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/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)))).
100 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc
101 g a2)) \to (leq g a1 a2))))
103 \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
104 A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n:
105 nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g
106 (asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda
107 (n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0))
108 (asucc g (ASort n1 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort
109 n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))
110 (\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1
111 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g
112 (ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2)))) (\lambda (H1: (leq g
113 (asucc g (ASort O n0)) (asucc g (ASort O n2)))).(let H2 \def (match H1 in leq
114 return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a
115 (ASort O (next g n0))) \to ((eq A a0 (ASort O (next g n2))) \to (leq g (ASort
116 O n0) (ASort O n2))))))) with [(leq_sort h1 h2 n3 n4 k H2) \Rightarrow
117 (\lambda (H3: (eq A (ASort h1 n3) (ASort O (next g n0)))).(\lambda (H4: (eq A
118 (ASort h2 n4) (ASort O (next g n2)))).((let H5 \def (f_equal A nat (\lambda
119 (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n5)
120 \Rightarrow n5 | (AHead _ _) \Rightarrow n3])) (ASort h1 n3) (ASort O (next g
121 n0)) H3) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e in A return
122 (\lambda (_: A).nat) with [(ASort n5 _) \Rightarrow n5 | (AHead _ _)
123 \Rightarrow h1])) (ASort h1 n3) (ASort O (next g n0)) H3) in (eq_ind nat O
124 (\lambda (n5: nat).((eq nat n3 (next g n0)) \to ((eq A (ASort h2 n4) (ASort O
125 (next g n2))) \to ((eq A (aplus g (ASort n5 n3) k) (aplus g (ASort h2 n4) k))
126 \to (leq g (ASort O n0) (ASort O n2)))))) (\lambda (H7: (eq nat n3 (next g
127 n0))).(eq_ind nat (next g n0) (\lambda (n5: nat).((eq A (ASort h2 n4) (ASort
128 O (next g n2))) \to ((eq A (aplus g (ASort O n5) k) (aplus g (ASort h2 n4)
129 k)) \to (leq g (ASort O n0) (ASort O n2))))) (\lambda (H8: (eq A (ASort h2
130 n4) (ASort O (next g n2)))).(let H9 \def (f_equal A nat (\lambda (e:
131 A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n5) \Rightarrow
132 n5 | (AHead _ _) \Rightarrow n4])) (ASort h2 n4) (ASort O (next g n2)) H8) in
133 ((let H10 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda
134 (_: A).nat) with [(ASort n5 _) \Rightarrow n5 | (AHead _ _) \Rightarrow h2]))
135 (ASort h2 n4) (ASort O (next g n2)) H8) in (eq_ind nat O (\lambda (n5:
136 nat).((eq nat n4 (next g n2)) \to ((eq A (aplus g (ASort O (next g n0)) k)
137 (aplus g (ASort n5 n4) k)) \to (leq g (ASort O n0) (ASort O n2))))) (\lambda
138 (H11: (eq nat n4 (next g n2))).(eq_ind nat (next g n2) (\lambda (n5:
139 nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O n5) k)) \to
140 (leq g (ASort O n0) (ASort O n2)))) (\lambda (H12: (eq A (aplus g (ASort O
141 (next g n0)) k) (aplus g (ASort O (next g n2)) k))).(let H13 \def (eq_ind_r A
142 (aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a (aplus g (ASort O
143 (next g n2)) k))) H12 (aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g n0
144 k)) in (let H14 \def (eq_ind_r A (aplus g (ASort O (next g n2)) k) (\lambda
145 (a: A).(eq A (aplus g (ASort O n0) (S k)) a)) H13 (aplus g (ASort O n2) (S
146 k)) (aplus_sort_O_S_simpl g n2 k)) in (leq_sort g O O n0 n2 (S k) H14)))) n4
147 (sym_eq nat n4 (next g n2) H11))) h2 (sym_eq nat h2 O H10))) H9))) n3 (sym_eq
148 nat n3 (next g n0) H7))) h1 (sym_eq nat h1 O H6))) H5)) H4 H2))) | (leq_head
149 a0 a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A (AHead a0 a4) (ASort O
150 (next g n0)))).(\lambda (H5: (eq A (AHead a3 a5) (ASort O (next g
151 n2)))).((let H6 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match e in A
152 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
153 _) \Rightarrow True])) I (ASort O (next g n0)) H4) in (False_ind ((eq A
154 (AHead a3 a5) (ASort O (next g n2))) \to ((leq g a0 a3) \to ((leq g a4 a5)
155 \to (leq g (ASort O n0) (ASort O n2))))) H6)) H5 H2 H3)))]) in (H2
156 (refl_equal A (ASort O (next g n0))) (refl_equal A (ASort O (next g n2))))))
157 (\lambda (n3: nat).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g
158 (ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2))))).(\lambda (H1: (leq
159 g (asucc g (ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H2 \def (match H1
160 in leq return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a
161 a0)).((eq A a (ASort O (next g n0))) \to ((eq A a0 (ASort n3 n2)) \to (leq g
162 (ASort O n0) (ASort (S n3) n2))))))) with [(leq_sort h1 h2 n4 n5 k H2)
163 \Rightarrow (\lambda (H3: (eq A (ASort h1 n4) (ASort O (next g
164 n0)))).(\lambda (H4: (eq A (ASort h2 n5) (ASort n3 n2))).((let H5 \def
165 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
166 [(ASort _ n6) \Rightarrow n6 | (AHead _ _) \Rightarrow n4])) (ASort h1 n4)
167 (ASort O (next g n0)) H3) in ((let H6 \def (f_equal A nat (\lambda (e:
168 A).(match e in A return (\lambda (_: A).nat) with [(ASort n6 _) \Rightarrow
169 n6 | (AHead _ _) \Rightarrow h1])) (ASort h1 n4) (ASort O (next g n0)) H3) in
170 (eq_ind nat O (\lambda (n6: nat).((eq nat n4 (next g n0)) \to ((eq A (ASort
171 h2 n5) (ASort n3 n2)) \to ((eq A (aplus g (ASort n6 n4) k) (aplus g (ASort h2
172 n5) k)) \to (leq g (ASort O n0) (ASort (S n3) n2)))))) (\lambda (H7: (eq nat
173 n4 (next g n0))).(eq_ind nat (next g n0) (\lambda (n6: nat).((eq A (ASort h2
174 n5) (ASort n3 n2)) \to ((eq A (aplus g (ASort O n6) k) (aplus g (ASort h2 n5)
175 k)) \to (leq g (ASort O n0) (ASort (S n3) n2))))) (\lambda (H8: (eq A (ASort
176 h2 n5) (ASort n3 n2))).(let H9 \def (f_equal A nat (\lambda (e: A).(match e
177 in A return (\lambda (_: A).nat) with [(ASort _ n6) \Rightarrow n6 | (AHead _
178 _) \Rightarrow n5])) (ASort h2 n5) (ASort n3 n2) H8) in ((let H10 \def
179 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
180 [(ASort n6 _) \Rightarrow n6 | (AHead _ _) \Rightarrow h2])) (ASort h2 n5)
181 (ASort n3 n2) H8) in (eq_ind nat n3 (\lambda (n6: nat).((eq nat n5 n2) \to
182 ((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n6 n5) k)) \to (leq
183 g (ASort O n0) (ASort (S n3) n2))))) (\lambda (H11: (eq nat n5 n2)).(eq_ind
184 nat n2 (\lambda (n6: nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g
185 (ASort n3 n6) k)) \to (leq g (ASort O n0) (ASort (S n3) n2)))) (\lambda (H12:
186 (eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n3 n2) k))).(let H13
187 \def (eq_ind_r A (aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a
188 (aplus g (ASort n3 n2) k))) H12 (aplus g (ASort O n0) (S k))
189 (aplus_sort_O_S_simpl g n0 k)) in (let H14 \def (eq_ind_r A (aplus g (ASort
190 n3 n2) k) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S k)) a)) H13 (aplus g
191 (ASort (S n3) n2) (S k)) (aplus_sort_S_S_simpl g n2 n3 k)) in (leq_sort g O
192 (S n3) n0 n2 (S k) H14)))) n5 (sym_eq nat n5 n2 H11))) h2 (sym_eq nat h2 n3
193 H10))) H9))) n4 (sym_eq nat n4 (next g n0) H7))) h1 (sym_eq nat h1 O H6)))
194 H5)) H4 H2))) | (leq_head a0 a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A
195 (AHead a0 a4) (ASort O (next g n0)))).(\lambda (H5: (eq A (AHead a3 a5)
196 (ASort n3 n2))).((let H6 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match
197 e in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
198 (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H4) in (False_ind
199 ((eq A (AHead a3 a5) (ASort n3 n2)) \to ((leq g a0 a3) \to ((leq g a4 a5) \to
200 (leq g (ASort O n0) (ASort (S n3) n2))))) H6)) H5 H2 H3)))]) in (H2
201 (refl_equal A (ASort O (next g n0))) (refl_equal A (ASort n3 n2))))))) n1
202 H0)) (\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0))
203 (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda
204 (H0: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(nat_ind
205 (\lambda (n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4
206 n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq
207 g (ASort n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4
208 n2))))) (\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort O
209 n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort O n2)))
210 \to (leq g (ASort n3 n0) (ASort O n2))))).(let H2 \def (match H1 in leq
211 return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a
212 (ASort n3 n0)) \to ((eq A a0 (ASort O (next g n2))) \to (leq g (ASort (S n3)
213 n0) (ASort O n2))))))) with [(leq_sort h1 h2 n4 n5 k H2) \Rightarrow (\lambda
214 (H3: (eq A (ASort h1 n4) (ASort n3 n0))).(\lambda (H4: (eq A (ASort h2 n5)
215 (ASort O (next g n2)))).((let H5 \def (f_equal A nat (\lambda (e: A).(match e
216 in A return (\lambda (_: A).nat) with [(ASort _ n6) \Rightarrow n6 | (AHead _
217 _) \Rightarrow n4])) (ASort h1 n4) (ASort n3 n0) H3) in ((let H6 \def
218 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
219 [(ASort n6 _) \Rightarrow n6 | (AHead _ _) \Rightarrow h1])) (ASort h1 n4)
220 (ASort n3 n0) H3) in (eq_ind nat n3 (\lambda (n6: nat).((eq nat n4 n0) \to
221 ((eq A (ASort h2 n5) (ASort O (next g n2))) \to ((eq A (aplus g (ASort n6 n4)
222 k) (aplus g (ASort h2 n5) k)) \to (leq g (ASort (S n3) n0) (ASort O n2))))))
223 (\lambda (H7: (eq nat n4 n0)).(eq_ind nat n0 (\lambda (n6: nat).((eq A (ASort
224 h2 n5) (ASort O (next g n2))) \to ((eq A (aplus g (ASort n3 n6) k) (aplus g
225 (ASort h2 n5) k)) \to (leq g (ASort (S n3) n0) (ASort O n2))))) (\lambda (H8:
226 (eq A (ASort h2 n5) (ASort O (next g n2)))).(let H9 \def (f_equal A nat
227 (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n6)
228 \Rightarrow n6 | (AHead _ _) \Rightarrow n5])) (ASort h2 n5) (ASort O (next g
229 n2)) H8) in ((let H10 \def (f_equal A nat (\lambda (e: A).(match e in A
230 return (\lambda (_: A).nat) with [(ASort n6 _) \Rightarrow n6 | (AHead _ _)
231 \Rightarrow h2])) (ASort h2 n5) (ASort O (next g n2)) H8) in (eq_ind nat O
232 (\lambda (n6: nat).((eq nat n5 (next g n2)) \to ((eq A (aplus g (ASort n3 n0)
233 k) (aplus g (ASort n6 n5) k)) \to (leq g (ASort (S n3) n0) (ASort O n2)))))
234 (\lambda (H11: (eq nat n5 (next g n2))).(eq_ind nat (next g n2) (\lambda (n6:
235 nat).((eq A (aplus g (ASort n3 n0) k) (aplus g (ASort O n6) k)) \to (leq g
236 (ASort (S n3) n0) (ASort O n2)))) (\lambda (H12: (eq A (aplus g (ASort n3 n0)
237 k) (aplus g (ASort O (next g n2)) k))).(let H13 \def (eq_ind_r A (aplus g
238 (ASort n3 n0) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n2)) k)))
239 H12 (aplus g (ASort (S n3) n0) (S k)) (aplus_sort_S_S_simpl g n0 n3 k)) in
240 (let H14 \def (eq_ind_r A (aplus g (ASort O (next g n2)) k) (\lambda (a:
241 A).(eq A (aplus g (ASort (S n3) n0) (S k)) a)) H13 (aplus g (ASort O n2) (S
242 k)) (aplus_sort_O_S_simpl g n2 k)) in (leq_sort g (S n3) O n0 n2 (S k)
243 H14)))) n5 (sym_eq nat n5 (next g n2) H11))) h2 (sym_eq nat h2 O H10))) H9)))
244 n4 (sym_eq nat n4 n0 H7))) h1 (sym_eq nat h1 n3 H6))) H5)) H4 H2))) |
245 (leq_head a0 a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A (AHead a0 a4)
246 (ASort n3 n0))).(\lambda (H5: (eq A (AHead a3 a5) (ASort O (next g
247 n2)))).((let H6 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match e in A
248 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
249 _) \Rightarrow True])) I (ASort n3 n0) H4) in (False_ind ((eq A (AHead a3 a5)
250 (ASort O (next g n2))) \to ((leq g a0 a3) \to ((leq g a4 a5) \to (leq g
251 (ASort (S n3) n0) (ASort O n2))))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A
252 (ASort n3 n0)) (refl_equal A (ASort O (next g n2))))))) (\lambda (n4:
253 nat).(\lambda (_: (((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4
254 n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq
255 g (ASort n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4
256 n2)))))).(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S
257 n4) n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort (S
258 n4) n2))) \to (leq g (ASort n3 n0) (ASort (S n4) n2))))).(let H2 \def (match
259 H1 in leq return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a
260 a0)).((eq A a (ASort n3 n0)) \to ((eq A a0 (ASort n4 n2)) \to (leq g (ASort
261 (S n3) n0) (ASort (S n4) n2))))))) with [(leq_sort h1 h2 n5 n6 k H2)
262 \Rightarrow (\lambda (H3: (eq A (ASort h1 n5) (ASort n3 n0))).(\lambda (H4:
263 (eq A (ASort h2 n6) (ASort n4 n2))).((let H5 \def (f_equal A nat (\lambda (e:
264 A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n7) \Rightarrow
265 n7 | (AHead _ _) \Rightarrow n5])) (ASort h1 n5) (ASort n3 n0) H3) in ((let
266 H6 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_:
267 A).nat) with [(ASort n7 _) \Rightarrow n7 | (AHead _ _) \Rightarrow h1]))
268 (ASort h1 n5) (ASort n3 n0) H3) in (eq_ind nat n3 (\lambda (n7: nat).((eq nat
269 n5 n0) \to ((eq A (ASort h2 n6) (ASort n4 n2)) \to ((eq A (aplus g (ASort n7
270 n5) k) (aplus g (ASort h2 n6) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4)
271 n2)))))) (\lambda (H7: (eq nat n5 n0)).(eq_ind nat n0 (\lambda (n7: nat).((eq
272 A (ASort h2 n6) (ASort n4 n2)) \to ((eq A (aplus g (ASort n3 n7) k) (aplus g
273 (ASort h2 n6) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))) (\lambda
274 (H8: (eq A (ASort h2 n6) (ASort n4 n2))).(let H9 \def (f_equal A nat (\lambda
275 (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n7)
276 \Rightarrow n7 | (AHead _ _) \Rightarrow n6])) (ASort h2 n6) (ASort n4 n2)
277 H8) in ((let H10 \def (f_equal A nat (\lambda (e: A).(match e in A return
278 (\lambda (_: A).nat) with [(ASort n7 _) \Rightarrow n7 | (AHead _ _)
279 \Rightarrow h2])) (ASort h2 n6) (ASort n4 n2) H8) in (eq_ind nat n4 (\lambda
280 (n7: nat).((eq nat n6 n2) \to ((eq A (aplus g (ASort n3 n0) k) (aplus g
281 (ASort n7 n6) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))) (\lambda
282 (H11: (eq nat n6 n2)).(eq_ind nat n2 (\lambda (n7: nat).((eq A (aplus g
283 (ASort n3 n0) k) (aplus g (ASort n4 n7) k)) \to (leq g (ASort (S n3) n0)
284 (ASort (S n4) n2)))) (\lambda (H12: (eq A (aplus g (ASort n3 n0) k) (aplus g
285 (ASort n4 n2) k))).(let H13 \def (eq_ind_r A (aplus g (ASort n3 n0) k)
286 (\lambda (a: A).(eq A a (aplus g (ASort n4 n2) k))) H12 (aplus g (ASort (S
287 n3) n0) (S k)) (aplus_sort_S_S_simpl g n0 n3 k)) in (let H14 \def (eq_ind_r A
288 (aplus g (ASort n4 n2) k) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S
289 k)) a)) H13 (aplus g (ASort (S n4) n2) (S k)) (aplus_sort_S_S_simpl g n2 n4
290 k)) in (leq_sort g (S n3) (S n4) n0 n2 (S k) H14)))) n6 (sym_eq nat n6 n2
291 H11))) h2 (sym_eq nat h2 n4 H10))) H9))) n5 (sym_eq nat n5 n0 H7))) h1
292 (sym_eq nat h1 n3 H6))) H5)) H4 H2))) | (leq_head a0 a3 H2 a4 a5 H3)
293 \Rightarrow (\lambda (H4: (eq A (AHead a0 a4) (ASort n3 n0))).(\lambda (H5:
294 (eq A (AHead a3 a5) (ASort n4 n2))).((let H6 \def (eq_ind A (AHead a0 a4)
295 (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
296 \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n3 n0) H4) in
297 (False_ind ((eq A (AHead a3 a5) (ASort n4 n2)) \to ((leq g a0 a3) \to ((leq g
298 a4 a5) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))) H6)) H5 H2 H3)))])
299 in (H2 (refl_equal A (ASort n3 n0)) (refl_equal A (ASort n4 n2)))))))) n1 H0
300 IHn)))) n H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n n0))
301 (asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda (H0:
302 (((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0)
303 a0)))).(\lambda (H1: (leq g (asucc g (ASort n n0)) (asucc g (AHead a
304 a0)))).(nat_ind (\lambda (n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g
305 a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 n0))
306 (asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1
307 n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a a0))))))
308 (\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O
309 n0) a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq
310 g (ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g
311 (AHead a a0)))).(let H5 \def (match H4 in leq return (\lambda (a3:
312 A).(\lambda (a4: A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (ASort O (next g
313 n0))) \to ((eq A a4 (AHead a (asucc g a0))) \to (leq g (ASort O n0) (AHead a
314 a0))))))) with [(leq_sort h1 h2 n1 n2 k H5) \Rightarrow (\lambda (H6: (eq A
315 (ASort h1 n1) (ASort O (next g n0)))).(\lambda (H7: (eq A (ASort h2 n2)
316 (AHead a (asucc g a0)))).((let H8 \def (f_equal A nat (\lambda (e: A).(match
317 e in A return (\lambda (_: A).nat) with [(ASort _ n3) \Rightarrow n3 | (AHead
318 _ _) \Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H6) in ((let H9
319 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat)
320 with [(ASort n3 _) \Rightarrow n3 | (AHead _ _) \Rightarrow h1])) (ASort h1
321 n1) (ASort O (next g n0)) H6) in (eq_ind nat O (\lambda (n3: nat).((eq nat n1
322 (next g n0)) \to ((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A
323 (aplus g (ASort n3 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0)
324 (AHead a a0)))))) (\lambda (H10: (eq nat n1 (next g n0))).(eq_ind nat (next g
325 n0) (\lambda (n3: nat).((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq
326 A (aplus g (ASort O n3) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0)
327 (AHead a a0))))) (\lambda (H11: (eq A (ASort h2 n2) (AHead a (asucc g
328 a0)))).(let H12 \def (eq_ind A (ASort h2 n2) (\lambda (e: A).(match e in A
329 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
330 \Rightarrow False])) I (AHead a (asucc g a0)) H11) in (False_ind ((eq A
331 (aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n2) k)) \to (leq g
332 (ASort O n0) (AHead a a0))) H12))) n1 (sym_eq nat n1 (next g n0) H10))) h1
333 (sym_eq nat h1 O H9))) H8)) H7 H5))) | (leq_head a3 a4 H5 a5 a6 H6)
334 \Rightarrow (\lambda (H7: (eq A (AHead a3 a5) (ASort O (next g
335 n0)))).(\lambda (H8: (eq A (AHead a4 a6) (AHead a (asucc g a0)))).((let H9
336 \def (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda
337 (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
338 True])) I (ASort O (next g n0)) H7) in (False_ind ((eq A (AHead a4 a6) (AHead
339 a (asucc g a0))) \to ((leq g a3 a4) \to ((leq g a5 a6) \to (leq g (ASort O
340 n0) (AHead a a0))))) H9)) H8 H5 H6)))]) in (H5 (refl_equal A (ASort O (next g
341 n0))) (refl_equal A (AHead a (asucc g a0)))))))) (\lambda (n1: nat).(\lambda
342 (_: (((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0)
343 a))) \to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1
344 n0) a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to
345 (leq g (ASort n1 n0) (AHead a a0))))))).(\lambda (_: (((leq g (asucc g (ASort
346 (S n1) n0)) (asucc g a)) \to (leq g (ASort (S n1) n0) a)))).(\lambda (_:
347 (((leq g (asucc g (ASort (S n1) n0)) (asucc g a0)) \to (leq g (ASort (S n1)
348 n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort (S n1) n0)) (asucc g (AHead a
349 a0)))).(let H5 \def (match H4 in leq return (\lambda (a3: A).(\lambda (a4:
350 A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (ASort n1 n0)) \to ((eq A a4 (AHead
351 a (asucc g a0))) \to (leq g (ASort (S n1) n0) (AHead a a0))))))) with
352 [(leq_sort h1 h2 n2 n3 k H5) \Rightarrow (\lambda (H6: (eq A (ASort h1 n2)
353 (ASort n1 n0))).(\lambda (H7: (eq A (ASort h2 n3) (AHead a (asucc g
354 a0)))).((let H8 \def (f_equal A nat (\lambda (e: A).(match e in A return
355 (\lambda (_: A).nat) with [(ASort _ n4) \Rightarrow n4 | (AHead _ _)
356 \Rightarrow n2])) (ASort h1 n2) (ASort n1 n0) H6) in ((let H9 \def (f_equal A
357 nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort n4
358 _) \Rightarrow n4 | (AHead _ _) \Rightarrow h1])) (ASort h1 n2) (ASort n1 n0)
359 H6) in (eq_ind nat n1 (\lambda (n4: nat).((eq nat n2 n0) \to ((eq A (ASort h2
360 n3) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n4 n2) k) (aplus g
361 (ASort h2 n3) k)) \to (leq g (ASort (S n1) n0) (AHead a a0)))))) (\lambda
362 (H10: (eq nat n2 n0)).(eq_ind nat n0 (\lambda (n4: nat).((eq A (ASort h2 n3)
363 (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n1 n4) k) (aplus g (ASort
364 h2 n3) k)) \to (leq g (ASort (S n1) n0) (AHead a a0))))) (\lambda (H11: (eq A
365 (ASort h2 n3) (AHead a (asucc g a0)))).(let H12 \def (eq_ind A (ASort h2 n3)
366 (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
367 \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0))
368 H11) in (False_ind ((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n3)
369 k)) \to (leq g (ASort (S n1) n0) (AHead a a0))) H12))) n2 (sym_eq nat n2 n0
370 H10))) h1 (sym_eq nat h1 n1 H9))) H8)) H7 H5))) | (leq_head a3 a4 H5 a5 a6
371 H6) \Rightarrow (\lambda (H7: (eq A (AHead a3 a5) (ASort n1 n0))).(\lambda
372 (H8: (eq A (AHead a4 a6) (AHead a (asucc g a0)))).((let H9 \def (eq_ind A
373 (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with
374 [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1
375 n0) H7) in (False_ind ((eq A (AHead a4 a6) (AHead a (asucc g a0))) \to ((leq
376 g a3 a4) \to ((leq g a5 a6) \to (leq g (ASort (S n1) n0) (AHead a a0)))))
377 H9)) H8 H5 H6)))]) in (H5 (refl_equal A (ASort n1 n0)) (refl_equal A (AHead a
378 (asucc g a0)))))))))) n H H0 H1)))))) a2)))) (\lambda (a: A).(\lambda (_:
379 ((\forall (a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq g a
380 a2))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0)
381 (asucc g a2)) \to (leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3:
382 A).((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))
383 (\lambda (n: nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a
384 a0)) (asucc g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g
385 (AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1
386 n0)))) (\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O
387 n0)))).(let H3 \def (match H2 in leq return (\lambda (a3: A).(\lambda (a4:
388 A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (AHead a (asucc g a0))) \to ((eq A
389 a4 (ASort O (next g n0))) \to (leq g (AHead a a0) (ASort O n0))))))) with
390 [(leq_sort h1 h2 n1 n2 k H3) \Rightarrow (\lambda (H4: (eq A (ASort h1 n1)
391 (AHead a (asucc g a0)))).(\lambda (H5: (eq A (ASort h2 n2) (ASort O (next g
392 n0)))).((let H6 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A
393 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
394 \Rightarrow False])) I (AHead a (asucc g a0)) H4) in (False_ind ((eq A (ASort
395 h2 n2) (ASort O (next g n0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g
396 (ASort h2 n2) k)) \to (leq g (AHead a a0) (ASort O n0)))) H6)) H5 H3))) |
397 (leq_head a3 a4 H3 a5 a6 H4) \Rightarrow (\lambda (H5: (eq A (AHead a3 a5)
398 (AHead a (asucc g a0)))).(\lambda (H6: (eq A (AHead a4 a6) (ASort O (next g
399 n0)))).((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return
400 (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a7)
401 \Rightarrow a7])) (AHead a3 a5) (AHead a (asucc g a0)) H5) in ((let H8 \def
402 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
403 [(ASort _ _) \Rightarrow a3 | (AHead a7 _) \Rightarrow a7])) (AHead a3 a5)
404 (AHead a (asucc g a0)) H5) in (eq_ind A a (\lambda (a7: A).((eq A a5 (asucc g
405 a0)) \to ((eq A (AHead a4 a6) (ASort O (next g n0))) \to ((leq g a7 a4) \to
406 ((leq g a5 a6) \to (leq g (AHead a a0) (ASort O n0))))))) (\lambda (H9: (eq A
407 a5 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a7: A).((eq A (AHead a4
408 a6) (ASort O (next g n0))) \to ((leq g a a4) \to ((leq g a7 a6) \to (leq g
409 (AHead a a0) (ASort O n0)))))) (\lambda (H10: (eq A (AHead a4 a6) (ASort O
410 (next g n0)))).(let H11 \def (eq_ind A (AHead a4 a6) (\lambda (e: A).(match e
411 in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
412 (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H10) in (False_ind
413 ((leq g a a4) \to ((leq g (asucc g a0) a6) \to (leq g (AHead a a0) (ASort O
414 n0)))) H11))) a5 (sym_eq A a5 (asucc g a0) H9))) a3 (sym_eq A a3 a H8))) H7))
415 H6 H3 H4)))]) in (H3 (refl_equal A (AHead a (asucc g a0))) (refl_equal A
416 (ASort O (next g n0)))))) (\lambda (n1: nat).(\lambda (_: (((leq g (asucc g
417 (AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1
418 n0))))).(\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort (S n1)
419 n0)))).(let H3 \def (match H2 in leq return (\lambda (a3: A).(\lambda (a4:
420 A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (AHead a (asucc g a0))) \to ((eq A
421 a4 (ASort n1 n0)) \to (leq g (AHead a a0) (ASort (S n1) n0))))))) with
422 [(leq_sort h1 h2 n2 n3 k H3) \Rightarrow (\lambda (H4: (eq A (ASort h1 n2)
423 (AHead a (asucc g a0)))).(\lambda (H5: (eq A (ASort h2 n3) (ASort n1
424 n0))).((let H6 \def (eq_ind A (ASort h1 n2) (\lambda (e: A).(match e in A
425 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
426 \Rightarrow False])) I (AHead a (asucc g a0)) H4) in (False_ind ((eq A (ASort
427 h2 n3) (ASort n1 n0)) \to ((eq A (aplus g (ASort h1 n2) k) (aplus g (ASort h2
428 n3) k)) \to (leq g (AHead a a0) (ASort (S n1) n0)))) H6)) H5 H3))) |
429 (leq_head a3 a4 H3 a5 a6 H4) \Rightarrow (\lambda (H5: (eq A (AHead a3 a5)
430 (AHead a (asucc g a0)))).(\lambda (H6: (eq A (AHead a4 a6) (ASort n1
431 n0))).((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return
432 (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a7)
433 \Rightarrow a7])) (AHead a3 a5) (AHead a (asucc g a0)) H5) in ((let H8 \def
434 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
435 [(ASort _ _) \Rightarrow a3 | (AHead a7 _) \Rightarrow a7])) (AHead a3 a5)
436 (AHead a (asucc g a0)) H5) in (eq_ind A a (\lambda (a7: A).((eq A a5 (asucc g
437 a0)) \to ((eq A (AHead a4 a6) (ASort n1 n0)) \to ((leq g a7 a4) \to ((leq g
438 a5 a6) \to (leq g (AHead a a0) (ASort (S n1) n0))))))) (\lambda (H9: (eq A a5
439 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a7: A).((eq A (AHead a4 a6)
440 (ASort n1 n0)) \to ((leq g a a4) \to ((leq g a7 a6) \to (leq g (AHead a a0)
441 (ASort (S n1) n0)))))) (\lambda (H10: (eq A (AHead a4 a6) (ASort n1
442 n0))).(let H11 \def (eq_ind A (AHead a4 a6) (\lambda (e: A).(match e in A
443 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
444 _) \Rightarrow True])) I (ASort n1 n0) H10) in (False_ind ((leq g a a4) \to
445 ((leq g (asucc g a0) a6) \to (leq g (AHead a a0) (ASort (S n1) n0)))) H11)))
446 a5 (sym_eq A a5 (asucc g a0) H9))) a3 (sym_eq A a3 a H8))) H7)) H6 H3 H4)))])
447 in (H3 (refl_equal A (AHead a (asucc g a0))) (refl_equal A (ASort n1
448 n0))))))) n H1)))) (\lambda (a3: A).(\lambda (_: (((leq g (asucc g (AHead a
449 a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda (a4: A).(\lambda
450 (_: (((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g (AHead a a0)
451 a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g (AHead a3
452 a4)))).(let H4 \def (match H3 in leq return (\lambda (a5: A).(\lambda (a6:
453 A).(\lambda (_: (leq ? a5 a6)).((eq A a5 (AHead a (asucc g a0))) \to ((eq A
454 a6 (AHead a3 (asucc g a4))) \to (leq g (AHead a a0) (AHead a3 a4))))))) with
455 [(leq_sort h1 h2 n1 n2 k H4) \Rightarrow (\lambda (H5: (eq A (ASort h1 n1)
456 (AHead a (asucc g a0)))).(\lambda (H6: (eq A (ASort h2 n2) (AHead a3 (asucc g
457 a4)))).((let H7 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A
458 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
459 \Rightarrow False])) I (AHead a (asucc g a0)) H5) in (False_ind ((eq A (ASort
460 h2 n2) (AHead a3 (asucc g a4))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g
461 (ASort h2 n2) k)) \to (leq g (AHead a a0) (AHead a3 a4)))) H7)) H6 H4))) |
462 (leq_head a5 a6 H4 a7 a8 H5) \Rightarrow (\lambda (H6: (eq A (AHead a5 a7)
463 (AHead a (asucc g a0)))).(\lambda (H7: (eq A (AHead a6 a8) (AHead a3 (asucc g
464 a4)))).((let H8 \def (f_equal A A (\lambda (e: A).(match e in A return
465 (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 | (AHead _ a9)
466 \Rightarrow a9])) (AHead a5 a7) (AHead a (asucc g a0)) H6) in ((let H9 \def
467 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
468 [(ASort _ _) \Rightarrow a5 | (AHead a9 _) \Rightarrow a9])) (AHead a5 a7)
469 (AHead a (asucc g a0)) H6) in (eq_ind A a (\lambda (a9: A).((eq A a7 (asucc g
470 a0)) \to ((eq A (AHead a6 a8) (AHead a3 (asucc g a4))) \to ((leq g a9 a6) \to
471 ((leq g a7 a8) \to (leq g (AHead a a0) (AHead a3 a4))))))) (\lambda (H10: (eq
472 A a7 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a9: A).((eq A (AHead a6
473 a8) (AHead a3 (asucc g a4))) \to ((leq g a a6) \to ((leq g a9 a8) \to (leq g
474 (AHead a a0) (AHead a3 a4)))))) (\lambda (H11: (eq A (AHead a6 a8) (AHead a3
475 (asucc g a4)))).(let H12 \def (f_equal A A (\lambda (e: A).(match e in A
476 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a8 | (AHead _ a9)
477 \Rightarrow a9])) (AHead a6 a8) (AHead a3 (asucc g a4)) H11) in ((let H13
478 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
479 with [(ASort _ _) \Rightarrow a6 | (AHead a9 _) \Rightarrow a9])) (AHead a6
480 a8) (AHead a3 (asucc g a4)) H11) in (eq_ind A a3 (\lambda (a9: A).((eq A a8
481 (asucc g a4)) \to ((leq g a a9) \to ((leq g (asucc g a0) a8) \to (leq g
482 (AHead a a0) (AHead a3 a4)))))) (\lambda (H14: (eq A a8 (asucc g
483 a4))).(eq_ind A (asucc g a4) (\lambda (a9: A).((leq g a a3) \to ((leq g
484 (asucc g a0) a9) \to (leq g (AHead a a0) (AHead a3 a4))))) (\lambda (H15:
485 (leq g a a3)).(\lambda (H16: (leq g (asucc g a0) (asucc g a4))).(leq_head g a
486 a3 H15 a0 a4 (H0 a4 H16)))) a8 (sym_eq A a8 (asucc g a4) H14))) a6 (sym_eq A
487 a6 a3 H13))) H12))) a7 (sym_eq A a7 (asucc g a0) H10))) a5 (sym_eq A a5 a
488 H9))) H8)) H7 H4 H5)))]) in (H4 (refl_equal A (AHead a (asucc g a0)))
489 (refl_equal A (AHead a3 (asucc g a4)))))))))) a2)))))) a1)).
492 \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g
495 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(ex A (\lambda (a1:
496 A).(leq g a0 (asucc g a1))))) (\lambda (n: nat).(\lambda (n0: nat).(ex_intro
497 A (\lambda (a0: A).(leq g (ASort n n0) (asucc g a0))) (ASort (S n) n0)
498 (leq_refl g (ASort n n0))))) (\lambda (a0: A).(\lambda (_: (ex A (\lambda
499 (a1: A).(leq g a0 (asucc g a1))))).(\lambda (a1: A).(\lambda (H0: (ex A
500 (\lambda (a2: A).(leq g a1 (asucc g a2))))).(let H1 \def H0 in (ex_ind A
501 (\lambda (a2: A).(leq g a1 (asucc g a2))) (ex A (\lambda (a2: A).(leq g
502 (AHead a0 a1) (asucc g a2)))) (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc
503 g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2)))
504 (AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1))))))
507 theorem leq_ahead_asucc_false:
508 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2)
509 (asucc g a1)) \to (\forall (P: Prop).P))))
511 \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
512 A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda
513 (n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead
514 (ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
515 \Rightarrow (ASort h n0)]))).(\lambda (P: Prop).(nat_ind (\lambda (n1:
516 nat).((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O
517 (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P)) (\lambda (H0: (leq g
518 (AHead (ASort O n0) a2) (ASort O (next g n0)))).(let H1 \def (match H0 in leq
519 return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a
520 (AHead (ASort O n0) a2)) \to ((eq A a0 (ASort O (next g n0))) \to P))))) with
521 [(leq_sort h1 h2 n1 n2 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n1)
522 (AHead (ASort O n0) a2))).(\lambda (H3: (eq A (ASort h2 n2) (ASort O (next g
523 n0)))).((let H4 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A
524 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
525 \Rightarrow False])) I (AHead (ASort O n0) a2) H2) in (False_ind ((eq A
526 (ASort h2 n2) (ASort O (next g n0))) \to ((eq A (aplus g (ASort h1 n1) k)
527 (aplus g (ASort h2 n2) k)) \to P)) H4)) H3 H1))) | (leq_head a0 a3 H1 a4 a5
528 H2) \Rightarrow (\lambda (H3: (eq A (AHead a0 a4) (AHead (ASort O n0)
529 a2))).(\lambda (H4: (eq A (AHead a3 a5) (ASort O (next g n0)))).((let H5 \def
530 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
531 [(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a0 a4)
532 (AHead (ASort O n0) a2) H3) in ((let H6 \def (f_equal A A (\lambda (e:
533 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 |
534 (AHead a _) \Rightarrow a])) (AHead a0 a4) (AHead (ASort O n0) a2) H3) in
535 (eq_ind A (ASort O n0) (\lambda (a: A).((eq A a4 a2) \to ((eq A (AHead a3 a5)
536 (ASort O (next g n0))) \to ((leq g a a3) \to ((leq g a4 a5) \to P)))))
537 (\lambda (H7: (eq A a4 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a3 a5)
538 (ASort O (next g n0))) \to ((leq g (ASort O n0) a3) \to ((leq g a a5) \to
539 P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort O (next g n0)))).(let H9 \def
540 (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda (_:
541 A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
542 True])) I (ASort O (next g n0)) H8) in (False_ind ((leq g (ASort O n0) a3)
543 \to ((leq g a2 a5) \to P)) H9))) a4 (sym_eq A a4 a2 H7))) a0 (sym_eq A a0
544 (ASort O n0) H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead (ASort O
545 n0) a2)) (refl_equal A (ASort O (next g n0)))))) (\lambda (n1: nat).(\lambda
546 (_: (((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O
547 (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P))).(\lambda (H0: (leq
548 g (AHead (ASort (S n1) n0) a2) (ASort n1 n0))).(let H1 \def (match H0 in leq
549 return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a
550 (AHead (ASort (S n1) n0) a2)) \to ((eq A a0 (ASort n1 n0)) \to P))))) with
551 [(leq_sort h1 h2 n2 n3 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n2)
552 (AHead (ASort (S n1) n0) a2))).(\lambda (H3: (eq A (ASort h2 n3) (ASort n1
553 n0))).((let H4 \def (eq_ind A (ASort h1 n2) (\lambda (e: A).(match e in A
554 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
555 \Rightarrow False])) I (AHead (ASort (S n1) n0) a2) H2) in (False_ind ((eq A
556 (ASort h2 n3) (ASort n1 n0)) \to ((eq A (aplus g (ASort h1 n2) k) (aplus g
557 (ASort h2 n3) k)) \to P)) H4)) H3 H1))) | (leq_head a0 a3 H1 a4 a5 H2)
558 \Rightarrow (\lambda (H3: (eq A (AHead a0 a4) (AHead (ASort (S n1) n0)
559 a2))).(\lambda (H4: (eq A (AHead a3 a5) (ASort n1 n0))).((let H5 \def
560 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
561 [(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a0 a4)
562 (AHead (ASort (S n1) n0) a2) H3) in ((let H6 \def (f_equal A A (\lambda (e:
563 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 |
564 (AHead a _) \Rightarrow a])) (AHead a0 a4) (AHead (ASort (S n1) n0) a2) H3)
565 in (eq_ind A (ASort (S n1) n0) (\lambda (a: A).((eq A a4 a2) \to ((eq A
566 (AHead a3 a5) (ASort n1 n0)) \to ((leq g a a3) \to ((leq g a4 a5) \to P)))))
567 (\lambda (H7: (eq A a4 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a3 a5)
568 (ASort n1 n0)) \to ((leq g (ASort (S n1) n0) a3) \to ((leq g a a5) \to P))))
569 (\lambda (H8: (eq A (AHead a3 a5) (ASort n1 n0))).(let H9 \def (eq_ind A
570 (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with
571 [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1
572 n0) H8) in (False_ind ((leq g (ASort (S n1) n0) a3) \to ((leq g a2 a5) \to
573 P)) H9))) a4 (sym_eq A a4 a2 H7))) a0 (sym_eq A a0 (ASort (S n1) n0) H6)))
574 H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead (ASort (S n1) n0) a2))
575 (refl_equal A (ASort n1 n0))))))) n H)))))) (\lambda (a: A).(\lambda (_:
576 ((\forall (a2: A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P:
577 Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead
578 a0 a2) (asucc g a0)) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda
579 (H1: (leq g (AHead (AHead a a0) a2) (AHead a (asucc g a0)))).(\lambda (P:
580 Prop).(let H2 \def (match H1 in leq return (\lambda (a3: A).(\lambda (a4:
581 A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (AHead (AHead a a0) a2)) \to ((eq A
582 a4 (AHead a (asucc g a0))) \to P))))) with [(leq_sort h1 h2 n1 n2 k H2)
583 \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead (AHead a a0)
584 a2))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a (asucc g a0)))).((let H5
585 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A return (\lambda
586 (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
587 False])) I (AHead (AHead a a0) a2) H3) in (False_ind ((eq A (ASort h2 n2)
588 (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort
589 h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a3 a4 H2 a5 a6 H3) \Rightarrow
590 (\lambda (H4: (eq A (AHead a3 a5) (AHead (AHead a a0) a2))).(\lambda (H5: (eq
591 A (AHead a4 a6) (AHead a (asucc g a0)))).((let H6 \def (f_equal A A (\lambda
592 (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow
593 a5 | (AHead _ a7) \Rightarrow a7])) (AHead a3 a5) (AHead (AHead a a0) a2) H4)
594 in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda
595 (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a7 _) \Rightarrow a7]))
596 (AHead a3 a5) (AHead (AHead a a0) a2) H4) in (eq_ind A (AHead a a0) (\lambda
597 (a7: A).((eq A a5 a2) \to ((eq A (AHead a4 a6) (AHead a (asucc g a0))) \to
598 ((leq g a7 a4) \to ((leq g a5 a6) \to P))))) (\lambda (H8: (eq A a5
599 a2)).(eq_ind A a2 (\lambda (a7: A).((eq A (AHead a4 a6) (AHead a (asucc g
600 a0))) \to ((leq g (AHead a a0) a4) \to ((leq g a7 a6) \to P)))) (\lambda (H9:
601 (eq A (AHead a4 a6) (AHead a (asucc g a0)))).(let H10 \def (f_equal A A
602 (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
603 \Rightarrow a6 | (AHead _ a7) \Rightarrow a7])) (AHead a4 a6) (AHead a (asucc
604 g a0)) H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e in A
605 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead a7 _)
606 \Rightarrow a7])) (AHead a4 a6) (AHead a (asucc g a0)) H9) in (eq_ind A a
607 (\lambda (a7: A).((eq A a6 (asucc g a0)) \to ((leq g (AHead a a0) a7) \to
608 ((leq g a2 a6) \to P)))) (\lambda (H12: (eq A a6 (asucc g a0))).(eq_ind A
609 (asucc g a0) (\lambda (a7: A).((leq g (AHead a a0) a) \to ((leq g a2 a7) \to
610 P))) (\lambda (H13: (leq g (AHead a a0) a)).(\lambda (_: (leq g a2 (asucc g
611 a0))).(leq_ahead_false_1 g a a0 H13 P))) a6 (sym_eq A a6 (asucc g a0) H12)))
612 a4 (sym_eq A a4 a H11))) H10))) a5 (sym_eq A a5 a2 H8))) a3 (sym_eq A a3
613 (AHead a a0) H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead (AHead a
614 a0) a2)) (refl_equal A (AHead a (asucc g a0)))))))))))) a1)).
616 theorem leq_asucc_false:
617 \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P:
620 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0)
621 a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda
622 (H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
623 \Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).(nat_ind
624 (\lambda (n1: nat).((leq g (match n1 with [O \Rightarrow (ASort O (next g
625 n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P)) (\lambda (H0:
626 (leq g (ASort O (next g n0)) (ASort O n0))).(let H1 \def (match H0 in leq
627 return (\lambda (a0: A).(\lambda (a1: A).(\lambda (_: (leq ? a0 a1)).((eq A
628 a0 (ASort O (next g n0))) \to ((eq A a1 (ASort O n0)) \to P))))) with
629 [(leq_sort h1 h2 n1 n2 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n1)
630 (ASort O (next g n0)))).(\lambda (H3: (eq A (ASort h2 n2) (ASort O
631 n0))).((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return
632 (\lambda (_: A).nat) with [(ASort _ n3) \Rightarrow n3 | (AHead _ _)
633 \Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H2) in ((let H5 \def
634 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
635 [(ASort n3 _) \Rightarrow n3 | (AHead _ _) \Rightarrow h1])) (ASort h1 n1)
636 (ASort O (next g n0)) H2) in (eq_ind nat O (\lambda (n3: nat).((eq nat n1
637 (next g n0)) \to ((eq A (ASort h2 n2) (ASort O n0)) \to ((eq A (aplus g
638 (ASort n3 n1) k) (aplus g (ASort h2 n2) k)) \to P)))) (\lambda (H6: (eq nat
639 n1 (next g n0))).(eq_ind nat (next g n0) (\lambda (n3: nat).((eq A (ASort h2
640 n2) (ASort O n0)) \to ((eq A (aplus g (ASort O n3) k) (aplus g (ASort h2 n2)
641 k)) \to P))) (\lambda (H7: (eq A (ASort h2 n2) (ASort O n0))).(let H8 \def
642 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
643 [(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow n2])) (ASort h2 n2)
644 (ASort O n0) H7) in ((let H9 \def (f_equal A nat (\lambda (e: A).(match e in
645 A return (\lambda (_: A).nat) with [(ASort n3 _) \Rightarrow n3 | (AHead _ _)
646 \Rightarrow h2])) (ASort h2 n2) (ASort O n0) H7) in (eq_ind nat O (\lambda
647 (n3: nat).((eq nat n2 n0) \to ((eq A (aplus g (ASort O (next g n0)) k) (aplus
648 g (ASort n3 n2) k)) \to P))) (\lambda (H10: (eq nat n2 n0)).(eq_ind nat n0
649 (\lambda (n3: nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O
650 n3) k)) \to P)) (\lambda (H11: (eq A (aplus g (ASort O (next g n0)) k) (aplus
651 g (ASort O n0) k))).(let H12 \def (eq_ind_r A (aplus g (ASort O (next g n0))
652 k) (\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) k))) H11 (aplus g (ASort O
653 n0) (S k)) (aplus_sort_O_S_simpl g n0 k)) in (let H_y \def (aplus_inj g (S k)
654 k (ASort O n0) H12) in (le_Sx_x k (eq_ind_r nat k (\lambda (n3: nat).(le n3
655 k)) (le_n k) (S k) H_y) P)))) n2 (sym_eq nat n2 n0 H10))) h2 (sym_eq nat h2 O
656 H9))) H8))) n1 (sym_eq nat n1 (next g n0) H6))) h1 (sym_eq nat h1 O H5)))
657 H4)) H3 H1))) | (leq_head a1 a2 H1 a3 a4 H2) \Rightarrow (\lambda (H3: (eq A
658 (AHead a1 a3) (ASort O (next g n0)))).(\lambda (H4: (eq A (AHead a2 a4)
659 (ASort O n0))).((let H5 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e
660 in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
661 (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H3) in (False_ind
662 ((eq A (AHead a2 a4) (ASort O n0)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to
663 P))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (ASort O (next g n0)))
664 (refl_equal A (ASort O n0))))) (\lambda (n1: nat).(\lambda (_: (((leq g
665 (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow
666 (ASort h n0)]) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (ASort n1 n0)
667 (ASort (S n1) n0))).(let H1 \def (match H0 in leq return (\lambda (a0:
668 A).(\lambda (a1: A).(\lambda (_: (leq ? a0 a1)).((eq A a0 (ASort n1 n0)) \to
669 ((eq A a1 (ASort (S n1) n0)) \to P))))) with [(leq_sort h1 h2 n2 n3 k H1)
670 \Rightarrow (\lambda (H2: (eq A (ASort h1 n2) (ASort n1 n0))).(\lambda (H3:
671 (eq A (ASort h2 n3) (ASort (S n1) n0))).((let H4 \def (f_equal A nat (\lambda
672 (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n4)
673 \Rightarrow n4 | (AHead _ _) \Rightarrow n2])) (ASort h1 n2) (ASort n1 n0)
674 H2) in ((let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return
675 (\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _)
676 \Rightarrow h1])) (ASort h1 n2) (ASort n1 n0) H2) in (eq_ind nat n1 (\lambda
677 (n4: nat).((eq nat n2 n0) \to ((eq A (ASort h2 n3) (ASort (S n1) n0)) \to
678 ((eq A (aplus g (ASort n4 n2) k) (aplus g (ASort h2 n3) k)) \to P))))
679 (\lambda (H6: (eq nat n2 n0)).(eq_ind nat n0 (\lambda (n4: nat).((eq A (ASort
680 h2 n3) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort n1 n4) k) (aplus g
681 (ASort h2 n3) k)) \to P))) (\lambda (H7: (eq A (ASort h2 n3) (ASort (S n1)
682 n0))).(let H8 \def (f_equal A nat (\lambda (e: A).(match e in A return
683 (\lambda (_: A).nat) with [(ASort _ n4) \Rightarrow n4 | (AHead _ _)
684 \Rightarrow n3])) (ASort h2 n3) (ASort (S n1) n0) H7) in ((let H9 \def
685 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
686 [(ASort n4 _) \Rightarrow n4 | (AHead _ _) \Rightarrow h2])) (ASort h2 n3)
687 (ASort (S n1) n0) H7) in (eq_ind nat (S n1) (\lambda (n4: nat).((eq nat n3
688 n0) \to ((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort n4 n3) k)) \to P)))
689 (\lambda (H10: (eq nat n3 n0)).(eq_ind nat n0 (\lambda (n4: nat).((eq A
690 (aplus g (ASort n1 n0) k) (aplus g (ASort (S n1) n4) k)) \to P)) (\lambda
691 (H11: (eq A (aplus g (ASort n1 n0) k) (aplus g (ASort (S n1) n0) k))).(let
692 H12 \def (eq_ind_r A (aplus g (ASort n1 n0) k) (\lambda (a0: A).(eq A a0
693 (aplus g (ASort (S n1) n0) k))) H11 (aplus g (ASort (S n1) n0) (S k))
694 (aplus_sort_S_S_simpl g n0 n1 k)) in (let H_y \def (aplus_inj g (S k) k
695 (ASort (S n1) n0) H12) in (le_Sx_x k (eq_ind_r nat k (\lambda (n4: nat).(le
696 n4 k)) (le_n k) (S k) H_y) P)))) n3 (sym_eq nat n3 n0 H10))) h2 (sym_eq nat
697 h2 (S n1) H9))) H8))) n2 (sym_eq nat n2 n0 H6))) h1 (sym_eq nat h1 n1 H5)))
698 H4)) H3 H1))) | (leq_head a1 a2 H1 a3 a4 H2) \Rightarrow (\lambda (H3: (eq A
699 (AHead a1 a3) (ASort n1 n0))).(\lambda (H4: (eq A (AHead a2 a4) (ASort (S n1)
700 n0))).((let H5 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e in A
701 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
702 _) \Rightarrow True])) I (ASort n1 n0) H3) in (False_ind ((eq A (AHead a2 a4)
703 (ASort (S n1) n0)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to P))) H5)) H4 H1
704 H2)))]) in (H1 (refl_equal A (ASort n1 n0)) (refl_equal A (ASort (S n1)
705 n0))))))) n H))))) (\lambda (a0: A).(\lambda (_: (((leq g (asucc g a0) a0)
706 \to (\forall (P: Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g
707 a1) a1) \to (\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g
708 a1)) (AHead a0 a1))).(\lambda (P: Prop).(let H2 \def (match H1 in leq return
709 (\lambda (a2: A).(\lambda (a3: A).(\lambda (_: (leq ? a2 a3)).((eq A a2
710 (AHead a0 (asucc g a1))) \to ((eq A a3 (AHead a0 a1)) \to P))))) with
711 [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1)
712 (AHead a0 (asucc g a1)))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a0
713 a1))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A
714 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
715 \Rightarrow False])) I (AHead a0 (asucc g a1)) H3) in (False_ind ((eq A
716 (ASort h2 n2) (AHead a0 a1)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g
717 (ASort h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a2 a3 H2 a4 a5 H3)
718 \Rightarrow (\lambda (H4: (eq A (AHead a2 a4) (AHead a0 (asucc g
719 a1)))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a0 a1))).((let H6 \def
720 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
721 [(ASort _ _) \Rightarrow a4 | (AHead _ a6) \Rightarrow a6])) (AHead a2 a4)
722 (AHead a0 (asucc g a1)) H4) in ((let H7 \def (f_equal A A (\lambda (e:
723 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a2 |
724 (AHead a6 _) \Rightarrow a6])) (AHead a2 a4) (AHead a0 (asucc g a1)) H4) in
725 (eq_ind A a0 (\lambda (a6: A).((eq A a4 (asucc g a1)) \to ((eq A (AHead a3
726 a5) (AHead a0 a1)) \to ((leq g a6 a3) \to ((leq g a4 a5) \to P))))) (\lambda
727 (H8: (eq A a4 (asucc g a1))).(eq_ind A (asucc g a1) (\lambda (a6: A).((eq A
728 (AHead a3 a5) (AHead a0 a1)) \to ((leq g a0 a3) \to ((leq g a6 a5) \to P))))
729 (\lambda (H9: (eq A (AHead a3 a5) (AHead a0 a1))).(let H10 \def (f_equal A A
730 (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
731 \Rightarrow a5 | (AHead _ a6) \Rightarrow a6])) (AHead a3 a5) (AHead a0 a1)
732 H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e in A return
733 (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a6 _)
734 \Rightarrow a6])) (AHead a3 a5) (AHead a0 a1) H9) in (eq_ind A a0 (\lambda
735 (a6: A).((eq A a5 a1) \to ((leq g a0 a6) \to ((leq g (asucc g a1) a5) \to
736 P)))) (\lambda (H12: (eq A a5 a1)).(eq_ind A a1 (\lambda (a6: A).((leq g a0
737 a0) \to ((leq g (asucc g a1) a6) \to P))) (\lambda (_: (leq g a0
738 a0)).(\lambda (H14: (leq g (asucc g a1) a1)).(H0 H14 P))) a5 (sym_eq A a5 a1
739 H12))) a3 (sym_eq A a3 a0 H11))) H10))) a4 (sym_eq A a4 (asucc g a1) H8))) a2
740 (sym_eq A a2 a0 H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead a0
741 (asucc g a1))) (refl_equal A (AHead a0 a1)))))))))) a)).
projects / helm.git / blob