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 "LambdaDelta-1/pr2/defs.ma".
19 include "LambdaDelta-1/pr0/props.ma".
21 include "LambdaDelta-1/getl/drop.ma".
23 include "LambdaDelta-1/getl/clear.ma".
26 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
27 (u: T).(\forall (f: F).(pr2 c (THead (Flat f) u t1) (THead (Flat f) u
30 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
31 t2)).(\lambda (u: T).(\lambda (f: F).(pr2_ind (\lambda (c0: C).(\lambda (t:
32 T).(\lambda (t0: T).(pr2 c0 (THead (Flat f) u t) (THead (Flat f) u t0)))))
33 (\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t0
34 t3)).(pr2_free c0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u
35 (pr0_refl u) t0 t3 H0 (Flat f))))))) (\lambda (c0: C).(\lambda (d:
36 C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
37 Abbr) u0))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (pr0 t0
38 t3)).(\lambda (t: T).(\lambda (H2: (subst0 i u0 t3 t)).(pr2_delta c0 d u0 i
39 H0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t0
40 t3 H1 (Flat f)) (THead (Flat f) u t) (subst0_snd (Flat f) u0 t t3 i H2
41 u)))))))))))) c t1 t2 H)))))).
44 \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall
45 (k: K).(\forall (t: T).(pr2 c (THead k u1 t) (THead k u2 t)))))))
47 \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1
48 u2)).(\lambda (k: K).(\lambda (t: T).(pr2_ind (\lambda (c0: C).(\lambda (t0:
49 T).(\lambda (t1: T).(pr2 c0 (THead k t0 t) (THead k t1 t))))) (\lambda (c0:
50 C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(pr2_free c0
51 (THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k))))))
52 (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
53 (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2:
54 T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t2
55 t0)).(pr2_delta c0 d u i H0 (THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H1
56 t t (pr0_refl t) k) (THead k t0 t) (subst0_fst u t0 t2 i H2 t k)))))))))))) c
60 \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall
61 (k: K).((pr2 (CHead c k u) t1 t2) \to (pr2 c (THead k u t1) (THead k u
64 \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
65 (k: K).(\lambda (H: (pr2 (CHead c k u) t1 t2)).(insert_eq C (CHead c k u)
66 (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead k u t1) (THead
67 k u t2))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0:
68 C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c k u)) \to (pr2 c
69 (THead k u t) (THead k u t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda
70 (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c k
71 u))).(pr2_free c (THead k u t3) (THead k u t4) (pr0_comp u u (pr0_refl u) t3
72 t4 H1 k))))))) (K_ind (\lambda (k0: K).(\forall (c0: C).(\forall (d:
73 C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Abbr) u0))
74 \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t:
75 T).((subst0 i u0 t4 t) \to ((eq C c0 (CHead c k0 u)) \to (pr2 c (THead k0 u
76 t3) (THead k0 u t)))))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (d:
77 C).(\lambda (u0: T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0
78 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4)
79 \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Bind b) u))
80 \to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u t)))))))))) (\lambda (H1:
81 (getl O c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4:
82 T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4
83 t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) u))).(let H5 \def (eq_ind C c0
84 (\lambda (c1: C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Bind b)
85 u) H4) in (let H6 \def (f_equal C C (\lambda (e: C).(match e in C return
86 (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow
87 c1])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c
88 (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind
89 Abbr) u0) H5))) in ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C
90 return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _)
91 \Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0)
92 \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0)
93 (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u
94 (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in ((let H8
95 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
96 with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d
97 (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr)
98 u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in
99 (\lambda (H9: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H11 \def (eq_ind T
100 u0 (\lambda (t0: T).(subst0 O t0 t4 t)) H3 u H8) in (eq_ind B Abbr (\lambda
101 (b0: B).(pr2 c (THead (Bind b0) u t3) (THead (Bind b0) u t))) (pr2_free c
102 (THead (Bind Abbr) u t3) (THead (Bind Abbr) u t) (pr0_delta u u (pr0_refl u)
103 t3 t4 H2 t H11)) b H9))))) H7)) H6)))))))))) (\lambda (n: nat).(\lambda (H1:
104 (((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4:
105 T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead
106 c (Bind b) u)) \to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u
107 t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda
108 (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda
109 (H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Bind b)
110 u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind
111 Abbr) u0))) H2 (CHead c (Bind b) u) H5) in (let H7 \def (eq_ind C c0 (\lambda
112 (c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall
113 (t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1
114 (CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u t5) (THead (Bind b) u
115 t0)))))))))) H1 (CHead c (Bind b) u) H5) in (pr2_delta c d u0 (r (Bind b) n)
116 (getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H6) (THead (Bind b) u t3)
117 (THead (Bind b) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Bind b)) (THead
118 (Bind b) u t) (subst0_snd (Bind b) u0 t t4 (r (Bind b) n) H4 u)))))))))))))
119 i)))))) (\lambda (f: F).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0:
120 T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 (CHead d (Bind
121 Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall
122 (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Flat f) u)) \to (pr2 c
123 (THead (Flat f) u t3) (THead (Flat f) u t)))))))))) (\lambda (H1: (getl O c0
124 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2:
125 (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 t)).(\lambda (H4:
126 (eq C c0 (CHead c (Flat f) u))).(let H5 \def (eq_ind C c0 (\lambda (c1:
127 C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Flat f) u) H4) in
128 (pr2_delta c d u0 O (getl_intro O c (CHead d (Bind Abbr) u0) c (drop_refl c)
129 (clear_gen_flat f c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Flat f)
130 u) (CHead d (Bind Abbr) u0) H5))) (THead (Flat f) u t3) (THead (Flat f) u t4)
131 (pr0_comp u u (pr0_refl u) t3 t4 H2 (Flat f)) (THead (Flat f) u t)
132 (subst0_snd (Flat f) u0 t t4 O H3 u)))))))))) (\lambda (n: nat).(\lambda (H1:
133 (((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4:
134 T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead
135 c (Flat f) u)) \to (pr2 c (THead (Flat f) u t3) (THead (Flat f) u
136 t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda
137 (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda
138 (H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Flat f)
139 u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind
140 Abbr) u0))) H2 (CHead c (Flat f) u) H5) in (let H7 \def (eq_ind C c0 (\lambda
141 (c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall
142 (t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1
143 (CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u t5) (THead (Flat f) u
144 t0)))))))))) H1 (CHead c (Flat f) u) H5) in (pr2_delta c d u0 (r (Flat f) n)
145 (getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H6) (THead (Flat f) u t3)
146 (THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Flat f)) (THead
147 (Flat f) u t) (subst0_snd (Flat f) u0 t t4 (r (Flat f) n) H4 u)))))))))))))
148 i)))))) k) y t1 t2 H0))) H)))))).
150 theorem clear_pr2_trans:
151 \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to
152 (\forall (c1: C).((clear c1 c2) \to (pr2 c1 t1 t2))))))
154 \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c2 t1
155 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\forall (c1:
156 C).((clear c1 c) \to (pr2 c1 t t0)))))) (\lambda (c: C).(\lambda (t3:
157 T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c1: C).(\lambda (_:
158 (clear c1 c)).(pr2_free c1 t3 t4 H0))))))) (\lambda (c: C).(\lambda (d:
159 C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind
160 Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3
161 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c1:
162 C).(\lambda (H3: (clear c1 c)).(pr2_delta c1 d u i (clear_getl_trans i c
163 (CHead d (Bind Abbr) u) H0 c1 H3) t3 t4 H1 t H2))))))))))))) c2 t1 t2 H)))).
166 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
167 (f: F).(\forall (v: T).(pr2 (CHead c (Flat f) v) t1 t2))))))
169 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
170 t2)).(\lambda (f: F).(\lambda (v: T).(pr2_ind (\lambda (c0: C).(\lambda (t:
171 T).(\lambda (t0: T).(pr2 (CHead c0 (Flat f) v) t t0)))) (\lambda (c0:
172 C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free
173 (CHead c0 (Flat f) v) t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
174 (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr)
175 u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda
176 (t: T).(\lambda (H2: (subst0 i u t4 t)).(pr2_delta (CHead c0 (Flat f) v) d u
177 i (getl_flat c0 (CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))) c
181 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
182 (k: K).(\forall (u: T).(pr2 (CTail k u c) t1 t2))))))
184 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
185 t2)).(\lambda (k: K).(\lambda (u: T).(pr2_ind (\lambda (c0: C).(\lambda (t:
186 T).(\lambda (t0: T).(pr2 (CTail k u c0) t t0)))) (\lambda (c0: C).(\lambda
187 (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free (CTail k u c0)
188 t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
189 nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3:
190 T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2:
191 (subst0 i u0 t4 t)).(pr2_delta (CTail k u c0) (CTail k u d) u0 i (getl_ctail
192 Abbr c0 d u0 i H0 k u) t3 t4 H1 t H2))))))))))) c t1 t2 H)))))).
195 \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1:
196 T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Bind b) v1) t1 t2) \to
197 (\forall (v2: T).(pr2 (CHead c (Bind b) v2) t1 t2))))))))
199 \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda
200 (v1: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind
201 b) v1) t1 t2)).(\lambda (v2: T).(insert_eq C (CHead c (Bind b) v1) (\lambda
202 (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 (CHead c (Bind b) v2) t1 t2))
203 (\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0:
204 C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) v1)) \to (pr2
205 (CHead c (Bind b) v2) t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda
206 (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b)
207 v1))).(pr2_free (CHead c (Bind b) v2) t3 t4 H2)))))) (\lambda (c0:
208 C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H2: (getl i c0
209 (CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3:
210 (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5:
211 (eq C c0 (CHead c (Bind b) v1))).(let H6 \def (eq_ind C c0 (\lambda (c1:
212 C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead c (Bind b) v1) H5) in
213 (nat_ind (\lambda (n: nat).((getl n (CHead c (Bind b) v1) (CHead d (Bind
214 Abbr) u)) \to ((subst0 n u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))
215 (\lambda (H7: (getl O (CHead c (Bind b) v1) (CHead d (Bind Abbr)
216 u))).(\lambda (H8: (subst0 O u t4 t)).(let H9 \def (f_equal C C (\lambda (e:
217 C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
218 (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b)
219 v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind
220 b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H10 \def (f_equal C B (\lambda
221 (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
222 Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
223 [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind
224 Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1
225 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H11
226 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
227 with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d
228 (Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr)
229 u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in
230 (\lambda (H12: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H14 \def (eq_ind
231 T u (\lambda (t0: T).(subst0 O t0 t4 t)) H8 v1 H11) in (let H15 \def
232 (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abbr))) H Abbr H12) in (eq_ind B
233 Abbr (\lambda (b0: B).(pr2 (CHead c (Bind b0) v2) t3 t)) (let H16 \def (match
234 (H15 (refl_equal B Abbr)) in False return (\lambda (_: False).(pr2 (CHead c
235 (Bind Abbr) v2) t3 t)) with []) in H16) b H12)))))) H10)) H9)))) (\lambda
236 (i0: nat).(\lambda (_: (((getl i0 (CHead c (Bind b) v1) (CHead d (Bind Abbr)
237 u)) \to ((subst0 i0 u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))).(\lambda
238 (H7: (getl (S i0) (CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda
239 (H8: (subst0 (S i0) u t4 t)).(pr2_delta (CHead c (Bind b) v2) d u (S i0)
240 (getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c
241 (CHead d (Bind Abbr) u) v1 i0 H7) v2) t3 t4 H3 t H8))))) i H6 H4)))))))))))))
242 y t1 t2 H1))) H0)))))))).
245 \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h
246 d c e) \to (\forall (t1: T).(\forall (t2: T).((pr2 e t1 t2) \to (pr2 c (lift
247 h d t1) (lift h d t2)))))))))
249 \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
250 (H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 e t1
251 t2)).(insert_eq C e (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c
252 (lift h d t1) (lift h d t2))) (\lambda (y: C).(\lambda (H1: (pr2 y t1
253 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 e)
254 \to (pr2 c (lift h d t) (lift h d t0)))))) (\lambda (c0: C).(\lambda (t3:
255 T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0
256 e)).(pr2_free c (lift h d t3) (lift h d t4) (pr0_lift t3 t4 H2 h d)))))))
257 (\lambda (c0: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
258 (H2: (getl i c0 (CHead d0 (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4:
259 T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4
260 t)).(\lambda (H5: (eq C c0 e)).(let H6 \def (eq_ind C c0 (\lambda (c1:
261 C).(getl i c1 (CHead d0 (Bind Abbr) u))) H2 e H5) in (lt_le_e i d (pr2 c
262 (lift h d t3) (lift h d t)) (\lambda (H7: (lt i d)).(let H8 \def
263 (drop_getl_trans_le i d (le_S_n i d (le_S (S i) d H7)) c e h H (CHead d0
264 (Bind Abbr) u) H6) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i
265 O c e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1)))
266 (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d0 (Bind Abbr) u)))) (pr2 c
267 (lift h d t3) (lift h d t)) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H9:
268 (drop i O c x0)).(\lambda (H10: (drop h (minus d i) x0 x1)).(\lambda (H11:
269 (clear x1 (CHead d0 (Bind Abbr) u))).(let H12 \def (eq_ind nat (minus d i)
270 (\lambda (n: nat).(drop h n x0 x1)) H10 (S (minus d (S i))) (minus_x_Sy d i
271 H7)) in (let H13 \def (drop_clear_S x1 x0 h (minus d (S i)) H12 Abbr d0 u
272 H11) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h
273 (minus d (S i)) u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c
274 (lift h d t3) (lift h d t)) (\lambda (x: C).(\lambda (H14: (clear x0 (CHead x
275 (Bind Abbr) (lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S
276 i)) x d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead
277 x (Bind Abbr) (lift h (minus d (S i)) u)) x0 H9 H14) (lift h d t3) (lift h d
278 t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_lt t4 t u i H4 d H7
279 h))))) H13)))))))) H8))) (\lambda (H7: (le d i)).(pr2_delta c d0 u (plus i h)
280 (drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H6 H7) (lift h d t3)
281 (lift h d t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_ge t4 t u i h
282 H4 d H7)))))))))))))))) y t1 t2 H1))) H0)))))))).