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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 *)
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15 (* This file was automatically generated: do not edit *********************)
17 include "Basic-1/csubt/defs.ma".
19 theorem csubt_gen_abbr:
20 \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csubt g
21 (CHead e1 (Bind Abbr) v) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2
22 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))))
24 \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
25 (H: (csubt g (CHead e1 (Bind Abbr) v) c2)).(insert_eq C (CHead e1 (Bind Abbr)
26 v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C (\lambda (e2:
27 C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))))
28 (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda (c:
29 C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda
30 (e2: C).(eq C c0 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1
31 e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind
32 Abbr) v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C
33 return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
34 \Rightarrow False])) I (CHead e1 (Bind Abbr) v) H1) in (False_ind (ex2 C
35 (\lambda (e2: C).(eq C (CSort n) (CHead e2 (Bind Abbr) v))) (\lambda (e2:
36 C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1:
37 (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2
38 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2:
39 C).(csubt g e1 e2)))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C
40 (CHead c1 k u) (CHead e1 (Bind Abbr) v))).(let H4 \def (f_equal C C (\lambda
41 (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1
42 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind Abbr) v) H3)
43 in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
44 (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0]))
45 (CHead c1 k u) (CHead e1 (Bind Abbr) v) H3) in ((let H6 \def (f_equal C T
46 (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
47 \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind
48 Abbr) v) H3) in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1
49 e1)).(eq_ind_r T v (\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k
50 t) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K
51 (Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v)
52 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def
53 (eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C
54 (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt
55 g e1 e2))))) H2 e1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g
56 c c3)) H1 e1 H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr)
57 v) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)) c3
58 (refl_equal C (CHead c3 (Bind Abbr) v)) H10))) k H7) u H6)))) H5))
59 H4))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1
60 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda
61 (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1
62 e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1:
63 T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1
64 (Bind Abbr) v))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda
65 (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
66 \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
67 (_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_:
68 B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void
69 \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr)
70 v) H4) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2)
71 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H5)))))))))))
72 (\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_:
73 (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3
74 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (u:
75 T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (_: (ty3 g c3 u
76 t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abbr)
77 v))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (ee: C).(match
78 ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
79 (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
80 [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr
81 \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat
82 _) \Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H5) in (False_ind (ex2 C
83 (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v)))
84 (\lambda (e2: C).(csubt g e1 e2))) H6))))))))))) y c2 H0))) H))))).
89 theorem csubt_gen_abst:
90 \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csubt g
91 (CHead e1 (Bind Abst) v1) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead
92 e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda
93 (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
94 C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g
95 e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))))
97 \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda
98 (H: (csubt g (CHead e1 (Bind Abst) v1) c2)).(insert_eq C (CHead e1 (Bind
99 Abst) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(or (ex2 C (\lambda
100 (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1
101 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
102 Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_:
103 C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3
104 g e2 v2 v1)))))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g
105 (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or
106 (ex2 C (\lambda (e2: C).(eq C c0 (CHead e2 (Bind Abst) v1))) (\lambda (e2:
107 C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c0
108 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1
109 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2:
110 C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (n: nat).(\lambda (H1:
111 (eq C (CSort n) (CHead e1 (Bind Abst) v1))).(let H2 \def (eq_ind C (CSort n)
112 (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
113 \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abst)
114 v1) H1) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CSort n) (CHead e2
115 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2:
116 C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 (Bind Abbr) v2)))) (\lambda
117 (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2:
118 T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))
119 H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1
120 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C
121 (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt
122 g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2
123 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))
124 (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda
125 (v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3:
126 (eq C (CHead c1 k u) (CHead e1 (Bind Abst) v1))).(let H4 \def (f_equal C C
127 (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
128 \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1
129 (Bind Abst) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in
130 C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _)
131 \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H3) in ((let H6
132 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
133 with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u)
134 (CHead e1 (Bind Abst) v1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda
135 (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(or (ex2 C (\lambda (e2:
136 C).(eq C (CHead c3 k t) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g
137 e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t)
138 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1
139 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2:
140 C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) (eq_ind_r K (Bind Abst) (\lambda
141 (k0: K).(or (ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v1) (CHead e2 (Bind
142 Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2:
143 C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abbr) v2))))
144 (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda
145 (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2
146 v1)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind
147 Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst)
148 v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda
149 (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
150 T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1)))
151 (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H8) in (let
152 H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (or_introl
153 (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst)
154 v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda
155 (v2: T).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abbr) v2)))) (\lambda
156 (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2:
157 T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))
158 (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind
159 Abst) v1))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3
160 (Bind Abst) v1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
161 C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1
162 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2
163 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2:
164 C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
165 C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g
166 e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2
167 v1)))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1:
168 T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1
169 (Bind Abst) v1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda
170 (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
171 \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
172 (_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_:
173 B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void
174 \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abst)
175 v1) H4) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b)
176 u2) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T
177 (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) u2) (CHead e2
178 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))
179 (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda
180 (v2: T).(ty3 g e2 v2 v1))))) H5))))))))))) (\lambda (c1: C).(\lambda (c3:
181 C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind
182 Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst)
183 v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda
184 (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
185 T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1)))
186 (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (u:
187 T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (H4: (ty3 g c3 u
188 t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst)
189 v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
190 (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c]))
191 (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H5) in ((let H7 \def
192 (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
193 [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind
194 Abst) t) (CHead e1 (Bind Abst) v1) H5) in (\lambda (H8: (eq C c1 e1)).(let H9
195 \def (eq_ind T t (\lambda (t0: T).(ty3 g c3 u t0)) H4 v1 H7) in (let H10 \def
196 (eq_ind T t (\lambda (t0: T).(ty3 g c1 u t0)) H3 v1 H7) in (let H11 \def
197 (eq_ind C c1 (\lambda (c: C).(ty3 g c u v1)) H10 e1 H8) in (let H12 \def
198 (eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or (ex2
199 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2:
200 C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3
201 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1
202 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2:
203 C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H8) in (let H13 \def (eq_ind
204 C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (or_intror (ex2 C (\lambda
205 (e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abst) v1))) (\lambda
206 (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C
207 (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
208 C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g
209 e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))
210 (ex4_2_intro C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind
211 Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt
212 g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2:
213 C).(\lambda (v2: T).(ty3 g e2 v2 v1))) c3 u (refl_equal C (CHead c3 (Bind
214 Abbr) u)) H13 H11 H9))))))))) H6))))))))))) y c2 H0))) H))))).
219 theorem csubt_gen_flat:
220 \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).(\forall
221 (f: F).((csubt g (CHead e1 (Flat f) v) c2) \to (ex2 C (\lambda (e2: C).(eq C
222 c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))))))))
224 \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
225 (f: F).(\lambda (H: (csubt g (CHead e1 (Flat f) v) c2)).(insert_eq C (CHead
226 e1 (Flat f) v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C
227 (\lambda (e2: C).(eq C c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g
228 e1 e2)))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda
229 (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C (\lambda
230 (e2: C).(eq C c0 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1
231 e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Flat f)
232 v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return
233 (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
234 \Rightarrow False])) I (CHead e1 (Flat f) v) H1) in (False_ind (ex2 C
235 (\lambda (e2: C).(eq C (CSort n) (CHead e2 (Flat f) v))) (\lambda (e2:
236 C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1:
237 (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Flat f) v)) \to (ex2 C
238 (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g
239 e1 e2)))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k
240 u) (CHead e1 (Flat f) v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
241 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _)
242 \Rightarrow c])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in ((let H5 \def
243 (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
244 [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u)
245 (CHead e1 (Flat f) v) H3) in ((let H6 \def (f_equal C T (\lambda (e:
246 C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
247 (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in
248 (\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v
249 (\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2 (Flat
250 f) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K (Flat f) (\lambda
251 (k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v) (CHead e2 (Flat f) v)))
252 (\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def (eq_ind C c1 (\lambda (c:
253 C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C (\lambda (e2: C).(eq C c3
254 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))))) H2 e1 H8) in
255 (let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in
256 (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Flat f) v) (CHead e2 (Flat f)
257 v))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 (Flat f)
258 v)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c3:
259 C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Flat f)
260 v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda
261 (e2: C).(csubt g e1 e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b
262 Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind
263 Void) u1) (CHead e1 (Flat f) v))).(let H5 \def (eq_ind C (CHead c1 (Bind
264 Void) u1) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with
265 [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return
266 (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
267 False])])) I (CHead e1 (Flat f) v) H4) in (False_ind (ex2 C (\lambda (e2:
268 C).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Flat f) v))) (\lambda (e2:
269 C).(csubt g e1 e2))) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda
270 (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Flat f) v)) \to (ex2
271 C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g
272 e1 e2)))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u
273 t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t)
274 (CHead e1 (Flat f) v))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t)
275 (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
276 \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
277 (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
278 False])])) I (CHead e1 (Flat f) v) H5) in (False_ind (ex2 C (\lambda (e2:
279 C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Flat f) v))) (\lambda (e2:
280 C).(csubt g e1 e2))) H6))))))))))) y c2 H0))) H)))))).
285 theorem csubt_gen_bind:
286 \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall
287 (v1: T).((csubt g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2:
288 B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2)))))
289 (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))))))
291 \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda
292 (v1: T).(\lambda (H: (csubt g (CHead e1 (Bind b1) v1) c2)).(insert_eq C
293 (CHead e1 (Bind b1) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_:
294 C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2
295 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
296 T).(csubt g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csubt g y
297 c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind
298 b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
299 T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
300 C).(\lambda (_: T).(csubt g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq
301 C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n)
302 (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
303 \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1)
304 v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
305 (v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
306 (e2: C).(\lambda (_: T).(csubt g e1 e2))))) H2)))) (\lambda (c1: C).(\lambda
307 (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1
308 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
309 (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
310 C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (k: K).(\lambda (u:
311 T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(let H4 \def
312 (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
313 [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u)
314 (CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e:
315 C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
316 (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3)
317 in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
318 (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
319 (CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind
320 b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T
321 (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t)
322 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
323 T).(csubt g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T
324 (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 v1)
325 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
326 T).(csubt g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c
327 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
328 C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_:
329 B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 H8) in (let
330 H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in
331 (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C
332 (CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
333 (e2: C).(\lambda (_: T).(csubt g e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3
334 (Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
335 C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1
336 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
337 C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_:
338 B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (b:
339 B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2:
340 T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1)
341 v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
342 (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c]))
343 (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def
344 (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
345 [(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k in K return
346 (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
347 Void])])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7
348 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
349 with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1
350 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B Void
351 b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind C c1 (\lambda (c:
352 C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2:
353 B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2)))))
354 (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1
355 H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H9)
356 in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 (CHead e1 (Bind
357 b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
358 T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
359 C).(\lambda (_: T).(csubt g e1 e2))))))) H10 Void H8) in (ex2_3_intro B C T
360 (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b)
361 u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
362 T).(csubt g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 (Bind b) u2))
363 H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1:
364 (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3
365 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2
366 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g
367 e1 e2)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u
368 t)).(\lambda (H4: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst)
369 t) (CHead e1 (Bind b1) v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match
370 e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _
371 _) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in
372 ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_:
373 C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k
374 in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _)
375 \Rightarrow Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in
376 ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
377 C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead
378 c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in (\lambda (H9: (eq B Abst
379 b1)).(\lambda (H10: (eq C c1 e1)).(let H11 \def (eq_ind T t (\lambda (t0:
380 T).(ty3 g c3 u t0)) H4 v1 H8) in (let H12 \def (eq_ind T t (\lambda (t0:
381 T).(ty3 g c1 u t0)) H3 v1 H8) in (let H13 \def (eq_ind C c1 (\lambda (c:
382 C).(ty3 g c u v1)) H12 e1 H10) in (let H14 \def (eq_ind C c1 (\lambda (c:
383 C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2:
384 B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2)))))
385 (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1
386 H10) in (let H15 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H10)
387 in (let H16 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b)
388 v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq
389 C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda
390 (_: T).(csubt g e1 e2))))))) H14 Abst H9) in (ex2_3_intro B C T (\lambda (b2:
391 B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2
392 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g
393 e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H15))))))))))
394 H7)) H6))))))))))) y c2 H0))) H)))))).