr3 c v w) \to (pr3 c (THead (Bind Abst) v
395 (THead (Flat Appl) (TLRef O) (lift (S O) O t))) t))))))
397 \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind
398 Abst) w u) in (\lambda (v: T).(\lambda (H: (pr3 c v w)).(eq_ind_r T (THead
399 (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr3 c
400 (THead (Bind Abst) v (THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w
401 u))) (pr3_head_12 c v w H (Bind Abst) (THead (Flat Appl) (TLRef O) (THead
402 (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) u (pr3_pr1 (THead (Flat
403 Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) u
404 (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u)) (THead (Flat
405 Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u)))
406 (pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef O)) (lift (S
407 O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u))) u (pr1_sing
408 (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind Abbr) (TLRef O)
409 (lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O) (pr0_refl (TLRef O))
410 (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u))
411 (lift (S O) O u) (subst1_lift_S u O O (le_n O))) u (pr1_pr0 (THead (Bind
412 Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr not_abbr_abst u u
413 (pr0_refl u) (TLRef O))))) (CHead c (Bind Abst) w))) (lift (S O) O (THead
414 (Bind Abst) w u)) (lift_bind Abst w u (S O) O))))))).