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 "Basic-1/leq/props.ma".
20 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g
21 (asucc g a1) (asucc g a2)))))
23 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
24 a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g (asucc g a) (asucc g
25 a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2:
26 nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g
27 (ASort h2 n2) k))).(nat_ind (\lambda (n: nat).((eq A (aplus g (ASort n n1) k)
28 (aplus g (ASort h2 n2) k)) \to (leq g (match n with [O \Rightarrow (ASort O
29 (next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow
30 (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) (\lambda (H1: (eq
31 A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda (n:
32 nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort n n2) k)) \to (leq g
33 (ASort O (next g n1)) (match n with [O \Rightarrow (ASort O (next g n2)) | (S
34 h) \Rightarrow (ASort h n2)])))) (\lambda (H2: (eq A (aplus g (ASort O n1) k)
35 (aplus g (ASort O n2) k))).(leq_sort g O O (next g n1) (next g n2) k (eq_ind
36 A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O
37 (next g n2)) k))) (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq
38 A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort O n2) k)
39 (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k))))
40 (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort O n1) k)
41 H2) (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)) (aplus g
42 (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))) (\lambda (h3:
43 nat).(\lambda (_: (((eq A (aplus g (ASort O n1) k) (aplus g (ASort h3 n2) k))
44 \to (leq g (ASort O (next g n1)) (match h3 with [O \Rightarrow (ASort O (next
45 g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H2: (eq A (aplus g
46 (ASort O n1) k) (aplus g (ASort (S h3) n2) k))).(leq_sort g O h3 (next g n1)
47 n2 k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g
48 (ASort h3 n2) k))) (eq_ind A (aplus g (ASort (S h3) n2) (S k)) (\lambda (a:
49 A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort (S h3)
50 n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort (S h3) n2)
51 k)))) (refl_equal A (asucc g (aplus g (ASort (S h3) n2) k))) (aplus g (ASort
52 O n1) k) H2) (aplus g (ASort h3 n2) k) (aplus_sort_S_S_simpl g n2 h3 k))
53 (aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))))) h2 H1))
54 (\lambda (h3: nat).(\lambda (IHh1: (((eq A (aplus g (ASort h3 n1) k) (aplus g
55 (ASort h2 n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next g
56 n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow (ASort
57 O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H1: (eq A
58 (aplus g (ASort (S h3) n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda
59 (n: nat).((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort n n2) k)) \to
60 ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort n n2) k)) \to (leq g
61 (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow
62 (ASort h n1)]) (match n with [O \Rightarrow (ASort O (next g n2)) | (S h)
63 \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match n with [O
64 \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))))
65 (\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort O n2)
66 k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort O n2) k))
67 \to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h)
68 \Rightarrow (ASort h n1)]) (ASort O (next g n2)))))).(leq_sort g h3 O n1
69 (next g n2) k (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A
70 (aplus g (ASort h3 n1) k) a)) (eq_ind A (aplus g (ASort (S h3) n1) (S k))
71 (\lambda (a: A).(eq A a (aplus g (ASort O n2) (S k)))) (eq_ind_r A (aplus g
72 (ASort O n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O
73 n2) k)))) (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort
74 (S h3) n1) k) H2) (aplus g (ASort h3 n1) k) (aplus_sort_S_S_simpl g n1 h3 k))
75 (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k))))) (\lambda
76 (h4: nat).(\lambda (_: (((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort
77 h4 n2) k)) \to ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort h4 n2) k))
78 \to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h)
79 \Rightarrow (ASort h n1)]) (match h4 with [O \Rightarrow (ASort O (next g
80 n2)) | (S h) \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match h4
81 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h
82 n2)])))))).(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort
83 (S h4) n2) k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g
84 (ASort (S h4) n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next
85 g n1)) | (S h) \Rightarrow (ASort h n1)]) (ASort h4 n2))))).(leq_sort g h3 h4
86 n1 n2 k (eq_ind A (aplus g (ASort (S h3) n1) (S k)) (\lambda (a: A).(eq A a
87 (aplus g (ASort h4 n2) k))) (eq_ind A (aplus g (ASort (S h4) n2) (S k))
88 (\lambda (a: A).(eq A (aplus g (ASort (S h3) n1) (S k)) a)) (eq_ind_r A
89 (aplus g (ASort (S h4) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g
90 (aplus g (ASort (S h4) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S
91 h4) n2) k))) (aplus g (ASort (S h3) n1) k) H2) (aplus g (ASort h4 n2) k)
92 (aplus_sort_S_S_simpl g n2 h4 k)) (aplus g (ASort h3 n1) k)
93 (aplus_sort_S_S_simpl g n1 h3 k))))))) h2 H1 IHh1)))) h1 H0))))))) (\lambda
94 (a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3 a4)).(\lambda (_: (leq g
95 (asucc g a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_:
96 (leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g
97 a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))).
103 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc
104 g a2)) \to (leq g a1 a2))))
106 \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
107 A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n:
108 nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g
109 (asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda
110 (n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0))
111 (asucc g (ASort n1 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort
112 n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))
113 (\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1
114 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g
115 (ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2)))) (\lambda (H1: (leq g
116 (asucc g (ASort O n0)) (asucc g (ASort O n2)))).(let H_x \def (leq_gen_sort1
117 g O (next g n0) (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind
118 nat nat nat (\lambda (n3: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A
119 (aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n3) k))))) (\lambda (n3:
120 nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort
121 h2 n3))))) (leq g (ASort O n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1:
122 nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0))
123 x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2))
124 (ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A
125 return (\lambda (_: A).nat) with [(ASort n3 _) \Rightarrow n3 | (AHead _ _)
126 \Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def
127 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
128 [(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow ((match g with [(mk_G
129 next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in
130 (\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n3:
131 nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 x0) x2))) H3
132 O H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n3: nat).(eq A (aplus g
133 (ASort O (next g n0)) x2) (aplus g (ASort O n3) x2))) H8 (next g n2) H6) in
134 (let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) (\lambda (a:
135 A).(eq A a (aplus g (ASort O (next g n2)) x2))) H9 (aplus g (ASort O n0) (S
136 x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g
137 (ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2))
138 a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in
139 (leq_sort g O O n0 n2 (S x2) H11))))))) H5))))))) H2)))) (\lambda (n3:
140 nat).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2)))
141 \to (leq g (ASort O n0) (ASort n3 n2))))).(\lambda (H1: (leq g (asucc g
142 (ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H_x \def (leq_gen_sort1 g O
143 (next g n0) (ASort n3 n2) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat
144 (\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
145 O (next g n0)) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda
146 (h2: nat).(\lambda (_: nat).(eq A (ASort n3 n2) (ASort h2 n4))))) (leq g
147 (ASort O n0) (ASort (S n3) n2)) (\lambda (x0: nat).(\lambda (x1:
148 nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0))
149 x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n3 n2) (ASort x1
150 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return
151 (\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _)
152 \Rightarrow n3])) (ASort n3 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A
153 nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _
154 n4) \Rightarrow n4 | (AHead _ _) \Rightarrow n2])) (ASort n3 n2) (ASort x1
155 x0) H4) in (\lambda (H7: (eq nat n3 x1)).(let H8 \def (eq_ind_r nat x1
156 (\lambda (n4: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort
157 n4 x0) x2))) H3 n3 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n4:
158 nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 n4) x2))) H8
159 n2 H6) in (let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2)
160 (\lambda (a: A).(eq A a (aplus g (ASort n3 n2) x2))) H9 (aplus g (ASort O n0)
161 (S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g
162 (ASort n3 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) a)) H10
163 (aplus g (ASort (S n3) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n3 x2)) in
164 (leq_sort g O (S n3) n0 n2 (S x2) H11))))))) H5))))))) H2)))))) n1 H0))
165 (\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0)) (asucc g
166 (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda (H0: (leq
167 g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(nat_ind (\lambda
168 (n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to
169 ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort
170 n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 n2)))))
171 (\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort O
172 n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort O n2)))
173 \to (leq g (ASort n3 n0) (ASort O n2))))).(let H_x \def (leq_gen_sort1 g n3
174 n0 (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat
175 (\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
176 n3 n0) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda (h2:
177 nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort h2 n4))))) (leq g
178 (ASort (S n3) n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1:
179 nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort n3 n0) x2) (aplus
180 g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) (ASort x1
181 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return
182 (\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _)
183 \Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def
184 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
185 [(ASort _ n4) \Rightarrow n4 | (AHead _ _) \Rightarrow ((match g with [(mk_G
186 next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in
187 (\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n4:
188 nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n4 x0) x2))) H3 O H7)
189 in (let H9 \def (eq_ind_r nat x0 (\lambda (n4: nat).(eq A (aplus g (ASort n3
190 n0) x2) (aplus g (ASort O n4) x2))) H8 (next g n2) H6) in (let H10 \def
191 (eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g
192 (ASort O (next g n2)) x2))) H9 (aplus g (ASort (S n3) n0) (S x2))
193 (aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g
194 (ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S
195 x2)) a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in
196 (leq_sort g (S n3) O n0 n2 (S x2) H11))))))) H5))))))) H2))))) (\lambda (n4:
197 nat).(\lambda (_: (((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4
198 n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq
199 g (ASort n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4
200 n2)))))).(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S
201 n4) n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort (S
202 n4) n2))) \to (leq g (ASort n3 n0) (ASort (S n4) n2))))).(let H_x \def
203 (leq_gen_sort1 g n3 n0 (ASort n4 n2) H1) in (let H2 \def H_x in (ex2_3_ind
204 nat nat nat (\lambda (n5: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A
205 (aplus g (ASort n3 n0) k) (aplus g (ASort h2 n5) k))))) (\lambda (n5:
206 nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort n4 n2) (ASort h2
207 n5))))) (leq g (ASort (S n3) n0) (ASort (S n4) n2)) (\lambda (x0:
208 nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g
209 (ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n4
210 n2) (ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A
211 return (\lambda (_: A).nat) with [(ASort n5 _) \Rightarrow n5 | (AHead _ _)
212 \Rightarrow n4])) (ASort n4 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A
213 nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _
214 n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2])) (ASort n4 n2) (ASort x1
215 x0) H4) in (\lambda (H7: (eq nat n4 x1)).(let H8 \def (eq_ind_r nat x1
216 (\lambda (n5: nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n5 x0)
217 x2))) H3 n4 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n5: nat).(eq A
218 (aplus g (ASort n3 n0) x2) (aplus g (ASort n4 n5) x2))) H8 n2 H6) in (let H10
219 \def (eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g
220 (ASort n4 n2) x2))) H9 (aplus g (ASort (S n3) n0) (S x2))
221 (aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g
222 (ASort n4 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S x2))
223 a)) H10 (aplus g (ASort (S n4) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n4 x2))
224 in (leq_sort g (S n3) (S n4) n0 n2 (S x2) H11))))))) H5))))))) H2))))))) n1
225 H0 IHn)))) n H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n
226 n0)) (asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda
227 (H0: (((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0)
228 a0)))).(\lambda (H1: (leq g (asucc g (ASort n n0)) (asucc g (AHead a
229 a0)))).(nat_ind (\lambda (n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g
230 a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 n0))
231 (asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1
232 n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a a0))))))
233 (\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O
234 n0) a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq
235 g (ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g
236 (AHead a a0)))).(let H_x \def (leq_gen_sort1 g O (next g n0) (AHead a (asucc
237 g a0)) H4) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2:
238 nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort O (next g
239 n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2:
240 nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0)) (ASort h2 n2))))) (leq g
241 (ASort O n0) (AHead a a0)) (\lambda (x0: nat).(\lambda (x1: nat).(\lambda
242 (x2: nat).(\lambda (_: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g
243 (ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1
244 x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda (ee: A).(match
245 ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
246 (AHead _ _) \Rightarrow True])) I (ASort x1 x0) H7) in (False_ind (leq g
247 (ASort O n0) (AHead a a0)) H8))))))) H5)))))) (\lambda (n1: nat).(\lambda (_:
248 (((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0) a)))
249 \to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1 n0)
250 a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to (leq g
251 (ASort n1 n0) (AHead a a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1)
252 n0)) (asucc g a)) \to (leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g
253 (asucc g (ASort (S n1) n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0)
254 a0)))).(\lambda (H4: (leq g (asucc g (ASort (S n1) n0)) (asucc g (AHead a
255 a0)))).(let H_x \def (leq_gen_sort1 g n1 n0 (AHead a (asucc g a0)) H4) in
256 (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2:
257 nat).(\lambda (k: nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2)
258 k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a
259 (asucc g a0)) (ASort h2 n2))))) (leq g (ASort (S n1) n0) (AHead a a0))
260 (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A
261 (aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A
262 (AHead a (asucc g a0)) (ASort x1 x0))).(let H8 \def (eq_ind A (AHead a (asucc
263 g a0)) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with
264 [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort x1
265 x0) H7) in (False_ind (leq g (ASort (S n1) n0) (AHead a a0)) H8)))))))
266 H5)))))))) n H H0 H1)))))) a2)))) (\lambda (a: A).(\lambda (_: ((\forall (a2:
267 A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2))))).(\lambda (a0:
268 A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0) (asucc g a2)) \to
269 (leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3: A).((leq g (asucc g
270 (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3))) (\lambda (n:
271 nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a a0)) (asucc g
272 (ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g (AHead a a0))
273 (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1 n0)))) (\lambda
274 (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O n0)))).(let H_x \def
275 (leq_gen_head1 g a (asucc g a0) (ASort O (next g n0)) H2) in (let H3 \def H_x
276 in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a a3))) (\lambda
277 (_: A).(\lambda (a4: A).(leq g (asucc g a0) a4))) (\lambda (a3: A).(\lambda
278 (a4: A).(eq A (ASort O (next g n0)) (AHead a3 a4)))) (leq g (AHead a a0)
279 (ASort O n0)) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a
280 x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda (H6: (eq A (ASort O (next
281 g n0)) (AHead x0 x1))).(let H7 \def (eq_ind A (ASort O (next g n0)) (\lambda
282 (ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _)
283 \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in
284 (False_ind (leq g (AHead a a0) (ASort O n0)) H7))))))) H3)))) (\lambda (n1:
285 nat).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0)))
286 \to (leq g (AHead a a0) (ASort n1 n0))))).(\lambda (H2: (leq g (asucc g
287 (AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H_x \def (leq_gen_head1 g a
288 (asucc g a0) (ASort n1 n0) H2) in (let H3 \def H_x in (ex3_2_ind A A (\lambda
289 (a3: A).(\lambda (_: A).(leq g a a3))) (\lambda (_: A).(\lambda (a4: A).(leq
290 g (asucc g a0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0)
291 (AHead a3 a4)))) (leq g (AHead a a0) (ASort (S n1) n0)) (\lambda (x0:
292 A).(\lambda (x1: A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g
293 a0) x1)).(\lambda (H6: (eq A (ASort n1 n0) (AHead x0 x1))).(let H7 \def
294 (eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_:
295 A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
296 False])) I (AHead x0 x1) H6) in (False_ind (leq g (AHead a a0) (ASort (S n1)
297 n0)) H7))))))) H3)))))) n H1)))) (\lambda (a3: A).(\lambda (_: (((leq g
298 (asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda
299 (a4: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g
300 (AHead a a0) a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g
301 (AHead a3 a4)))).(let H_x \def (leq_gen_head1 g a (asucc g a0) (AHead a3
302 (asucc g a4)) H3) in (let H4 \def H_x in (ex3_2_ind A A (\lambda (a5:
303 A).(\lambda (_: A).(leq g a a5))) (\lambda (_: A).(\lambda (a6: A).(leq g
304 (asucc g a0) a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 (asucc g
305 a4)) (AHead a5 a6)))) (leq g (AHead a a0) (AHead a3 a4)) (\lambda (x0:
306 A).(\lambda (x1: A).(\lambda (H5: (leq g a x0)).(\lambda (H6: (leq g (asucc g
307 a0) x1)).(\lambda (H7: (eq A (AHead a3 (asucc g a4)) (AHead x0 x1))).(let H8
308 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
309 with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow a5])) (AHead a3
310 (asucc g a4)) (AHead x0 x1) H7) in ((let H9 \def (f_equal A A (\lambda (e:
311 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow
312 ((let rec asucc (g0: G) (l: A) on l: A \def (match l with [(ASort n0 n)
313 \Rightarrow (match n0 with [O \Rightarrow (ASort O (next g0 n)) | (S h)
314 \Rightarrow (ASort h n)]) | (AHead a5 a6) \Rightarrow (AHead a5 (asucc g0
315 a6))]) in asucc) g a4) | (AHead _ a5) \Rightarrow a5])) (AHead a3 (asucc g
316 a4)) (AHead x0 x1) H7) in (\lambda (H10: (eq A a3 x0)).(let H11 \def
317 (eq_ind_r A x1 (\lambda (a5: A).(leq g (asucc g a0) a5)) H6 (asucc g a4) H9)
318 in (let H12 \def (eq_ind_r A x0 (\lambda (a5: A).(leq g a a5)) H5 a3 H10) in
319 (leq_head g a a3 H12 a0 a4 (H0 a4 H11)))))) H8))))))) H4)))))))) a2))))))
326 \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g
329 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(ex A (\lambda (a1:
330 A).(leq g a0 (asucc g a1))))) (\lambda (n: nat).(\lambda (n0: nat).(ex_intro
331 A (\lambda (a0: A).(leq g (ASort n n0) (asucc g a0))) (ASort (S n) n0)
332 (leq_refl g (ASort n n0))))) (\lambda (a0: A).(\lambda (_: (ex A (\lambda
333 (a1: A).(leq g a0 (asucc g a1))))).(\lambda (a1: A).(\lambda (H0: (ex A
334 (\lambda (a2: A).(leq g a1 (asucc g a2))))).(let H1 \def H0 in (ex_ind A
335 (\lambda (a2: A).(leq g a1 (asucc g a2))) (ex A (\lambda (a2: A).(leq g
336 (AHead a0 a1) (asucc g a2)))) (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc
337 g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2)))
338 (AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1))))))
344 theorem leq_ahead_asucc_false:
345 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2)
346 (asucc g a1)) \to (\forall (P: Prop).P))))
348 \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
349 A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda
350 (n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead
351 (ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
352 \Rightarrow (ASort h n0)]))).(\lambda (P: Prop).(nat_ind (\lambda (n1:
353 nat).((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O
354 (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P)) (\lambda (H0: (leq g
355 (AHead (ASort O n0) a2) (ASort O (next g n0)))).(let H_x \def (leq_gen_head1
356 g (ASort O n0) a2 (ASort O (next g n0)) H0) in (let H1 \def H_x in (ex3_2_ind
357 A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_:
358 A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A
359 (ASort O (next g n0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1:
360 A).(\lambda (_: (leq g (ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda
361 (H4: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H5 \def (eq_ind A
362 (ASort O (next g n0)) (\lambda (ee: A).(match ee in A return (\lambda (_:
363 A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
364 False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))) (\lambda (n1:
365 nat).(\lambda (_: (((leq g (AHead (ASort n1 n0) a2) (match n1 with [O
366 \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to
367 P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort n1 n0))).(let
368 H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 (ASort n1 n0) H0) in (let H1
369 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort (S
370 n1) n0) a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3:
371 A).(\lambda (a4: A).(eq A (ASort n1 n0) (AHead a3 a4)))) P (\lambda (x0:
372 A).(\lambda (x1: A).(\lambda (_: (leq g (ASort (S n1) n0) x0)).(\lambda (_:
373 (leq g a2 x1)).(\lambda (H4: (eq A (ASort n1 n0) (AHead x0 x1))).(let H5 \def
374 (eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_:
375 A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
376 False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n H))))))
377 (\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a a2) (asucc g
378 a)) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall
379 (a2: A).((leq g (AHead a0 a2) (asucc g a0)) \to (\forall (P:
380 Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead (AHead a a0) a2)
381 (AHead a (asucc g a0)))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g
382 (AHead a a0) a2 (AHead a (asucc g a0)) H1) in (let H2 \def H_x in (ex3_2_ind
383 A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) (\lambda (_:
384 A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A
385 (AHead a (asucc g a0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1:
386 A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2
387 x1)).(\lambda (H5: (eq A (AHead a (asucc g a0)) (AHead x0 x1))).(let H6 \def
388 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
389 [(ASort _ _) \Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a (asucc g
390 a0)) (AHead x0 x1) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e
391 in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow ((let rec asucc
392 (g0: G) (l: A) on l: A \def (match l with [(ASort n0 n) \Rightarrow (match n0
393 with [O \Rightarrow (ASort O (next g0 n)) | (S h) \Rightarrow (ASort h n)]) |
394 (AHead a3 a4) \Rightarrow (AHead a3 (asucc g0 a4))]) in asucc) g a0) | (AHead
395 _ a3) \Rightarrow a3])) (AHead a (asucc g a0)) (AHead x0 x1) H5) in (\lambda
396 (H8: (eq A a x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3))
397 H4 (asucc g a0) H7) in (let H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g
398 (AHead a a0) a3)) H3 a H8) in (leq_ahead_false_1 g a a0 H10 P))))) H6)))))))
404 theorem leq_asucc_false:
405 \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P:
408 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0)
409 a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda
410 (H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
411 \Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).(nat_ind
412 (\lambda (n1: nat).((leq g (match n1 with [O \Rightarrow (ASort O (next g
413 n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P)) (\lambda (H0:
414 (leq g (ASort O (next g n0)) (ASort O n0))).(let H_x \def (leq_gen_sort1 g O
415 (next g n0) (ASort O n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat
416 (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
417 O (next g n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda
418 (h2: nat).(\lambda (_: nat).(eq A (ASort O n0) (ASort h2 n2))))) P (\lambda
419 (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g
420 (ASort O (next g n0)) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A
421 (ASort O n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e:
422 A).(match e in A return (\lambda (_: A).nat) with [(ASort n1 _) \Rightarrow
423 n1 | (AHead _ _) \Rightarrow O])) (ASort O n0) (ASort x1 x0) H3) in ((let H5
424 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat)
425 with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) \Rightarrow n0])) (ASort O
426 n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat O x1)).(let H7 \def (eq_ind_r
427 nat x1 (\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g
428 (ASort n1 x0) x2))) H2 O H6) in (let H8 \def (eq_ind_r nat x0 (\lambda (n1:
429 nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort O n1) x2))) H7
430 n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2)
431 (\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) x2))) H8 (aplus g (ASort O
432 n0) (S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H_y \def (aplus_inj g (S
433 x2) x2 (ASort O n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n1:
434 nat).(le n1 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))) (\lambda
435 (n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow (ASort O (next
436 g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P))).(\lambda
437 (H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let H_x \def (leq_gen_sort1 g
438 n1 n0 (ASort (S n1) n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat
439 (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
440 n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2:
441 nat).(\lambda (_: nat).(eq A (ASort (S n1) n0) (ASort h2 n2))))) P (\lambda
442 (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g
443 (ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A (ASort (S
444 n1) n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e: A).(match e
445 in A return (\lambda (_: A).nat) with [(ASort n2 _) \Rightarrow n2 | (AHead _
446 _) \Rightarrow (S n1)])) (ASort (S n1) n0) (ASort x1 x0) H3) in ((let H5 \def
447 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
448 [(ASort _ n2) \Rightarrow n2 | (AHead _ _) \Rightarrow n0])) (ASort (S n1)
449 n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat (S n1) x1)).(let H7 \def
450 (eq_ind_r nat x1 (\lambda (n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g
451 (ASort n2 x0) x2))) H2 (S n1) H6) in (let H8 \def (eq_ind_r nat x0 (\lambda
452 (n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort (S n1) n2) x2)))
453 H7 n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort n1 n0) x2) (\lambda
454 (a0: A).(eq A a0 (aplus g (ASort (S n1) n0) x2))) H8 (aplus g (ASort (S n1)
455 n0) (S x2)) (aplus_sort_S_S_simpl g n0 n1 x2)) in (let H_y \def (aplus_inj g
456 (S x2) x2 (ASort (S n1) n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n2:
457 nat).(le n2 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))))) n H)))))
458 (\lambda (a0: A).(\lambda (_: (((leq g (asucc g a0) a0) \to (\forall (P:
459 Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g a1) a1) \to
460 (\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g a1)) (AHead
461 a0 a1))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g a0 (asucc g a1)
462 (AHead a0 a1) H1) in (let H2 \def H_x in (ex3_2_ind A A (\lambda (a3:
463 A).(\lambda (_: A).(leq g a0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g
464 (asucc g a1) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a0 a1)
465 (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (H3: (leq g a0
466 x0)).(\lambda (H4: (leq g (asucc g a1) x1)).(\lambda (H5: (eq A (AHead a0 a1)
467 (AHead x0 x1))).(let H6 \def (f_equal A A (\lambda (e: A).(match e in A
468 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a2 _)
469 \Rightarrow a2])) (AHead a0 a1) (AHead x0 x1) H5) in ((let H7 \def (f_equal A
470 A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
471 \Rightarrow a1 | (AHead _ a2) \Rightarrow a2])) (AHead a0 a1) (AHead x0 x1)
472 H5) in (\lambda (H8: (eq A a0 x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a2:
473 A).(leq g (asucc g a1) a2)) H4 a1 H7) in (let H10 \def (eq_ind_r A x0
474 (\lambda (a2: A).(leq g a0 a2)) H3 a0 H8) in (H0 H9 P))))) H6)))))))
projects / helm.git / blob