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/Level-1/LambdaDelta/pr3/props".
19 include "pr3/defs.ma".
21 include "pr2/props.ma".
23 theorem clear_pr3_trans:
24 \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr3 c2 t1 t2) \to
25 (\forall (c1: C).((clear c1 c2) \to (pr3 c1 t1 t2))))))
27 \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c2 t1
28 t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(pr3_ind c2 (\lambda (t:
29 T).(\lambda (t0: T).(pr3 c1 t t0))) (\lambda (t: T).(pr3_refl c1 t)) (\lambda
30 (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c2 t4 t3)).(\lambda (t5:
31 T).(\lambda (_: (pr3 c2 t3 t5)).(\lambda (H3: (pr3 c1 t3 t5)).(pr3_sing c1 t3
32 t4 (clear_pr2_trans c2 t4 t3 H1 c1 H0) t5 H3))))))) t1 t2 H)))))).
35 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pr3 c
38 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
39 t2)).(pr3_sing c t2 t1 H t2 (pr3_refl c t2))))).
42 \forall (t2: T).(\forall (t1: T).(\forall (c: C).((pr3 c t1 t2) \to (\forall
43 (t3: T).((pr3 c t2 t3) \to (pr3 c t1 t3))))))
45 \lambda (t2: T).(\lambda (t1: T).(\lambda (c: C).(\lambda (H: (pr3 c t1
46 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3: T).((pr3 c t0
47 t3) \to (pr3 c t t3))))) (\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr3
48 c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3
49 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall
50 (t5: T).((pr3 c t4 t5) \to (pr3 c t0 t5))))).(\lambda (t5: T).(\lambda (H3:
51 (pr3 c t4 t5)).(pr3_sing c t0 t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
54 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
55 (u: T).(\forall (f: F).(pr3 c (THead (Flat f) u t1) (THead (Flat f) u
58 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
59 t2)).(\lambda (u: T).(\lambda (f: F).(pr3_ind c (\lambda (t: T).(\lambda (t0:
60 T).(pr3 c (THead (Flat f) u t) (THead (Flat f) u t0)))) (\lambda (t:
61 T).(pr3_refl c (THead (Flat f) u t))) (\lambda (t0: T).(\lambda (t3:
62 T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0
63 t4)).(\lambda (H2: (pr3 c (THead (Flat f) u t0) (THead (Flat f) u
64 t4))).(pr3_sing c (THead (Flat f) u t0) (THead (Flat f) u t3) (pr2_thin_dx c
65 t3 t0 H0 u f) (THead (Flat f) u t4) H2))))))) t1 t2 H)))))).
68 \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
69 (k: K).(\forall (t: T).(pr3 c (THead k u1 t) (THead k u2 t)))))))
71 \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
72 u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (k: K).(\forall
73 (t1: T).(pr3 c (THead k t t1) (THead k t0 t1)))))) (\lambda (t: T).(\lambda
74 (k: K).(\lambda (t0: T).(pr3_refl c (THead k t t0))))) (\lambda (t2:
75 T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda
76 (_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (k: K).(\forall (t: T).(pr3 c
77 (THead k t2 t) (THead k t3 t)))))).(\lambda (k: K).(\lambda (t: T).(pr3_sing
78 c (THead k t2 t) (THead k t1 t) (pr2_head_1 c t1 t2 H0 k t) (THead k t3 t)
79 (H2 k t)))))))))) u1 u2 H)))).
82 \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall
83 (k: K).((pr3 (CHead c k u) t1 t2) \to (pr3 c (THead k u t1) (THead k u
86 \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
87 (k: K).(\lambda (H: (pr3 (CHead c k u) t1 t2)).(pr3_ind (CHead c k u)
88 (\lambda (t: T).(\lambda (t0: T).(pr3 c (THead k u t) (THead k u t0))))
89 (\lambda (t: T).(pr3_refl c (THead k u t))) (\lambda (t0: T).(\lambda (t3:
90 T).(\lambda (H0: (pr2 (CHead c k u) t3 t0)).(\lambda (t4: T).(\lambda (_:
91 (pr3 (CHead c k u) t0 t4)).(\lambda (H2: (pr3 c (THead k u t0) (THead k u
92 t4))).(pr3_sing c (THead k u t0) (THead k u t3) (pr2_head_2 c u t3 t0 k H0)
93 (THead k u t4) H2))))))) t1 t2 H)))))).
96 \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
97 (k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u1) t1 t2) \to (pr3
98 c (THead k u1 t1) (THead k u2 t2)))))))))
100 \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
101 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3
102 (CHead c k u1) t1 t2)).(pr3_t (THead k u1 t2) (THead k u1 t1) c (pr3_head_2 c
103 u1 t1 t2 k H0) (THead k u2 t2) (pr3_head_1 c u1 u2 H k t2))))))))).
106 \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
107 (k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u2) t1 t2) \to (pr3
108 c (THead k u1 t1) (THead k u2 t2)))))))))
110 \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
111 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3
112 (CHead c k u2) t1 t2)).(pr3_t (THead k u2 t1) (THead k u1 t1) c (pr3_head_1 c
113 u1 u2 H k t1) (THead k u2 t2) (pr3_head_2 c u2 t1 t2 k H0))))))))).
116 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
117 (f: F).(\forall (v: T).(pr3 (CHead c (Flat f) v) t1 t2))))))
119 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
120 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (f: F).(\forall (v:
121 T).(pr3 (CHead c (Flat f) v) t t0))))) (\lambda (t: T).(\lambda (f:
122 F).(\lambda (v: T).(pr3_refl (CHead c (Flat f) v) t)))) (\lambda (t3:
123 T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda
124 (_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (f: F).(\forall (v: T).(pr3 (CHead
125 c (Flat f) v) t3 t5))))).(\lambda (f: F).(\lambda (v: T).(pr3_sing (CHead c
126 (Flat f) v) t3 t4 (pr2_cflat c t4 t3 H0 f v) t5 (H2 f v)))))))))) t1 t2 H)))).
128 theorem pr3_pr0_pr2_t:
129 \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (c: C).(\forall
130 (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3
131 (CHead c k u1) t1 t2))))))))
133 \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (c:
134 C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2
135 (CHead c k u2) t1 t2)).(let H1 \def (match H0 in pr2 return (\lambda (c0:
136 C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0
137 (CHead c k u2)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr3 (CHead c k u1) t1
138 t2)))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0
139 (CHead c k u2))).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3
140 t2)).(eq_ind C (CHead c k u2) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2)
141 \to ((pr0 t0 t3) \to (pr3 (CHead c k u1) t1 t2))))) (\lambda (H5: (eq T t0
142 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr3
143 (CHead c k u1) t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda
144 (t: T).((pr0 t1 t) \to (pr3 (CHead c k u1) t1 t2))) (\lambda (H7: (pr0 t1
145 t2)).(pr3_pr2 (CHead c k u1) t1 t2 (pr2_free (CHead c k u1) t1 t2 H7))) t3
146 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 (CHead c k u2)
147 H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t0 t3 H2 t H3) \Rightarrow (\lambda
148 (H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq
149 T t t2)).(eq_ind C (CHead c k u2) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t
150 t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i
151 u t3 t) \to (pr3 (CHead c k u1) t1 t2))))))) (\lambda (H7: (eq T t0
152 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c k u2)
153 (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr3
154 (CHead c k u1) t1 t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda
155 (t4: T).((getl i (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to
156 ((subst0 i u t3 t4) \to (pr3 (CHead c k u1) t1 t2))))) (\lambda (H9: (getl i
157 (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda
158 (H11: (subst0 i u t3 t2)).(nat_ind (\lambda (n: nat).((getl n (CHead c k u2)
159 (CHead d (Bind Abbr) u)) \to ((subst0 n u t3 t2) \to (pr3 (CHead c k u1) t1
160 t2)))) (\lambda (H12: (getl O (CHead c k u2) (CHead d (Bind Abbr)
161 u))).(\lambda (H13: (subst0 O u t3 t2)).(K_ind (\lambda (k0: K).((getl O
162 (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to (pr3 (CHead c k0 u1) t1 t2)))
163 (\lambda (b: B).(\lambda (H14: (getl O (CHead c (Bind b) u2) (CHead d (Bind
164 Abbr) u))).(let H15 \def (f_equal C C (\lambda (e: C).(match e in C return
165 (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow
166 c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c
167 (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind
168 Abbr) u) H14))) in ((let H16 \def (f_equal C B (\lambda (e: C).(match e in C
169 return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _)
170 \Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0)
171 \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u)
172 (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2
173 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H14))) in ((let H17
174 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
175 with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d
176 (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr)
177 u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H14))) in
178 (\lambda (H18: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H20 \def (eq_ind
179 T u (\lambda (t4: T).(subst0 O t4 t3 t2)) H13 u2 H17) in (eq_ind B Abbr
180 (\lambda (b0: B).(pr3 (CHead c (Bind b0) u1) t1 t2)) (ex2_ind T (\lambda (t4:
181 T).(subst0 O u1 t3 t4)) (\lambda (t4: T).(pr0 t4 t2)) (pr3 (CHead c (Bind
182 Abbr) u1) t1 t2) (\lambda (x: T).(\lambda (H21: (subst0 O u1 t3 x)).(\lambda
183 (H22: (pr0 x t2)).(pr3_sing (CHead c (Bind Abbr) u1) x t1 (pr2_delta (CHead c
184 (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 t3 H10 x H21) t2 (pr3_pr2
185 (CHead c (Bind Abbr) u1) x t2 (pr2_free (CHead c (Bind Abbr) u1) x t2
186 H22)))))) (pr0_subst0_back u2 t3 t2 O H20 u1 H)) b H18))))) H16)) H15))))
187 (\lambda (f: F).(\lambda (H14: (getl O (CHead c (Flat f) u2) (CHead d (Bind
188 Abbr) u))).(pr3_pr2 (CHead c (Flat f) u1) t1 t2 (pr2_cflat c t1 t2 (pr2_delta
189 c d u O (getl_intro O c (CHead d (Bind Abbr) u) c (drop_refl c)
190 (clear_gen_flat f c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Flat f)
191 u2) (CHead d (Bind Abbr) u) H14))) t1 t3 H10 t2 H13) f u1)))) k H12)))
192 (\lambda (i0: nat).(\lambda (IHi: (((getl i0 (CHead c k u2) (CHead d (Bind
193 Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c k u1) t1
194 t2))))).(\lambda (H12: (getl (S i0) (CHead c k u2) (CHead d (Bind Abbr)
195 u))).(\lambda (H13: (subst0 (S i0) u t3 t2)).(K_ind (\lambda (k0: K).((getl
196 (S i0) (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((((getl i0 (CHead c k0
197 u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c k0
198 u1) t1 t2)))) \to (pr3 (CHead c k0 u1) t1 t2)))) (\lambda (b: B).(\lambda
199 (H14: (getl (S i0) (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(\lambda
200 (_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0
201 u t3 t2) \to (pr3 (CHead c (Bind b) u1) t1 t2))))).(pr3_pr2 (CHead c (Bind b)
202 u1) t1 t2 (pr2_delta (CHead c (Bind b) u1) d u (S i0) (getl_head (Bind b) i0
203 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c (CHead d (Bind Abbr) u) u2
204 i0 H14) u1) t1 t3 H10 t2 H13))))) (\lambda (f: F).(\lambda (H14: (getl (S i0)
205 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(\lambda (_: (((getl i0
206 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to
207 (pr3 (CHead c (Flat f) u1) t1 t2))))).(pr3_pr2 (CHead c (Flat f) u1) t1 t2
208 (pr2_cflat c t1 t2 (pr2_delta c d u (r (Flat f) i0) (getl_gen_S (Flat f) c
209 (CHead d (Bind Abbr) u) u2 i0 H14) t1 t3 H10 t2 H13) f u1))))) k H12 IHi)))))
210 i H9 H11)))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0
211 (CHead c k u2) H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C (CHead c k u2))
212 (refl_equal T t1) (refl_equal T t2)))))))))).
214 theorem pr3_pr2_pr2_t:
215 \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall
216 (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3
217 (CHead c k u1) t1 t2))))))))
219 \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1
220 u2)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda (t:
221 T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t u1)
222 \to ((eq T t0 u2) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2
223 (CHead c k u2) t1 t2) \to (pr3 (CHead c k u1) t1 t2)))))))))))) with
224 [(pr2_free c0 t1 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2:
225 (eq T t1 u1)).(\lambda (H3: (eq T t2 u2)).(eq_ind C c (\lambda (_: C).((eq T
226 t1 u1) \to ((eq T t2 u2) \to ((pr0 t1 t2) \to (\forall (t3: T).(\forall (t4:
227 T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3
228 t4))))))))) (\lambda (H4: (eq T t1 u1)).(eq_ind T u1 (\lambda (t: T).((eq T
229 t2 u2) \to ((pr0 t t2) \to (\forall (t3: T).(\forall (t4: T).(\forall (k:
230 K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4)))))))) (\lambda
231 (H5: (eq T t2 u2)).(eq_ind T u2 (\lambda (t: T).((pr0 u1 t) \to (\forall (t3:
232 T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3
233 (CHead c k u1) t3 t4))))))) (\lambda (H6: (pr0 u1 u2)).(\lambda (t3:
234 T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H7: (pr2 (CHead c k u2) t3
235 t4)).(pr3_pr0_pr2_t u1 u2 H6 c t3 t4 k H7)))))) t2 (sym_eq T t2 u2 H5))) t1
236 (sym_eq T t1 u1 H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u
237 i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq
238 T t1 u1)).(\lambda (H5: (eq T t u2)).(eq_ind C c (\lambda (c1: C).((eq T t1
239 u1) \to ((eq T t u2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1
240 t2) \to ((subst0 i u t2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k:
241 K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4)))))))))))
242 (\lambda (H6: (eq T t1 u1)).(eq_ind T u1 (\lambda (t0: T).((eq T t u2) \to
243 ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t)
244 \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3
245 t4) \to (pr3 (CHead c k u1) t3 t4)))))))))) (\lambda (H7: (eq T t
246 u2)).(eq_ind T u2 (\lambda (t0: T).((getl i c (CHead d (Bind Abbr) u)) \to
247 ((pr0 u1 t2) \to ((subst0 i u t2 t0) \to (\forall (t3: T).(\forall (t4:
248 T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3
249 t4))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9:
250 (pr0 u1 t2)).(\lambda (H10: (subst0 i u t2 u2)).(\lambda (t3: T).(\lambda
251 (t0: T).(\lambda (k: K).(\lambda (H11: (pr2 (CHead c k u2) t3 t0)).(let H12
252 \def (match H11 in pr2 return (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5:
253 T).(\lambda (_: (pr2 c1 t4 t5)).((eq C c1 (CHead c k u2)) \to ((eq T t4 t3)
254 \to ((eq T t5 t0) \to (pr3 (CHead c k u1) t3 t0)))))))) with [(pr2_free c1 t4
255 t5 H12) \Rightarrow (\lambda (H13: (eq C c1 (CHead c k u2))).(\lambda (H14:
256 (eq T t4 t3)).(\lambda (H15: (eq T t5 t0)).(eq_ind C (CHead c k u2) (\lambda
257 (_: C).((eq T t4 t3) \to ((eq T t5 t0) \to ((pr0 t4 t5) \to (pr3 (CHead c k
258 u1) t3 t0))))) (\lambda (H16: (eq T t4 t3)).(eq_ind T t3 (\lambda (t6:
259 T).((eq T t5 t0) \to ((pr0 t6 t5) \to (pr3 (CHead c k u1) t3 t0)))) (\lambda
260 (H17: (eq T t5 t0)).(eq_ind T t0 (\lambda (t6: T).((pr0 t3 t6) \to (pr3
261 (CHead c k u1) t3 t0))) (\lambda (H18: (pr0 t3 t0)).(pr3_pr2 (CHead c k u1)
262 t3 t0 (pr2_free (CHead c k u1) t3 t0 H18))) t5 (sym_eq T t5 t0 H17))) t4
263 (sym_eq T t4 t3 H16))) c1 (sym_eq C c1 (CHead c k u2) H13) H14 H15 H12)))) |
264 (pr2_delta c1 d0 u0 i0 H12 t4 t5 H13 t6 H14) \Rightarrow (\lambda (H15: (eq C
265 c1 (CHead c k u2))).(\lambda (H16: (eq T t4 t3)).(\lambda (H17: (eq T t6
266 t0)).(eq_ind C (CHead c k u2) (\lambda (c2: C).((eq T t4 t3) \to ((eq T t6
267 t0) \to ((getl i0 c2 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t4 t5) \to ((subst0
268 i0 u0 t5 t6) \to (pr3 (CHead c k u1) t3 t0))))))) (\lambda (H18: (eq T t4
269 t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t6 t0) \to ((getl i0 (CHead c k u2)
270 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t7 t5) \to ((subst0 i0 u0 t5 t6) \to
271 (pr3 (CHead c k u1) t3 t0)))))) (\lambda (H19: (eq T t6 t0)).(eq_ind T t0
272 (\lambda (t7: T).((getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to
273 ((pr0 t3 t5) \to ((subst0 i0 u0 t5 t7) \to (pr3 (CHead c k u1) t3 t0)))))
274 (\lambda (H20: (getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda
275 (H21: (pr0 t3 t5)).(\lambda (H22: (subst0 i0 u0 t5 t0)).((match i0 in nat
276 return (\lambda (n: nat).((getl n (CHead c k u2) (CHead d0 (Bind Abbr) u0))
277 \to ((subst0 n u0 t5 t0) \to (pr3 (CHead c k u1) t3 t0)))) with [O
278 \Rightarrow (\lambda (H23: (getl O (CHead c k u2) (CHead d0 (Bind Abbr)
279 u0))).(\lambda (H24: (subst0 O u0 t5 t0)).((match k in K return (\lambda (k0:
280 K).((clear (CHead c k0 u2) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c k0
281 u1) t3 t0))) with [(Bind b) \Rightarrow (\lambda (H25: (clear (CHead c (Bind
282 b) u2) (CHead d0 (Bind Abbr) u0))).(let H26 \def (f_equal C C (\lambda (e:
283 C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 |
284 (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b)
285 u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H25)) in ((let H27 \def
286 (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
287 [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K
288 return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
289 \Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2)
290 (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H25)) in ((let H28 \def
291 (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
292 [(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind
293 Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0)
294 u2 H25)) in (\lambda (H29: (eq B Abbr b)).(\lambda (_: (eq C d0 c)).(let H31
295 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t0)) H24 u2 H28) in
296 (eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c (Bind b0) u1) t3 t0)) (ex2_ind
297 T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(subst0 (S (plus i
298 O)) u t7 t0)) (pr3 (CHead c (Bind Abbr) u1) t3 t0) (\lambda (x: T).(\lambda
299 (H32: (subst0 O t2 t5 x)).(\lambda (H33: (subst0 (S (plus i O)) u x t0)).(let
300 H34 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O
301 i))) in (let H35 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n
302 u x t0)) H33 (S i) H34) in (ex2_ind T (\lambda (t7: T).(subst0 O u1 t5 t7))
303 (\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c (Bind Abbr) u1) t3 t0) (\lambda
304 (x0: T).(\lambda (H36: (subst0 O u1 t5 x0)).(\lambda (H37: (pr0 x0
305 x)).(pr3_sing (CHead c (Bind Abbr) u1) x0 t3 (pr2_delta (CHead c (Bind Abbr)
306 u1) c u1 O (getl_refl Abbr c u1) t3 t5 H21 x0 H36) t0 (pr3_pr2 (CHead c (Bind
307 Abbr) u1) x0 t0 (pr2_delta (CHead c (Bind Abbr) u1) d u (S i)
308 (getl_clear_bind Abbr (CHead c (Bind Abbr) u1) c u1 (clear_bind Abbr c u1)
309 (CHead d (Bind Abbr) u) i H8) x0 x H37 t0 H35)))))) (pr0_subst0_back t2 t5 x
310 O H32 u1 H9))))))) (subst0_subst0 t5 t0 u2 O H31 t2 u i H10)) b H29)))))
311 H27)) H26))) | (Flat f) \Rightarrow (\lambda (H25: (clear (CHead c (Flat f)
312 u2) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c (Flat f) u1) t3 t0
313 (pr2_cflat c t3 t0 (pr2_delta c d0 u0 O (getl_intro O c (CHead d0 (Bind Abbr)
314 u0) c (drop_refl c) (clear_gen_flat f c (CHead d0 (Bind Abbr) u0) u2 H25)) t3
315 t5 H21 t0 H24) f u1)))]) (getl_gen_O (CHead c k u2) (CHead d0 (Bind Abbr) u0)
316 H23)))) | (S n) \Rightarrow (\lambda (H23: (getl (S n) (CHead c k u2) (CHead
317 d0 (Bind Abbr) u0))).(\lambda (H24: (subst0 (S n) u0 t5 t0)).((match k in K
318 return (\lambda (k0: K).((getl (S n) (CHead c k0 u2) (CHead d0 (Bind Abbr)
319 u0)) \to (pr3 (CHead c k0 u1) t3 t0))) with [(Bind b) \Rightarrow (\lambda
320 (H25: (getl (S n) (CHead c (Bind b) u2) (CHead d0 (Bind Abbr) u0))).(pr3_pr2
321 (CHead c (Bind b) u1) t3 t0 (pr2_delta (CHead c (Bind b) u1) d0 u0 (S n)
322 (getl_head (Bind b) n c (CHead d0 (Bind Abbr) u0) (getl_gen_S (Bind b) c
323 (CHead d0 (Bind Abbr) u0) u2 n H25) u1) t3 t5 H21 t0 H24))) | (Flat f)
324 \Rightarrow (\lambda (H25: (getl (S n) (CHead c (Flat f) u2) (CHead d0 (Bind
325 Abbr) u0))).(pr3_pr2 (CHead c (Flat f) u1) t3 t0 (pr2_cflat c t3 t0
326 (pr2_delta c d0 u0 (r (Flat f) n) (getl_gen_S (Flat f) c (CHead d0 (Bind
327 Abbr) u0) u2 n H25) t3 t5 H21 t0 H24) f u1)))]) H23)))]) H20 H22)))) t6
328 (sym_eq T t6 t0 H19))) t4 (sym_eq T t4 t3 H18))) c1 (sym_eq C c1 (CHead c k
329 u2) H15) H16 H17 H12 H13 H14))))]) in (H12 (refl_equal C (CHead c k u2))
330 (refl_equal T t3) (refl_equal T t0)))))))))) t (sym_eq T t u2 H7))) t1
331 (sym_eq T t1 u1 H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0
332 (refl_equal C c) (refl_equal T u1) (refl_equal T u2)))))).
334 theorem pr3_pr2_pr3_t:
335 \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall
336 (k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u1 u2) \to
337 (pr3 (CHead c k u1) t1 t2))))))))
339 \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
340 (k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2)
341 (\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u1 u2) \to (pr3
342 (CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c
343 u1 u2)).(pr3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3:
344 T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_:
345 (pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u1 u2)
346 \to (pr3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u1
347 u2)).(pr3_t t0 t3 (CHead c k u1) (pr3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2
348 u1 H3)))))))))) t1 t2 H)))))).
350 theorem pr3_pr3_pr3_t:
351 \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
352 (t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u2) t1 t2) \to (pr3
353 (CHead c k u1) t1 t2))))))))
355 \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
356 u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall
357 (t2: T).(\forall (k: K).((pr3 (CHead c k t0) t1 t2) \to (pr3 (CHead c k t) t1
358 t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k:
359 K).(\lambda (H0: (pr3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda
360 (t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2
361 t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pr3
362 (CHead c k t3) t4 t5) \to (pr3 (CHead c k t2) t4 t5))))))).(\lambda (t0:
363 T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pr3 (CHead c k t3) t0
364 t4)).(pr3_pr2_pr3_t c t2 t0 t4 k (H2 t0 t4 k H3) t1 H0))))))))))) u1 u2 H)))).
367 \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h
368 d c e) \to (\forall (t1: T).(\forall (t2: T).((pr3 e t1 t2) \to (pr3 c (lift
369 h d t1) (lift h d t2)))))))))
371 \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
372 (H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 e t1
373 t2)).(pr3_ind e (\lambda (t: T).(\lambda (t0: T).(pr3 c (lift h d t) (lift h
374 d t0)))) (\lambda (t: T).(pr3_refl c (lift h d t))) (\lambda (t0: T).(\lambda
375 (t3: T).(\lambda (H1: (pr2 e t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 e t0
376 t4)).(\lambda (H3: (pr3 c (lift h d t0) (lift h d t4))).(pr3_sing c (lift h d
377 t0) (lift h d t3) (pr2_lift c e h d H t3 t0 H1) (lift h d t4) H3))))))) t1 t2