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/csubt/clear.ma".
19 include "LambdaDelta-1/csubt/drop.ma".
21 include "LambdaDelta-1/getl/clear.ma".
23 theorem csubt_getl_abbr:
24 \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall
25 (n: nat).((getl n c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g
26 c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n
27 c2 (CHead d2 (Bind Abbr) u)))))))))))
29 \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda
30 (n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abbr) u))).(let H0 \def
31 (getl_gen_all c1 (CHead d1 (Bind Abbr) u) n H) in (ex2_ind C (\lambda (e:
32 C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u)))
33 (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1
34 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))) (\lambda (x:
35 C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind
36 Abbr) u))).(C_ind (\lambda (c: C).((drop n O c1 c) \to ((clear c (CHead d1
37 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda
38 (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr)
39 u))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 (CSort n0))).(\lambda
40 (H4: (clear (CSort n0) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1
41 (Bind Abbr) u) n0 H4 (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda
42 (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr)
43 u)))))))))) (\lambda (x0: C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0
44 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 c2) \to (ex2 C
45 (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind
46 Abbr) u)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: (drop n O c1
47 (CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 (Bind Abbr)
48 u))).(K_ind (\lambda (k0: K).((drop n O c1 (CHead x0 k0 t)) \to ((clear
49 (CHead x0 k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1
50 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2
51 (CHead d2 (Bind Abbr) u))))))))) (\lambda (b: B).(\lambda (H5: (drop n O c1
52 (CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1
53 (Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C
54 return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _)
55 \Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t)
56 (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def
57 (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
58 [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K
59 return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
60 \Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t)
61 (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def
62 (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
63 [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind
64 Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u)
65 t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda
66 (c2: C).(\lambda (H12: (csubt g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda
67 (t0: T).(drop n O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def
68 (eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) u))) H13 Abbr
69 H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1 (CHead c
70 (Bind Abbr) u))) H14 d1 H11) in (ex2_ind C (\lambda (d2: C).(csubt g d1 d2))
71 (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2:
72 C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))))
73 (\lambda (x1: C).(\lambda (H16: (csubt g d1 x1)).(\lambda (H17: (drop n O c2
74 (CHead x1 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2))
75 (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) x1 H16 (getl_intro n
76 c2 (CHead x1 (Bind Abbr) u) (CHead x1 (Bind Abbr) u) H17 (clear_bind Abbr x1
77 u)))))) (csubt_drop_abbr g n c1 c2 H12 d1 u H15)))))))))) H8)) H7)))))
78 (\lambda (f: F).(\lambda (H5: (drop n O c1 (CHead x0 (Flat f) t))).(\lambda
79 (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr) u))).(let H7 \def H5
80 in (unintro C c1 (\lambda (c: C).((drop n O c (CHead x0 (Flat f) t)) \to
81 (\forall (c2: C).((csubt g c c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2))
82 (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))))) (nat_ind (\lambda
83 (n0: nat).(\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall
84 (c2: C).((csubt g x1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2))
85 (\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u))))))))) (\lambda (x1:
86 C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2:
87 C).(\lambda (H9: (csubt g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c:
88 C).(csubt g c c2)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat
89 f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abbr) u)
90 (clear_gen_flat f x0 (CHead d1 (Bind Abbr) u) t H6) f t) in (let H11 \def
91 (csubt_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u)
92 H_y) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead d1 (Bind Abbr) u) e2))
93 (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2))
94 (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x2:
95 C).(\lambda (H12: (csubt g (CHead d1 (Bind Abbr) u) x2)).(\lambda (H13:
96 (clear c2 x2)).(let H14 \def (csubt_gen_abbr g d1 x2 u H12) in (ex2_ind C
97 (\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csubt
98 g d1 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O
99 c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x3: C).(\lambda (H15: (eq C x2
100 (CHead x3 (Bind Abbr) u))).(\lambda (H16: (csubt g d1 x3)).(let H17 \def
101 (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) u) H15)
102 in (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2
103 (CHead d2 (Bind Abbr) u))) x3 H16 (getl_intro O c2 (CHead x3 (Bind Abbr) u)
104 c2 (drop_refl c2) H17)))))) H14))))) H11)))))))) (\lambda (n0: nat).(\lambda
105 (H8: ((\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall
106 (c2: C).((csubt g x1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2))
107 (\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u)))))))))).(\lambda (x1:
108 C).(\lambda (H9: (drop (S n0) O x1 (CHead x0 (Flat f) t))).(\lambda (c2:
109 C).(\lambda (H10: (csubt g x1 c2)).(let H11 \def (drop_clear x1 (CHead x0
110 (Flat f) t) n0 H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e:
111 C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) (\lambda (_:
112 B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead x0 (Flat f) t)))))
113 (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2
114 (CHead d2 (Bind Abbr) u)))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4:
115 T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0
116 O x3 (CHead x0 (Flat f) t))).(let H14 \def (csubt_clear_conf g x1 c2 H10
117 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead
118 x3 (Bind x2) x4) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2:
119 C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr)
120 u)))) (\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 (Bind x2) x4)
121 x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def (csubt_gen_bind g x2 x3 x5
122 x4 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
123 T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
124 C).(\lambda (_: T).(csubt g x3 e2)))) (ex2 C (\lambda (d2: C).(csubt g d1
125 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda
126 (x6: B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7
127 (Bind x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def (eq_ind C x5
128 (\lambda (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21
129 \def (H8 x3 H13 x7 H19) in (ex2_ind C (\lambda (d2: C).(csubt g d1 d2))
130 (\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2:
131 C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr)
132 u)))) (\lambda (x9: C).(\lambda (H22: (csubt g d1 x9)).(\lambda (H23: (getl
133 n0 x7 (CHead x9 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1
134 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u))) x9 H22
135 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) u) n0 H23)))))
136 H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))) k H3 H4))))))) x H1
139 theorem csubt_getl_abst:
140 \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (t: T).(\forall
141 (n: nat).((getl n c1 (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g
142 c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
143 C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
144 C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
145 c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
148 \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (t: T).(\lambda
149 (n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abst) t))).(let H0 \def
150 (getl_gen_all c1 (CHead d1 (Bind Abst) t) n H) in (ex2_ind C (\lambda (e:
151 C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) t)))
152 (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1
153 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T
154 (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
155 (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u:
156 T).(ty3 g d2 u t))))))) (\lambda (x: C).(\lambda (H1: (drop n O c1
157 x)).(\lambda (H2: (clear x (CHead d1 (Bind Abst) t))).(C_ind (\lambda (c:
158 C).((drop n O c1 c) \to ((clear c (CHead d1 (Bind Abst) t)) \to (\forall (c2:
159 C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda
160 (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
161 C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
162 c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
163 t)))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 (CSort
164 n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abst)
165 t))).(clear_gen_sort (CHead d1 (Bind Abst) t) n0 H4 (\forall (c2: C).((csubt
166 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
167 C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
168 C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
169 c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
170 t))))))))))) (\lambda (x0: C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0
171 (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2
172 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2
173 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1
174 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u))))
175 (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))).(\lambda (k:
176 K).(\lambda (t0: T).(\lambda (H3: (drop n O c1 (CHead x0 k t0))).(\lambda
177 (H4: (clear (CHead x0 k t0) (CHead d1 (Bind Abst) t))).(K_ind (\lambda (k0:
178 K).((drop n O c1 (CHead x0 k0 t0)) \to ((clear (CHead x0 k0 t0) (CHead d1
179 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda
180 (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst)
181 t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda
182 (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2:
183 C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (b: B).(\lambda (H5:
184 (drop n O c1 (CHead x0 (Bind b) t0))).(\lambda (H6: (clear (CHead x0 (Bind b)
185 t0) (CHead d1 (Bind Abst) t))).(let H7 \def (f_equal C C (\lambda (e:
186 C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 |
187 (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) t) (CHead x0 (Bind b)
188 t0) (clear_gen_bind b x0 (CHead d1 (Bind Abst) t) t0 H6)) in ((let H8 \def
189 (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
190 [(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 in K
191 return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
192 \Rightarrow Abst])])) (CHead d1 (Bind Abst) t) (CHead x0 (Bind b) t0)
193 (clear_gen_bind b x0 (CHead d1 (Bind Abst) t) t0 H6)) in ((let H9 \def
194 (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
195 [(CSort _) \Rightarrow t | (CHead _ _ t1) \Rightarrow t1])) (CHead d1 (Bind
196 Abst) t) (CHead x0 (Bind b) t0) (clear_gen_bind b x0 (CHead d1 (Bind Abst) t)
197 t0 H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1
198 x0)).(\lambda (c2: C).(\lambda (H12: (csubt g c1 c2)).(let H13 \def (eq_ind_r
199 T t0 (\lambda (t1: T).(drop n O c1 (CHead x0 (Bind b) t1))) H5 t H9) in (let
200 H14 \def (eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) t)))
201 H13 Abst H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1
202 (CHead c (Bind Abst) t))) H14 d1 H11) in (or_ind (ex2 C (\lambda (d2:
203 C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t))))
204 (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2:
205 C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2:
206 C).(\lambda (u: T).(ty3 g d2 u t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1
207 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T
208 (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
209 (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u:
210 T).(ty3 g d2 u t))))) (\lambda (H16: (ex2 C (\lambda (d2: C).(csubt g d1 d2))
211 (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t))))).(ex2_ind C
212 (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2
213 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
214 C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
215 C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
216 c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
217 t))))) (\lambda (x1: C).(\lambda (H17: (csubt g d1 x1)).(\lambda (H18: (drop
218 n O c2 (CHead x1 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g
219 d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T
220 (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
221 (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u:
222 T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda
223 (d2: C).(getl n c2 (CHead d2 (Bind Abst) t))) x1 H17 (getl_intro n c2 (CHead
224 x1 (Bind Abst) t) (CHead x1 (Bind Abst) t) H18 (clear_bind Abst x1 t)))))))
225 H16)) (\lambda (H16: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1
226 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr)
227 u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T
228 (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
229 (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u:
230 T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda
231 (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
232 C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
233 c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
234 t))))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (H17: (csubt g d1
235 x1)).(\lambda (H18: (drop n O c2 (CHead x1 (Bind Abbr) x2))).(\lambda (H19:
236 (ty3 g x1 x2 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda
237 (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
238 C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
239 c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
240 t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2)))
241 (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u))))
242 (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x1 x2 H17 (getl_intro n c2
243 (CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2) H18 (clear_bind Abbr x1
244 x2)) H19))))))) H16)) (csubt_drop_abst g n c1 c2 H12 d1 t H15)))))))))) H8))
245 H7))))) (\lambda (f: F).(\lambda (H5: (drop n O c1 (CHead x0 (Flat f)
246 t0))).(\lambda (H6: (clear (CHead x0 (Flat f) t0) (CHead d1 (Bind Abst)
247 t))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop n O c (CHead x0
248 (Flat f) t0)) \to (\forall (c2: C).((csubt g c c2) \to (or (ex2 C (\lambda
249 (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst)
250 t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda
251 (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2:
252 C).(\lambda (u: T).(ty3 g d2 u t))))))))) (nat_ind (\lambda (n0:
253 nat).(\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t0)) \to (\forall
254 (c2: C).((csubt g x1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2))
255 (\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda
256 (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u:
257 T).(getl n0 c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u:
258 T).(ty3 g d2 u t)))))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead
259 x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H9: (csubt g x1 c2)).(let H10
260 \def (eq_ind C x1 (\lambda (c: C).(csubt g c c2)) H9 (CHead x0 (Flat f) t0)
261 (drop_gen_refl x1 (CHead x0 (Flat f) t0) H8)) in (let H_y \def (clear_flat x0
262 (CHead d1 (Bind Abst) t) (clear_gen_flat f x0 (CHead d1 (Bind Abst) t) t0 H6)
263 f t0) in (let H11 \def (csubt_clear_conf g (CHead x0 (Flat f) t0) c2 H10
264 (CHead d1 (Bind Abst) t) H_y) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead
265 d1 (Bind Abst) t) e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda
266 (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst)
267 t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda
268 (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2:
269 C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x2: C).(\lambda (H12: (csubt
270 g (CHead d1 (Bind Abst) t) x2)).(\lambda (H13: (clear c2 x2)).(let H14 \def
271 (csubt_gen_abst g d1 x2 t H12) in (or_ind (ex2 C (\lambda (e2: C).(eq C x2
272 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2))) (ex3_2 C T
273 (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 (Bind Abbr) v2))))
274 (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (e2: C).(\lambda
275 (v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2))
276 (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda
277 (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u:
278 T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u:
279 T).(ty3 g d2 u t))))) (\lambda (H15: (ex2 C (\lambda (e2: C).(eq C x2 (CHead
280 e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2)))).(ex2_ind C (\lambda
281 (e2: C).(eq C x2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2))
282 (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2
283 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
284 T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2
285 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
286 (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst) t))).(\lambda
287 (H17: (csubt g d1 x3)).(let H18 \def (eq_ind C x2 (\lambda (c: C).(clear c2
288 c)) H13 (CHead x3 (Bind Abst) t) H16) in (or_introl (ex2 C (\lambda (d2:
289 C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))))
290 (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2:
291 C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2:
292 C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g
293 d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))) x3 H17
294 (getl_intro O c2 (CHead x3 (Bind Abst) t) c2 (drop_refl c2) H18))))))) H15))
295 (\lambda (H15: (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead
296 e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2)))
297 (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))))).(ex3_2_ind C T (\lambda
298 (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
299 C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g
300 e2 v2 t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
301 C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
302 C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O
303 c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
304 t))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3
305 (Bind Abbr) x4))).(\lambda (H17: (csubt g d1 x3)).(\lambda (H18: (ty3 g x3 x4
306 t)).(let H19 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3
307 (Bind Abbr) x4) H16) in (or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2))
308 (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda
309 (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u:
310 T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u:
311 T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt
312 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr)
313 u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x3 x4 H17 (getl_intro
314 O c2 (CHead x3 (Bind Abbr) x4) c2 (drop_refl c2) H19) H18)))))))) H15))
315 H14))))) H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x1:
316 C).((drop n0 O x1 (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g x1
317 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl
318 n0 c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
319 T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2
320 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
321 t))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n0) O x1 (CHead x0 (Flat
322 f) t0))).(\lambda (c2: C).(\lambda (H10: (csubt g x1 c2)).(let H11 \def
323 (drop_clear x1 (CHead x0 (Flat f) t0) n0 H9) in (ex2_3_ind B C T (\lambda (b:
324 B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v)))))
325 (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead x0 (Flat
326 f) t0))))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
327 C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
328 C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl
329 (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g
330 d2 u t))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12:
331 (clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0 O x3 (CHead x0
332 (Flat f) t0))).(let H14 \def (csubt_clear_conf g x1 c2 H10 (CHead x3 (Bind
333 x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead x3 (Bind x2) x4)
334 e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1
335 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T
336 (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
337 (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda
338 (u: T).(ty3 g d2 u t))))) (\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3
339 (Bind x2) x4) x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def
340 (csubt_gen_bind g x2 x3 x5 x4 H15) in (ex2_3_ind B C T (\lambda (b2:
341 B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind b2) v2)))))
342 (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g x3 e2)))) (or (ex2
343 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead
344 d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1
345 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr)
346 u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x6:
347 B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind
348 x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def (eq_ind C x5 (\lambda
349 (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3
350 H13 x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
351 C).(getl n0 x7 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
352 C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl
353 n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2
354 u t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl
355 (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda
356 (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2
357 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
358 t))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2:
359 C).(getl n0 x7 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2:
360 C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abst) t)))
361 (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2
362 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
363 T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead
364 d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
365 (\lambda (x9: C).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24: (getl n0 x7
366 (CHead x9 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1
367 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T
368 (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
369 (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda
370 (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2))
371 (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t))) x9 H23
372 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abst) t) n0 H24)))))) H22))
373 (\lambda (H22: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2)))
374 (\lambda (d2: C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u))))
375 (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda
376 (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u:
377 T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u:
378 T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda
379 (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
380 C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl
381 (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g
382 d2 u t))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (H23: (csubt g d1
383 x9)).(\lambda (H24: (getl n0 x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H25:
384 (ty3 g x9 x10 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2))
385 (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T
386 (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda
387 (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda
388 (u: T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_:
389 T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead
390 d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x9 x10
391 H23 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) x10) n0 H24)
392 H25))))))) H22)) H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))) k H3
393 H4))))))) x H1 H2)))) H0))))))).