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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/leq/defs.ma".
19 theorem leq_gen_sort1:
20 \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq
21 g (ASort h1 n1) a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2:
22 nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
23 k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2
24 (ASort h2 n2))))))))))
26 \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2:
27 A).(\lambda (H: (leq g (ASort h1 n1) a2)).(insert_eq A (ASort h1 n1) (\lambda
28 (a: A).(leq g a a2)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2:
29 nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a k) (aplus g (ASort
30 h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A
31 a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g y a2)).(leq_ind g
32 (\lambda (a: A).(\lambda (a0: A).((eq A a (ASort h1 n1)) \to (ex2_3 nat nat
33 nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a
34 k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2:
35 nat).(\lambda (_: nat).(eq A a0 (ASort h2 n2))))))))) (\lambda (h0:
36 nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k:
37 nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2)
38 k))).(\lambda (H2: (eq A (ASort h0 n0) (ASort h1 n1))).(let H3 \def (f_equal
39 A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort
40 n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) (ASort h1
41 n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return
42 (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
43 \Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h0
44 h1)).(let H6 \def (eq_ind nat n0 (\lambda (n: nat).(eq A (aplus g (ASort h0
45 n) k) (aplus g (ASort h2 n2) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n:
46 nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0:
47 nat).(eq A (aplus g (ASort h0 n) k0) (aplus g (ASort h3 n3) k0))))) (\lambda
48 (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) (ASort h3
49 n3))))))) (let H7 \def (eq_ind nat h0 (\lambda (n: nat).(eq A (aplus g (ASort
50 n n1) k) (aplus g (ASort h2 n2) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda
51 (n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
52 (k0: nat).(eq A (aplus g (ASort n n1) k0) (aplus g (ASort h3 n3) k0)))))
53 (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2)
54 (ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3:
55 nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3
56 n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A
57 (ASort h2 n2) (ASort h3 n3))))) n2 h2 k H7 (refl_equal A (ASort h2 n2))) h0
58 H5)) n0 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_:
59 (leq g a1 a3)).(\lambda (_: (((eq A a1 (ASort h1 n1)) \to (ex2_3 nat nat nat
60 (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a1 k)
61 (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
62 (_: nat).(eq A a3 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5:
63 A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a4 (ASort h1 n1)) \to
64 (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k:
65 nat).(eq A (aplus g a4 k) (aplus g (ASort h2 n2) k))))) (\lambda (n2:
66 nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a5 (ASort h2
67 n2))))))))).(\lambda (H5: (eq A (AHead a1 a4) (ASort h1 n1))).(let H6 \def
68 (eq_ind A (AHead a1 a4) (\lambda (ee: A).(match ee in A return (\lambda (_:
69 A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
70 True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2:
71 nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (AHead a1 a4) k)
72 (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
73 (_: nat).(eq A (AHead a3 a5) (ASort h2 n2)))))) H6))))))))))) y a2 H0)))
79 theorem leq_gen_head1:
80 \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g
81 (AHead a1 a2) a) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a1
82 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3:
83 A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))))))
85 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda
86 (H: (leq g (AHead a1 a2) a)).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq
87 g a0 a)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g
88 a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3:
89 A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0:
90 (leq g y a)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a0 (AHead a1
91 a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda
92 (_: A).(\lambda (a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq
93 A a3 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1:
94 nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort
95 h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h1 n1)
96 (AHead a1 a2))).(let H3 \def (eq_ind A (ASort h1 n1) (\lambda (ee: A).(match
97 ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True |
98 (AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A
99 (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda
100 (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h2 n2)
101 (AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1:
102 (leq g a0 a3)).(\lambda (H2: (((eq A a0 (AHead a1 a2)) \to (ex3_2 A A
103 (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda (_: A).(\lambda
104 (a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq A a3 (AHead a4
105 a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4
106 a5)).(\lambda (H4: (((eq A a4 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6:
107 A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2
108 a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A a5 (AHead a6
109 a7)))))))).(\lambda (H5: (eq A (AHead a0 a4) (AHead a1 a2))).(let H6 \def
110 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
111 [(ASort _ _) \Rightarrow a0 | (AHead a6 _) \Rightarrow a6])) (AHead a0 a4)
112 (AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A
113 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a6)
114 \Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H5) in (\lambda (H8: (eq A a0
115 a1)).(let H9 \def (eq_ind A a4 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to
116 (ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_:
117 A).(\lambda (a8: A).(leq g a2 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A
118 a5 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a4 (\lambda (a6:
119 A).(leq g a6 a5)) H3 a2 H7) in (let H11 \def (eq_ind A a0 (\lambda (a6:
120 A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_:
121 A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8))) (\lambda
122 (a7: A).(\lambda (a8: A).(eq A a3 (AHead a7 a8))))))) H2 a1 H8) in (let H12
123 \def (eq_ind A a0 (\lambda (a6: A).(leq g a6 a3)) H1 a1 H8) in (ex3_2_intro A
124 A (\lambda (a6: A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda
125 (a7: A).(leq g a2 a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a3 a5)
126 (AHead a6 a7)))) a3 a5 H12 H10 (refl_equal A (AHead a3 a5)))))))))
127 H6))))))))))) y a H0))) H))))).
132 theorem leq_gen_sort2:
133 \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq
134 g a2 (ASort h1 n1)) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2:
135 nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) (aplus g (ASort h1 n1)
136 k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2
137 (ASort h2 n2))))))))))
139 \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2:
140 A).(\lambda (H: (leq g a2 (ASort h1 n1))).(insert_eq A (ASort h1 n1) (\lambda
141 (a: A).(leq g a2 a)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2:
142 nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k)
143 (aplus g a k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq
144 A a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g a2 y)).(leq_ind
145 g (\lambda (a: A).(\lambda (a0: A).((eq A a0 (ASort h1 n1)) \to (ex2_3 nat
146 nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus
147 g (ASort h2 n2) k) (aplus g a0 k))))) (\lambda (n2: nat).(\lambda (h2:
148 nat).(\lambda (_: nat).(eq A a (ASort h2 n2))))))))) (\lambda (h0:
149 nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k:
150 nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2)
151 k))).(\lambda (H2: (eq A (ASort h2 n2) (ASort h1 n1))).(let H3 \def (f_equal
152 A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort
153 n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n2) (ASort h1
154 n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return
155 (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
156 \Rightarrow n2])) (ASort h2 n2) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h2
157 h1)).(let H6 \def (eq_ind nat n2 (\lambda (n: nat).(eq A (aplus g (ASort h0
158 n0) k) (aplus g (ASort h2 n) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n:
159 nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0:
160 nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h2 n) k0))))) (\lambda
161 (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) (ASort h3
162 n3))))))) (let H7 \def (eq_ind nat h2 (\lambda (n: nat).(eq A (aplus g (ASort
163 h0 n0) k) (aplus g (ASort n n1) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda
164 (n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
165 (k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort n n1) k0)))))
166 (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0)
167 (ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3:
168 nat).(\lambda (k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h1
169 n1) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A
170 (ASort h0 n0) (ASort h3 n3))))) n0 h0 k H7 (refl_equal A (ASort h0 n0))) h2
171 H5)) n2 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_:
172 (leq g a1 a3)).(\lambda (_: (((eq A a3 (ASort h1 n1)) \to (ex2_3 nat nat nat
173 (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
174 h2 n2) k) (aplus g a3 k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
175 (_: nat).(eq A a1 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5:
176 A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a5 (ASort h1 n1)) \to
177 (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k:
178 nat).(eq A (aplus g (ASort h2 n2) k) (aplus g a5 k))))) (\lambda (n2:
179 nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a4 (ASort h2
180 n2))))))))).(\lambda (H5: (eq A (AHead a3 a5) (ASort h1 n1))).(let H6 \def
181 (eq_ind A (AHead a3 a5) (\lambda (ee: A).(match ee in A return (\lambda (_:
182 A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
183 True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2:
184 nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k)
185 (aplus g (AHead a3 a5) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
186 (_: nat).(eq A (AHead a1 a4) (ASort h2 n2)))))) H6))))))))))) a2 y H0)))
192 theorem leq_gen_head2:
193 \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g a
194 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a3
195 a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3:
196 A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))))))
198 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda
199 (H: (leq g a (AHead a1 a2))).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq
200 g a a0)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g
201 a3 a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3:
202 A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0:
203 (leq g a y)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a3 (AHead a1
204 a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda
205 (_: A).(\lambda (a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq
206 A a0 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1:
207 nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort
208 h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h2 n2)
209 (AHead a1 a2))).(let H3 \def (eq_ind A (ASort h2 n2) (\lambda (ee: A).(match
210 ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True |
211 (AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A
212 (\lambda (a3: A).(\lambda (_: A).(leq g a3 a1))) (\lambda (_: A).(\lambda
213 (a4: A).(leq g a4 a2))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h1 n1)
214 (AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1:
215 (leq g a0 a3)).(\lambda (H2: (((eq A a3 (AHead a1 a2)) \to (ex3_2 A A
216 (\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda (_: A).(\lambda
217 (a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq A a0 (AHead a4
218 a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4
219 a5)).(\lambda (H4: (((eq A a5 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6:
220 A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda (a7: A).(leq g a7
221 a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A a4 (AHead a6
222 a7)))))))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a1 a2))).(let H6 \def
223 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
224 [(ASort _ _) \Rightarrow a3 | (AHead a6 _) \Rightarrow a6])) (AHead a3 a5)
225 (AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A
226 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a6)
227 \Rightarrow a6])) (AHead a3 a5) (AHead a1 a2) H5) in (\lambda (H8: (eq A a3
228 a1)).(let H9 \def (eq_ind A a5 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to
229 (ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a7 a1))) (\lambda (_:
230 A).(\lambda (a8: A).(leq g a8 a2))) (\lambda (a7: A).(\lambda (a8: A).(eq A
231 a4 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a5 (\lambda (a6:
232 A).(leq g a4 a6)) H3 a2 H7) in (let H11 \def (eq_ind A a3 (\lambda (a6:
233 A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_:
234 A).(leq g a7 a1))) (\lambda (_: A).(\lambda (a8: A).(leq g a8 a2))) (\lambda
235 (a7: A).(\lambda (a8: A).(eq A a0 (AHead a7 a8))))))) H2 a1 H8) in (let H12
236 \def (eq_ind A a3 (\lambda (a6: A).(leq g a0 a6)) H1 a1 H8) in (ex3_2_intro A
237 A (\lambda (a6: A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda
238 (a7: A).(leq g a7 a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a0 a4)
239 (AHead a6 a7)))) a0 a4 H12 H10 (refl_equal A (AHead a0 a4)))))))))
240 H6))))))))))) a y H0))) H))))).