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 "leq/props.ma".
19 include "aplus/props.ma".
22 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g
23 (asucc g a1) (asucc g a2)))))
25 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
26 a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g (asucc g a) (asucc g
27 a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2:
28 nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g
29 (ASort h2 n2) k))).(nat_ind (\lambda (n: nat).((eq A (aplus g (ASort n n1) k)
30 (aplus g (ASort h2 n2) k)) \to (leq g (match n with [O \Rightarrow (ASort O
31 (next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow
32 (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) (\lambda (H1: (eq
33 A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda (n:
34 nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort n n2) k)) \to (leq g
35 (ASort O (next g n1)) (match n with [O \Rightarrow (ASort O (next g n2)) | (S
36 h) \Rightarrow (ASort h n2)])))) (\lambda (H2: (eq A (aplus g (ASort O n1) k)
37 (aplus g (ASort O n2) k))).(leq_sort g O O (next g n1) (next g n2) k (eq_ind
38 A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O
39 (next g n2)) k))) (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq
40 A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort O n2) k)
41 (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k))))
42 (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort O n1) k)
43 H2) (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)) (aplus g
44 (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))) (\lambda (h3:
45 nat).(\lambda (_: (((eq A (aplus g (ASort O n1) k) (aplus g (ASort h3 n2) k))
46 \to (leq g (ASort O (next g n1)) (match h3 with [O \Rightarrow (ASort O (next
47 g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H2: (eq A (aplus g
48 (ASort O n1) k) (aplus g (ASort (S h3) n2) k))).(leq_sort g O h3 (next g n1)
49 n2 k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g
50 (ASort h3 n2) k))) (eq_ind A (aplus g (ASort (S h3) n2) (S k)) (\lambda (a:
51 A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort (S h3)
52 n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort (S h3) n2)
53 k)))) (refl_equal A (asucc g (aplus g (ASort (S h3) n2) k))) (aplus g (ASort
54 O n1) k) H2) (aplus g (ASort h3 n2) k) (aplus_sort_S_S_simpl g n2 h3 k))
55 (aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))))) h2 H1))
56 (\lambda (h3: nat).(\lambda (IHh1: (((eq A (aplus g (ASort h3 n1) k) (aplus g
57 (ASort h2 n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next g
58 n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow (ASort
59 O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H1: (eq A
60 (aplus g (ASort (S h3) n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda
61 (n: nat).((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort n n2) k)) \to
62 ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort n n2) k)) \to (leq g
63 (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow
64 (ASort h n1)]) (match n with [O \Rightarrow (ASort O (next g n2)) | (S h)
65 \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match n with [O
66 \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))))
67 (\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort O n2)
68 k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort O n2) k))
69 \to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h)
70 \Rightarrow (ASort h n1)]) (ASort O (next g n2)))))).(leq_sort g h3 O n1
71 (next g n2) k (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A
72 (aplus g (ASort h3 n1) k) a)) (eq_ind A (aplus g (ASort (S h3) n1) (S k))
73 (\lambda (a: A).(eq A a (aplus g (ASort O n2) (S k)))) (eq_ind_r A (aplus g
74 (ASort O n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O
75 n2) k)))) (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort
76 (S h3) n1) k) H2) (aplus g (ASort h3 n1) k) (aplus_sort_S_S_simpl g n1 h3 k))
77 (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k))))) (\lambda
78 (h4: nat).(\lambda (_: (((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort
79 h4 n2) k)) \to ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort h4 n2) k))
80 \to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h)
81 \Rightarrow (ASort h n1)]) (match h4 with [O \Rightarrow (ASort O (next g
82 n2)) | (S h) \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match h4
83 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h
84 n2)])))))).(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort
85 (S h4) n2) k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g
86 (ASort (S h4) n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next
87 g n1)) | (S h) \Rightarrow (ASort h n1)]) (ASort h4 n2))))).(leq_sort g h3 h4
88 n1 n2 k (eq_ind A (aplus g (ASort (S h3) n1) (S k)) (\lambda (a: A).(eq A a
89 (aplus g (ASort h4 n2) k))) (eq_ind A (aplus g (ASort (S h4) n2) (S k))
90 (\lambda (a: A).(eq A (aplus g (ASort (S h3) n1) (S k)) a)) (eq_ind_r A
91 (aplus g (ASort (S h4) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g
92 (aplus g (ASort (S h4) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S
93 h4) n2) k))) (aplus g (ASort (S h3) n1) k) H2) (aplus g (ASort h4 n2) k)
94 (aplus_sort_S_S_simpl g n2 h4 k)) (aplus g (ASort h3 n1) k)
95 (aplus_sort_S_S_simpl g n1 h3 k))))))) h2 H1 IHh1)))) h1 H0))))))) (\lambda
96 (a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3 a4)).(\lambda (_: (leq g
97 (asucc g a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_:
98 (leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g
99 a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))).
102 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc
103 g a2)) \to (leq g a1 a2))))
105 \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
106 A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n:
107 nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g
108 (asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda
109 (n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0))
110 (asucc g (ASort n1 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort
111 n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))
112 (\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1
113 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g
114 (ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2)))) (\lambda (H1: (leq g
115 (asucc g (ASort O n0)) (asucc g (ASort O n2)))).(let H2 \def (match H1 in leq
116 return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a
117 (ASort O (next g n0))) \to ((eq A a0 (ASort O (next g n2))) \to (leq g (ASort
118 O n0) (ASort O n2))))))) with [(leq_sort h1 h2 n3 n4 k H2) \Rightarrow
119 (\lambda (H3: (eq A (ASort h1 n3) (ASort O (next g n0)))).(\lambda (H4: (eq A
120 (ASort h2 n4) (ASort O (next g n2)))).((let H5 \def (f_equal A nat (\lambda
121 (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n5)
122 \Rightarrow n5 | (AHead _ _) \Rightarrow n3])) (ASort h1 n3) (ASort O (next g
123 n0)) H3) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e in A return
124 (\lambda (_: A).nat) with [(ASort n5 _) \Rightarrow n5 | (AHead _ _)
125 \Rightarrow h1])) (ASort h1 n3) (ASort O (next g n0)) H3) in (eq_ind nat O
126 (\lambda (n5: nat).((eq nat n3 (next g n0)) \to ((eq A (ASort h2 n4) (ASort O
127 (next g n2))) \to ((eq A (aplus g (ASort n5 n3) k) (aplus g (ASort h2 n4) k))
128 \to (leq g (ASort O n0) (ASort O n2)))))) (\lambda (H7: (eq nat n3 (next g
129 n0))).(eq_ind nat (next g n0) (\lambda (n5: nat).((eq A (ASort h2 n4) (ASort
130 O (next g n2))) \to ((eq A (aplus g (ASort O n5) k) (aplus g (ASort h2 n4)
131 k)) \to (leq g (ASort O n0) (ASort O n2))))) (\lambda (H8: (eq A (ASort h2
132 n4) (ASort O (next g n2)))).(let H9 \def (f_equal A nat (\lambda (e:
133 A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n5) \Rightarrow
134 n5 | (AHead _ _) \Rightarrow n4])) (ASort h2 n4) (ASort O (next g n2)) H8) in
135 ((let H10 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda
136 (_: A).nat) with [(ASort n5 _) \Rightarrow n5 | (AHead _ _) \Rightarrow h2]))
137 (ASort h2 n4) (ASort O (next g n2)) H8) in (eq_ind nat O (\lambda (n5:
138 nat).((eq nat n4 (next g n2)) \to ((eq A (aplus g (ASort O (next g n0)) k)
139 (aplus g (ASort n5 n4) k)) \to (leq g (ASort O n0) (ASort O n2))))) (\lambda
140 (H11: (eq nat n4 (next g n2))).(eq_ind nat (next g n2) (\lambda (n5:
141 nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O n5) k)) \to
142 (leq g (ASort O n0) (ASort O n2)))) (\lambda (H12: (eq A (aplus g (ASort O
143 (next g n0)) k) (aplus g (ASort O (next g n2)) k))).(let H13 \def (eq_ind_r A
144 (aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a (aplus g (ASort O
145 (next g n2)) k))) H12 (aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g n0
146 k)) in (let H14 \def (eq_ind_r A (aplus g (ASort O (next g n2)) k) (\lambda
147 (a: A).(eq A (aplus g (ASort O n0) (S k)) a)) H13 (aplus g (ASort O n2) (S
148 k)) (aplus_sort_O_S_simpl g n2 k)) in (leq_sort g O O n0 n2 (S k) H14)))) n4
149 (sym_eq nat n4 (next g n2) H11))) h2 (sym_eq nat h2 O H10))) H9))) n3 (sym_eq
150 nat n3 (next g n0) H7))) h1 (sym_eq nat h1 O H6))) H5)) H4 H2))) | (leq_head
151 a0 a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A (AHead a0 a4) (ASort O
152 (next g n0)))).(\lambda (H5: (eq A (AHead a3 a5) (ASort O (next g
153 n2)))).((let H6 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match e in A
154 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
155 _) \Rightarrow True])) I (ASort O (next g n0)) H4) in (False_ind ((eq A
156 (AHead a3 a5) (ASort O (next g n2))) \to ((leq g a0 a3) \to ((leq g a4 a5)
157 \to (leq g (ASort O n0) (ASort O n2))))) H6)) H5 H2 H3)))]) in (H2
158 (refl_equal A (ASort O (next g n0))) (refl_equal A (ASort O (next g n2))))))
159 (\lambda (n3: nat).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g
160 (ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2))))).(\lambda (H1: (leq
161 g (asucc g (ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H2 \def (match H1
162 in leq return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a
163 a0)).((eq A a (ASort O (next g n0))) \to ((eq A a0 (ASort n3 n2)) \to (leq g
164 (ASort O n0) (ASort (S n3) n2))))))) with [(leq_sort h1 h2 n4 n5 k H2)
165 \Rightarrow (\lambda (H3: (eq A (ASort h1 n4) (ASort O (next g
166 n0)))).(\lambda (H4: (eq A (ASort h2 n5) (ASort n3 n2))).((let H5 \def
167 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
168 [(ASort _ n6) \Rightarrow n6 | (AHead _ _) \Rightarrow n4])) (ASort h1 n4)
169 (ASort O (next g n0)) H3) in ((let H6 \def (f_equal A nat (\lambda (e:
170 A).(match e in A return (\lambda (_: A).nat) with [(ASort n6 _) \Rightarrow
171 n6 | (AHead _ _) \Rightarrow h1])) (ASort h1 n4) (ASort O (next g n0)) H3) in
172 (eq_ind nat O (\lambda (n6: nat).((eq nat n4 (next g n0)) \to ((eq A (ASort
173 h2 n5) (ASort n3 n2)) \to ((eq A (aplus g (ASort n6 n4) k) (aplus g (ASort h2
174 n5) k)) \to (leq g (ASort O n0) (ASort (S n3) n2)))))) (\lambda (H7: (eq nat
175 n4 (next g n0))).(eq_ind nat (next g n0) (\lambda (n6: nat).((eq A (ASort h2
176 n5) (ASort n3 n2)) \to ((eq A (aplus g (ASort O n6) k) (aplus g (ASort h2 n5)
177 k)) \to (leq g (ASort O n0) (ASort (S n3) n2))))) (\lambda (H8: (eq A (ASort
178 h2 n5) (ASort n3 n2))).(let H9 \def (f_equal A nat (\lambda (e: A).(match e
179 in A return (\lambda (_: A).nat) with [(ASort _ n6) \Rightarrow n6 | (AHead _
180 _) \Rightarrow n5])) (ASort h2 n5) (ASort n3 n2) H8) in ((let H10 \def
181 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
182 [(ASort n6 _) \Rightarrow n6 | (AHead _ _) \Rightarrow h2])) (ASort h2 n5)
183 (ASort n3 n2) H8) in (eq_ind nat n3 (\lambda (n6: nat).((eq nat n5 n2) \to
184 ((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n6 n5) k)) \to (leq
185 g (ASort O n0) (ASort (S n3) n2))))) (\lambda (H11: (eq nat n5 n2)).(eq_ind
186 nat n2 (\lambda (n6: nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g
187 (ASort n3 n6) k)) \to (leq g (ASort O n0) (ASort (S n3) n2)))) (\lambda (H12:
188 (eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n3 n2) k))).(let H13
189 \def (eq_ind_r A (aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a
190 (aplus g (ASort n3 n2) k))) H12 (aplus g (ASort O n0) (S k))
191 (aplus_sort_O_S_simpl g n0 k)) in (let H14 \def (eq_ind_r A (aplus g (ASort
192 n3 n2) k) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S k)) a)) H13 (aplus g
193 (ASort (S n3) n2) (S k)) (aplus_sort_S_S_simpl g n2 n3 k)) in (leq_sort g O
194 (S n3) n0 n2 (S k) H14)))) n5 (sym_eq nat n5 n2 H11))) h2 (sym_eq nat h2 n3
195 H10))) H9))) n4 (sym_eq nat n4 (next g n0) H7))) h1 (sym_eq nat h1 O H6)))
196 H5)) H4 H2))) | (leq_head a0 a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A
197 (AHead a0 a4) (ASort O (next g n0)))).(\lambda (H5: (eq A (AHead a3 a5)
198 (ASort n3 n2))).((let H6 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match
199 e in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
200 (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H4) in (False_ind
201 ((eq A (AHead a3 a5) (ASort n3 n2)) \to ((leq g a0 a3) \to ((leq g a4 a5) \to
202 (leq g (ASort O n0) (ASort (S n3) n2))))) H6)) H5 H2 H3)))]) in (H2
203 (refl_equal A (ASort O (next g n0))) (refl_equal A (ASort n3 n2))))))) n1
204 H0)) (\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0))
205 (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda
206 (H0: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(nat_ind
207 (\lambda (n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4
208 n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq
209 g (ASort n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4
210 n2))))) (\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort O
211 n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort O n2)))
212 \to (leq g (ASort n3 n0) (ASort O n2))))).(let H2 \def (match H1 in leq
213 return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a
214 (ASort n3 n0)) \to ((eq A a0 (ASort O (next g n2))) \to (leq g (ASort (S n3)
215 n0) (ASort O n2))))))) with [(leq_sort h1 h2 n4 n5 k H2) \Rightarrow (\lambda
216 (H3: (eq A (ASort h1 n4) (ASort n3 n0))).(\lambda (H4: (eq A (ASort h2 n5)
217 (ASort O (next g n2)))).((let H5 \def (f_equal A nat (\lambda (e: A).(match e
218 in A return (\lambda (_: A).nat) with [(ASort _ n6) \Rightarrow n6 | (AHead _
219 _) \Rightarrow n4])) (ASort h1 n4) (ASort n3 n0) H3) in ((let H6 \def
220 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
221 [(ASort n6 _) \Rightarrow n6 | (AHead _ _) \Rightarrow h1])) (ASort h1 n4)
222 (ASort n3 n0) H3) in (eq_ind nat n3 (\lambda (n6: nat).((eq nat n4 n0) \to
223 ((eq A (ASort h2 n5) (ASort O (next g n2))) \to ((eq A (aplus g (ASort n6 n4)
224 k) (aplus g (ASort h2 n5) k)) \to (leq g (ASort (S n3) n0) (ASort O n2))))))
225 (\lambda (H7: (eq nat n4 n0)).(eq_ind nat n0 (\lambda (n6: nat).((eq A (ASort
226 h2 n5) (ASort O (next g n2))) \to ((eq A (aplus g (ASort n3 n6) k) (aplus g
227 (ASort h2 n5) k)) \to (leq g (ASort (S n3) n0) (ASort O n2))))) (\lambda (H8:
228 (eq A (ASort h2 n5) (ASort O (next g n2)))).(let H9 \def (f_equal A nat
229 (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n6)
230 \Rightarrow n6 | (AHead _ _) \Rightarrow n5])) (ASort h2 n5) (ASort O (next g
231 n2)) H8) in ((let H10 \def (f_equal A nat (\lambda (e: A).(match e in A
232 return (\lambda (_: A).nat) with [(ASort n6 _) \Rightarrow n6 | (AHead _ _)
233 \Rightarrow h2])) (ASort h2 n5) (ASort O (next g n2)) H8) in (eq_ind nat O
234 (\lambda (n6: nat).((eq nat n5 (next g n2)) \to ((eq A (aplus g (ASort n3 n0)
235 k) (aplus g (ASort n6 n5) k)) \to (leq g (ASort (S n3) n0) (ASort O n2)))))
236 (\lambda (H11: (eq nat n5 (next g n2))).(eq_ind nat (next g n2) (\lambda (n6:
237 nat).((eq A (aplus g (ASort n3 n0) k) (aplus g (ASort O n6) k)) \to (leq g
238 (ASort (S n3) n0) (ASort O n2)))) (\lambda (H12: (eq A (aplus g (ASort n3 n0)
239 k) (aplus g (ASort O (next g n2)) k))).(let H13 \def (eq_ind_r A (aplus g
240 (ASort n3 n0) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n2)) k)))
241 H12 (aplus g (ASort (S n3) n0) (S k)) (aplus_sort_S_S_simpl g n0 n3 k)) in
242 (let H14 \def (eq_ind_r A (aplus g (ASort O (next g n2)) k) (\lambda (a:
243 A).(eq A (aplus g (ASort (S n3) n0) (S k)) a)) H13 (aplus g (ASort O n2) (S
244 k)) (aplus_sort_O_S_simpl g n2 k)) in (leq_sort g (S n3) O n0 n2 (S k)
245 H14)))) n5 (sym_eq nat n5 (next g n2) H11))) h2 (sym_eq nat h2 O H10))) H9)))
246 n4 (sym_eq nat n4 n0 H7))) h1 (sym_eq nat h1 n3 H6))) H5)) H4 H2))) |
247 (leq_head a0 a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A (AHead a0 a4)
248 (ASort n3 n0))).(\lambda (H5: (eq A (AHead a3 a5) (ASort O (next g
249 n2)))).((let H6 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match e in A
250 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
251 _) \Rightarrow True])) I (ASort n3 n0) H4) in (False_ind ((eq A (AHead a3 a5)
252 (ASort O (next g n2))) \to ((leq g a0 a3) \to ((leq g a4 a5) \to (leq g
253 (ASort (S n3) n0) (ASort O n2))))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A
254 (ASort n3 n0)) (refl_equal A (ASort O (next g n2))))))) (\lambda (n4:
255 nat).(\lambda (_: (((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4
256 n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq
257 g (ASort n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4
258 n2)))))).(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S
259 n4) n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort (S
260 n4) n2))) \to (leq g (ASort n3 n0) (ASort (S n4) n2))))).(let H2 \def (match
261 H1 in leq return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a
262 a0)).((eq A a (ASort n3 n0)) \to ((eq A a0 (ASort n4 n2)) \to (leq g (ASort
263 (S n3) n0) (ASort (S n4) n2))))))) with [(leq_sort h1 h2 n5 n6 k H2)
264 \Rightarrow (\lambda (H3: (eq A (ASort h1 n5) (ASort n3 n0))).(\lambda (H4:
265 (eq A (ASort h2 n6) (ASort n4 n2))).((let H5 \def (f_equal A nat (\lambda (e:
266 A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n7) \Rightarrow
267 n7 | (AHead _ _) \Rightarrow n5])) (ASort h1 n5) (ASort n3 n0) H3) in ((let
268 H6 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_:
269 A).nat) with [(ASort n7 _) \Rightarrow n7 | (AHead _ _) \Rightarrow h1]))
270 (ASort h1 n5) (ASort n3 n0) H3) in (eq_ind nat n3 (\lambda (n7: nat).((eq nat
271 n5 n0) \to ((eq A (ASort h2 n6) (ASort n4 n2)) \to ((eq A (aplus g (ASort n7
272 n5) k) (aplus g (ASort h2 n6) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4)
273 n2)))))) (\lambda (H7: (eq nat n5 n0)).(eq_ind nat n0 (\lambda (n7: nat).((eq
274 A (ASort h2 n6) (ASort n4 n2)) \to ((eq A (aplus g (ASort n3 n7) k) (aplus g
275 (ASort h2 n6) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))) (\lambda
276 (H8: (eq A (ASort h2 n6) (ASort n4 n2))).(let H9 \def (f_equal A nat (\lambda
277 (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n7)
278 \Rightarrow n7 | (AHead _ _) \Rightarrow n6])) (ASort h2 n6) (ASort n4 n2)
279 H8) in ((let H10 \def (f_equal A nat (\lambda (e: A).(match e in A return
280 (\lambda (_: A).nat) with [(ASort n7 _) \Rightarrow n7 | (AHead _ _)
281 \Rightarrow h2])) (ASort h2 n6) (ASort n4 n2) H8) in (eq_ind nat n4 (\lambda
282 (n7: nat).((eq nat n6 n2) \to ((eq A (aplus g (ASort n3 n0) k) (aplus g
283 (ASort n7 n6) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))) (\lambda
284 (H11: (eq nat n6 n2)).(eq_ind nat n2 (\lambda (n7: nat).((eq A (aplus g
285 (ASort n3 n0) k) (aplus g (ASort n4 n7) k)) \to (leq g (ASort (S n3) n0)
286 (ASort (S n4) n2)))) (\lambda (H12: (eq A (aplus g (ASort n3 n0) k) (aplus g
287 (ASort n4 n2) k))).(let H13 \def (eq_ind_r A (aplus g (ASort n3 n0) k)
288 (\lambda (a: A).(eq A a (aplus g (ASort n4 n2) k))) H12 (aplus g (ASort (S
289 n3) n0) (S k)) (aplus_sort_S_S_simpl g n0 n3 k)) in (let H14 \def (eq_ind_r A
290 (aplus g (ASort n4 n2) k) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S
291 k)) a)) H13 (aplus g (ASort (S n4) n2) (S k)) (aplus_sort_S_S_simpl g n2 n4
292 k)) in (leq_sort g (S n3) (S n4) n0 n2 (S k) H14)))) n6 (sym_eq nat n6 n2
293 H11))) h2 (sym_eq nat h2 n4 H10))) H9))) n5 (sym_eq nat n5 n0 H7))) h1
294 (sym_eq nat h1 n3 H6))) H5)) H4 H2))) | (leq_head a0 a3 H2 a4 a5 H3)
295 \Rightarrow (\lambda (H4: (eq A (AHead a0 a4) (ASort n3 n0))).(\lambda (H5:
296 (eq A (AHead a3 a5) (ASort n4 n2))).((let H6 \def (eq_ind A (AHead a0 a4)
297 (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
298 \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n3 n0) H4) in
299 (False_ind ((eq A (AHead a3 a5) (ASort n4 n2)) \to ((leq g a0 a3) \to ((leq g
300 a4 a5) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))) H6)) H5 H2 H3)))])
301 in (H2 (refl_equal A (ASort n3 n0)) (refl_equal A (ASort n4 n2)))))))) n1 H0
302 IHn)))) n H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n n0))
303 (asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda (H0:
304 (((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0)
305 a0)))).(\lambda (H1: (leq g (asucc g (ASort n n0)) (asucc g (AHead a
306 a0)))).(nat_ind (\lambda (n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g
307 a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 n0))
308 (asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1
309 n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a a0))))))
310 (\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O
311 n0) a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq
312 g (ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g
313 (AHead a a0)))).(let H5 \def (match H4 in leq return (\lambda (a3:
314 A).(\lambda (a4: A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (ASort O (next g
315 n0))) \to ((eq A a4 (AHead a (asucc g a0))) \to (leq g (ASort O n0) (AHead a
316 a0))))))) with [(leq_sort h1 h2 n1 n2 k H5) \Rightarrow (\lambda (H6: (eq A
317 (ASort h1 n1) (ASort O (next g n0)))).(\lambda (H7: (eq A (ASort h2 n2)
318 (AHead a (asucc g a0)))).((let H8 \def (f_equal A nat (\lambda (e: A).(match
319 e in A return (\lambda (_: A).nat) with [(ASort _ n3) \Rightarrow n3 | (AHead
320 _ _) \Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H6) in ((let H9
321 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat)
322 with [(ASort n3 _) \Rightarrow n3 | (AHead _ _) \Rightarrow h1])) (ASort h1
323 n1) (ASort O (next g n0)) H6) in (eq_ind nat O (\lambda (n3: nat).((eq nat n1
324 (next g n0)) \to ((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A
325 (aplus g (ASort n3 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0)
326 (AHead a a0)))))) (\lambda (H10: (eq nat n1 (next g n0))).(eq_ind nat (next g
327 n0) (\lambda (n3: nat).((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq
328 A (aplus g (ASort O n3) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0)
329 (AHead a a0))))) (\lambda (H11: (eq A (ASort h2 n2) (AHead a (asucc g
330 a0)))).(let H12 \def (eq_ind A (ASort h2 n2) (\lambda (e: A).(match e in A
331 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
332 \Rightarrow False])) I (AHead a (asucc g a0)) H11) in (False_ind ((eq A
333 (aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n2) k)) \to (leq g
334 (ASort O n0) (AHead a a0))) H12))) n1 (sym_eq nat n1 (next g n0) H10))) h1
335 (sym_eq nat h1 O H9))) H8)) H7 H5))) | (leq_head a3 a4 H5 a5 a6 H6)
336 \Rightarrow (\lambda (H7: (eq A (AHead a3 a5) (ASort O (next g
337 n0)))).(\lambda (H8: (eq A (AHead a4 a6) (AHead a (asucc g a0)))).((let H9
338 \def (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda
339 (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
340 True])) I (ASort O (next g n0)) H7) in (False_ind ((eq A (AHead a4 a6) (AHead
341 a (asucc g a0))) \to ((leq g a3 a4) \to ((leq g a5 a6) \to (leq g (ASort O
342 n0) (AHead a a0))))) H9)) H8 H5 H6)))]) in (H5 (refl_equal A (ASort O (next g
343 n0))) (refl_equal A (AHead a (asucc g a0)))))))) (\lambda (n1: nat).(\lambda
344 (_: (((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0)
345 a))) \to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1
346 n0) a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to
347 (leq g (ASort n1 n0) (AHead a a0))))))).(\lambda (_: (((leq g (asucc g (ASort
348 (S n1) n0)) (asucc g a)) \to (leq g (ASort (S n1) n0) a)))).(\lambda (_:
349 (((leq g (asucc g (ASort (S n1) n0)) (asucc g a0)) \to (leq g (ASort (S n1)
350 n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort (S n1) n0)) (asucc g (AHead a
351 a0)))).(let H5 \def (match H4 in leq return (\lambda (a3: A).(\lambda (a4:
352 A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (ASort n1 n0)) \to ((eq A a4 (AHead
353 a (asucc g a0))) \to (leq g (ASort (S n1) n0) (AHead a a0))))))) with
354 [(leq_sort h1 h2 n2 n3 k H5) \Rightarrow (\lambda (H6: (eq A (ASort h1 n2)
355 (ASort n1 n0))).(\lambda (H7: (eq A (ASort h2 n3) (AHead a (asucc g
356 a0)))).((let H8 \def (f_equal A nat (\lambda (e: A).(match e in A return
357 (\lambda (_: A).nat) with [(ASort _ n4) \Rightarrow n4 | (AHead _ _)
358 \Rightarrow n2])) (ASort h1 n2) (ASort n1 n0) H6) in ((let H9 \def (f_equal A
359 nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort n4
360 _) \Rightarrow n4 | (AHead _ _) \Rightarrow h1])) (ASort h1 n2) (ASort n1 n0)
361 H6) in (eq_ind nat n1 (\lambda (n4: nat).((eq nat n2 n0) \to ((eq A (ASort h2
362 n3) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n4 n2) k) (aplus g
363 (ASort h2 n3) k)) \to (leq g (ASort (S n1) n0) (AHead a a0)))))) (\lambda
364 (H10: (eq nat n2 n0)).(eq_ind nat n0 (\lambda (n4: nat).((eq A (ASort h2 n3)
365 (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n1 n4) k) (aplus g (ASort
366 h2 n3) k)) \to (leq g (ASort (S n1) n0) (AHead a a0))))) (\lambda (H11: (eq A
367 (ASort h2 n3) (AHead a (asucc g a0)))).(let H12 \def (eq_ind A (ASort h2 n3)
368 (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
369 \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0))
370 H11) in (False_ind ((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n3)
371 k)) \to (leq g (ASort (S n1) n0) (AHead a a0))) H12))) n2 (sym_eq nat n2 n0
372 H10))) h1 (sym_eq nat h1 n1 H9))) H8)) H7 H5))) | (leq_head a3 a4 H5 a5 a6
373 H6) \Rightarrow (\lambda (H7: (eq A (AHead a3 a5) (ASort n1 n0))).(\lambda
374 (H8: (eq A (AHead a4 a6) (AHead a (asucc g a0)))).((let H9 \def (eq_ind A
375 (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with
376 [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1
377 n0) H7) in (False_ind ((eq A (AHead a4 a6) (AHead a (asucc g a0))) \to ((leq
378 g a3 a4) \to ((leq g a5 a6) \to (leq g (ASort (S n1) n0) (AHead a a0)))))
379 H9)) H8 H5 H6)))]) in (H5 (refl_equal A (ASort n1 n0)) (refl_equal A (AHead a
380 (asucc g a0)))))))))) n H H0 H1)))))) a2)))) (\lambda (a: A).(\lambda (_:
381 ((\forall (a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq g a
382 a2))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0)
383 (asucc g a2)) \to (leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3:
384 A).((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))
385 (\lambda (n: nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a
386 a0)) (asucc g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g
387 (AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1
388 n0)))) (\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O
389 n0)))).(let H3 \def (match H2 in leq return (\lambda (a3: A).(\lambda (a4:
390 A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (AHead a (asucc g a0))) \to ((eq A
391 a4 (ASort O (next g n0))) \to (leq g (AHead a a0) (ASort O n0))))))) with
392 [(leq_sort h1 h2 n1 n2 k H3) \Rightarrow (\lambda (H4: (eq A (ASort h1 n1)
393 (AHead a (asucc g a0)))).(\lambda (H5: (eq A (ASort h2 n2) (ASort O (next g
394 n0)))).((let H6 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A
395 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
396 \Rightarrow False])) I (AHead a (asucc g a0)) H4) in (False_ind ((eq A (ASort
397 h2 n2) (ASort O (next g n0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g
398 (ASort h2 n2) k)) \to (leq g (AHead a a0) (ASort O n0)))) H6)) H5 H3))) |
399 (leq_head a3 a4 H3 a5 a6 H4) \Rightarrow (\lambda (H5: (eq A (AHead a3 a5)
400 (AHead a (asucc g a0)))).(\lambda (H6: (eq A (AHead a4 a6) (ASort O (next g
401 n0)))).((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return
402 (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a7)
403 \Rightarrow a7])) (AHead a3 a5) (AHead a (asucc g a0)) H5) in ((let H8 \def
404 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
405 [(ASort _ _) \Rightarrow a3 | (AHead a7 _) \Rightarrow a7])) (AHead a3 a5)
406 (AHead a (asucc g a0)) H5) in (eq_ind A a (\lambda (a7: A).((eq A a5 (asucc g
407 a0)) \to ((eq A (AHead a4 a6) (ASort O (next g n0))) \to ((leq g a7 a4) \to
408 ((leq g a5 a6) \to (leq g (AHead a a0) (ASort O n0))))))) (\lambda (H9: (eq A
409 a5 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a7: A).((eq A (AHead a4
410 a6) (ASort O (next g n0))) \to ((leq g a a4) \to ((leq g a7 a6) \to (leq g
411 (AHead a a0) (ASort O n0)))))) (\lambda (H10: (eq A (AHead a4 a6) (ASort O
412 (next g n0)))).(let H11 \def (eq_ind A (AHead a4 a6) (\lambda (e: A).(match e
413 in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
414 (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H10) in (False_ind
415 ((leq g a a4) \to ((leq g (asucc g a0) a6) \to (leq g (AHead a a0) (ASort O
416 n0)))) H11))) a5 (sym_eq A a5 (asucc g a0) H9))) a3 (sym_eq A a3 a H8))) H7))
417 H6 H3 H4)))]) in (H3 (refl_equal A (AHead a (asucc g a0))) (refl_equal A
418 (ASort O (next g n0)))))) (\lambda (n1: nat).(\lambda (_: (((leq g (asucc g
419 (AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1
420 n0))))).(\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort (S n1)
421 n0)))).(let H3 \def (match H2 in leq return (\lambda (a3: A).(\lambda (a4:
422 A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (AHead a (asucc g a0))) \to ((eq A
423 a4 (ASort n1 n0)) \to (leq g (AHead a a0) (ASort (S n1) n0))))))) with
424 [(leq_sort h1 h2 n2 n3 k H3) \Rightarrow (\lambda (H4: (eq A (ASort h1 n2)
425 (AHead a (asucc g a0)))).(\lambda (H5: (eq A (ASort h2 n3) (ASort n1
426 n0))).((let H6 \def (eq_ind A (ASort h1 n2) (\lambda (e: A).(match e in A
427 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
428 \Rightarrow False])) I (AHead a (asucc g a0)) H4) in (False_ind ((eq A (ASort
429 h2 n3) (ASort n1 n0)) \to ((eq A (aplus g (ASort h1 n2) k) (aplus g (ASort h2
430 n3) k)) \to (leq g (AHead a a0) (ASort (S n1) n0)))) H6)) H5 H3))) |
431 (leq_head a3 a4 H3 a5 a6 H4) \Rightarrow (\lambda (H5: (eq A (AHead a3 a5)
432 (AHead a (asucc g a0)))).(\lambda (H6: (eq A (AHead a4 a6) (ASort n1
433 n0))).((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return
434 (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a7)
435 \Rightarrow a7])) (AHead a3 a5) (AHead a (asucc g a0)) H5) in ((let H8 \def
436 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
437 [(ASort _ _) \Rightarrow a3 | (AHead a7 _) \Rightarrow a7])) (AHead a3 a5)
438 (AHead a (asucc g a0)) H5) in (eq_ind A a (\lambda (a7: A).((eq A a5 (asucc g
439 a0)) \to ((eq A (AHead a4 a6) (ASort n1 n0)) \to ((leq g a7 a4) \to ((leq g
440 a5 a6) \to (leq g (AHead a a0) (ASort (S n1) n0))))))) (\lambda (H9: (eq A a5
441 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a7: A).((eq A (AHead a4 a6)
442 (ASort n1 n0)) \to ((leq g a a4) \to ((leq g a7 a6) \to (leq g (AHead a a0)
443 (ASort (S n1) n0)))))) (\lambda (H10: (eq A (AHead a4 a6) (ASort n1
444 n0))).(let H11 \def (eq_ind A (AHead a4 a6) (\lambda (e: A).(match e in A
445 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
446 _) \Rightarrow True])) I (ASort n1 n0) H10) in (False_ind ((leq g a a4) \to
447 ((leq g (asucc g a0) a6) \to (leq g (AHead a a0) (ASort (S n1) n0)))) H11)))
448 a5 (sym_eq A a5 (asucc g a0) H9))) a3 (sym_eq A a3 a H8))) H7)) H6 H3 H4)))])
449 in (H3 (refl_equal A (AHead a (asucc g a0))) (refl_equal A (ASort n1
450 n0))))))) n H1)))) (\lambda (a3: A).(\lambda (_: (((leq g (asucc g (AHead a
451 a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda (a4: A).(\lambda
452 (_: (((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g (AHead a a0)
453 a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g (AHead a3
454 a4)))).(let H4 \def (match H3 in leq return (\lambda (a5: A).(\lambda (a6:
455 A).(\lambda (_: (leq ? a5 a6)).((eq A a5 (AHead a (asucc g a0))) \to ((eq A
456 a6 (AHead a3 (asucc g a4))) \to (leq g (AHead a a0) (AHead a3 a4))))))) with
457 [(leq_sort h1 h2 n1 n2 k H4) \Rightarrow (\lambda (H5: (eq A (ASort h1 n1)
458 (AHead a (asucc g a0)))).(\lambda (H6: (eq A (ASort h2 n2) (AHead a3 (asucc g
459 a4)))).((let H7 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A
460 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
461 \Rightarrow False])) I (AHead a (asucc g a0)) H5) in (False_ind ((eq A (ASort
462 h2 n2) (AHead a3 (asucc g a4))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g
463 (ASort h2 n2) k)) \to (leq g (AHead a a0) (AHead a3 a4)))) H7)) H6 H4))) |
464 (leq_head a5 a6 H4 a7 a8 H5) \Rightarrow (\lambda (H6: (eq A (AHead a5 a7)
465 (AHead a (asucc g a0)))).(\lambda (H7: (eq A (AHead a6 a8) (AHead a3 (asucc g
466 a4)))).((let H8 \def (f_equal A A (\lambda (e: A).(match e in A return
467 (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 | (AHead _ a9)
468 \Rightarrow a9])) (AHead a5 a7) (AHead a (asucc g a0)) H6) in ((let H9 \def
469 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
470 [(ASort _ _) \Rightarrow a5 | (AHead a9 _) \Rightarrow a9])) (AHead a5 a7)
471 (AHead a (asucc g a0)) H6) in (eq_ind A a (\lambda (a9: A).((eq A a7 (asucc g
472 a0)) \to ((eq A (AHead a6 a8) (AHead a3 (asucc g a4))) \to ((leq g a9 a6) \to
473 ((leq g a7 a8) \to (leq g (AHead a a0) (AHead a3 a4))))))) (\lambda (H10: (eq
474 A a7 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a9: A).((eq A (AHead a6
475 a8) (AHead a3 (asucc g a4))) \to ((leq g a a6) \to ((leq g a9 a8) \to (leq g
476 (AHead a a0) (AHead a3 a4)))))) (\lambda (H11: (eq A (AHead a6 a8) (AHead a3
477 (asucc g a4)))).(let H12 \def (f_equal A A (\lambda (e: A).(match e in A
478 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a8 | (AHead _ a9)
479 \Rightarrow a9])) (AHead a6 a8) (AHead a3 (asucc g a4)) H11) in ((let H13
480 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
481 with [(ASort _ _) \Rightarrow a6 | (AHead a9 _) \Rightarrow a9])) (AHead a6
482 a8) (AHead a3 (asucc g a4)) H11) in (eq_ind A a3 (\lambda (a9: A).((eq A a8
483 (asucc g a4)) \to ((leq g a a9) \to ((leq g (asucc g a0) a8) \to (leq g
484 (AHead a a0) (AHead a3 a4)))))) (\lambda (H14: (eq A a8 (asucc g
485 a4))).(eq_ind A (asucc g a4) (\lambda (a9: A).((leq g a a3) \to ((leq g
486 (asucc g a0) a9) \to (leq g (AHead a a0) (AHead a3 a4))))) (\lambda (H15:
487 (leq g a a3)).(\lambda (H16: (leq g (asucc g a0) (asucc g a4))).(leq_head g a
488 a3 H15 a0 a4 (H0 a4 H16)))) a8 (sym_eq A a8 (asucc g a4) H14))) a6 (sym_eq A
489 a6 a3 H13))) H12))) a7 (sym_eq A a7 (asucc g a0) H10))) a5 (sym_eq A a5 a
490 H9))) H8)) H7 H4 H5)))]) in (H4 (refl_equal A (AHead a (asucc g a0)))
491 (refl_equal A (AHead a3 (asucc g a4)))))))))) a2)))))) a1)).
494 \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g
497 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(ex A (\lambda (a1:
498 A).(leq g a0 (asucc g a1))))) (\lambda (n: nat).(\lambda (n0: nat).(ex_intro
499 A (\lambda (a0: A).(leq g (ASort n n0) (asucc g a0))) (ASort (S n) n0)
500 (leq_refl g (ASort n n0))))) (\lambda (a0: A).(\lambda (_: (ex A (\lambda
501 (a1: A).(leq g a0 (asucc g a1))))).(\lambda (a1: A).(\lambda (H0: (ex A
502 (\lambda (a2: A).(leq g a1 (asucc g a2))))).(let H1 \def H0 in (ex_ind A
503 (\lambda (a2: A).(leq g a1 (asucc g a2))) (ex A (\lambda (a2: A).(leq g
504 (AHead a0 a1) (asucc g a2)))) (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc
505 g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2)))
506 (AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1))))))
509 theorem leq_ahead_asucc_false:
510 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2)
511 (asucc g a1)) \to (\forall (P: Prop).P))))
513 \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
514 A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda
515 (n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead
516 (ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
517 \Rightarrow (ASort h n0)]))).(\lambda (P: Prop).(nat_ind (\lambda (n1:
518 nat).((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O
519 (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P)) (\lambda (H0: (leq g
520 (AHead (ASort O n0) a2) (ASort O (next g n0)))).(let H1 \def (match H0 in leq
521 return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a
522 (AHead (ASort O n0) a2)) \to ((eq A a0 (ASort O (next g n0))) \to P))))) with
523 [(leq_sort h1 h2 n1 n2 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n1)
524 (AHead (ASort O n0) a2))).(\lambda (H3: (eq A (ASort h2 n2) (ASort O (next g
525 n0)))).((let H4 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A
526 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
527 \Rightarrow False])) I (AHead (ASort O n0) a2) H2) in (False_ind ((eq A
528 (ASort h2 n2) (ASort O (next g n0))) \to ((eq A (aplus g (ASort h1 n1) k)
529 (aplus g (ASort h2 n2) k)) \to P)) H4)) H3 H1))) | (leq_head a0 a3 H1 a4 a5
530 H2) \Rightarrow (\lambda (H3: (eq A (AHead a0 a4) (AHead (ASort O n0)
531 a2))).(\lambda (H4: (eq A (AHead a3 a5) (ASort O (next g n0)))).((let H5 \def
532 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
533 [(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a0 a4)
534 (AHead (ASort O n0) a2) H3) in ((let H6 \def (f_equal A A (\lambda (e:
535 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 |
536 (AHead a _) \Rightarrow a])) (AHead a0 a4) (AHead (ASort O n0) a2) H3) in
537 (eq_ind A (ASort O n0) (\lambda (a: A).((eq A a4 a2) \to ((eq A (AHead a3 a5)
538 (ASort O (next g n0))) \to ((leq g a a3) \to ((leq g a4 a5) \to P)))))
539 (\lambda (H7: (eq A a4 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a3 a5)
540 (ASort O (next g n0))) \to ((leq g (ASort O n0) a3) \to ((leq g a a5) \to
541 P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort O (next g n0)))).(let H9 \def
542 (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda (_:
543 A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
544 True])) I (ASort O (next g n0)) H8) in (False_ind ((leq g (ASort O n0) a3)
545 \to ((leq g a2 a5) \to P)) H9))) a4 (sym_eq A a4 a2 H7))) a0 (sym_eq A a0
546 (ASort O n0) H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead (ASort O
547 n0) a2)) (refl_equal A (ASort O (next g n0)))))) (\lambda (n1: nat).(\lambda
548 (_: (((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O
549 (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P))).(\lambda (H0: (leq
550 g (AHead (ASort (S n1) n0) a2) (ASort n1 n0))).(let H1 \def (match H0 in leq
551 return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a
552 (AHead (ASort (S n1) n0) a2)) \to ((eq A a0 (ASort n1 n0)) \to P))))) with
553 [(leq_sort h1 h2 n2 n3 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n2)
554 (AHead (ASort (S n1) n0) a2))).(\lambda (H3: (eq A (ASort h2 n3) (ASort n1
555 n0))).((let H4 \def (eq_ind A (ASort h1 n2) (\lambda (e: A).(match e in A
556 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
557 \Rightarrow False])) I (AHead (ASort (S n1) n0) a2) H2) in (False_ind ((eq A
558 (ASort h2 n3) (ASort n1 n0)) \to ((eq A (aplus g (ASort h1 n2) k) (aplus g
559 (ASort h2 n3) k)) \to P)) H4)) H3 H1))) | (leq_head a0 a3 H1 a4 a5 H2)
560 \Rightarrow (\lambda (H3: (eq A (AHead a0 a4) (AHead (ASort (S n1) n0)
561 a2))).(\lambda (H4: (eq A (AHead a3 a5) (ASort n1 n0))).((let H5 \def
562 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
563 [(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a0 a4)
564 (AHead (ASort (S n1) n0) a2) H3) in ((let H6 \def (f_equal A A (\lambda (e:
565 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 |
566 (AHead a _) \Rightarrow a])) (AHead a0 a4) (AHead (ASort (S n1) n0) a2) H3)
567 in (eq_ind A (ASort (S n1) n0) (\lambda (a: A).((eq A a4 a2) \to ((eq A
568 (AHead a3 a5) (ASort n1 n0)) \to ((leq g a a3) \to ((leq g a4 a5) \to P)))))
569 (\lambda (H7: (eq A a4 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a3 a5)
570 (ASort n1 n0)) \to ((leq g (ASort (S n1) n0) a3) \to ((leq g a a5) \to P))))
571 (\lambda (H8: (eq A (AHead a3 a5) (ASort n1 n0))).(let H9 \def (eq_ind A
572 (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with
573 [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1
574 n0) H8) in (False_ind ((leq g (ASort (S n1) n0) a3) \to ((leq g a2 a5) \to
575 P)) H9))) a4 (sym_eq A a4 a2 H7))) a0 (sym_eq A a0 (ASort (S n1) n0) H6)))
576 H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead (ASort (S n1) n0) a2))
577 (refl_equal A (ASort n1 n0))))))) n H)))))) (\lambda (a: A).(\lambda (_:
578 ((\forall (a2: A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P:
579 Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead
580 a0 a2) (asucc g a0)) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda
581 (H1: (leq g (AHead (AHead a a0) a2) (AHead a (asucc g a0)))).(\lambda (P:
582 Prop).(let H2 \def (match H1 in leq return (\lambda (a3: A).(\lambda (a4:
583 A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (AHead (AHead a a0) a2)) \to ((eq A
584 a4 (AHead a (asucc g a0))) \to P))))) with [(leq_sort h1 h2 n1 n2 k H2)
585 \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead (AHead a a0)
586 a2))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a (asucc g a0)))).((let H5
587 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A return (\lambda
588 (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
589 False])) I (AHead (AHead a a0) a2) H3) in (False_ind ((eq A (ASort h2 n2)
590 (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort
591 h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a3 a4 H2 a5 a6 H3) \Rightarrow
592 (\lambda (H4: (eq A (AHead a3 a5) (AHead (AHead a a0) a2))).(\lambda (H5: (eq
593 A (AHead a4 a6) (AHead a (asucc g a0)))).((let H6 \def (f_equal A A (\lambda
594 (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow
595 a5 | (AHead _ a7) \Rightarrow a7])) (AHead a3 a5) (AHead (AHead a a0) a2) H4)
596 in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda
597 (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a7 _) \Rightarrow a7]))
598 (AHead a3 a5) (AHead (AHead a a0) a2) H4) in (eq_ind A (AHead a a0) (\lambda
599 (a7: A).((eq A a5 a2) \to ((eq A (AHead a4 a6) (AHead a (asucc g a0))) \to
600 ((leq g a7 a4) \to ((leq g a5 a6) \to P))))) (\lambda (H8: (eq A a5
601 a2)).(eq_ind A a2 (\lambda (a7: A).((eq A (AHead a4 a6) (AHead a (asucc g
602 a0))) \to ((leq g (AHead a a0) a4) \to ((leq g a7 a6) \to P)))) (\lambda (H9:
603 (eq A (AHead a4 a6) (AHead a (asucc g a0)))).(let H10 \def (f_equal A A
604 (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
605 \Rightarrow a6 | (AHead _ a7) \Rightarrow a7])) (AHead a4 a6) (AHead a (asucc
606 g a0)) H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e in A
607 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead a7 _)
608 \Rightarrow a7])) (AHead a4 a6) (AHead a (asucc g a0)) H9) in (eq_ind A a
609 (\lambda (a7: A).((eq A a6 (asucc g a0)) \to ((leq g (AHead a a0) a7) \to
610 ((leq g a2 a6) \to P)))) (\lambda (H12: (eq A a6 (asucc g a0))).(eq_ind A
611 (asucc g a0) (\lambda (a7: A).((leq g (AHead a a0) a) \to ((leq g a2 a7) \to
612 P))) (\lambda (H13: (leq g (AHead a a0) a)).(\lambda (_: (leq g a2 (asucc g
613 a0))).(leq_ahead_false_1 g a a0 H13 P))) a6 (sym_eq A a6 (asucc g a0) H12)))
614 a4 (sym_eq A a4 a H11))) H10))) a5 (sym_eq A a5 a2 H8))) a3 (sym_eq A a3
615 (AHead a a0) H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead (AHead a
616 a0) a2)) (refl_equal A (AHead a (asucc g a0)))))))))))) a1)).
618 theorem leq_asucc_false:
619 \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P:
622 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0)
623 a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda
624 (H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
625 \Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).(nat_ind
626 (\lambda (n1: nat).((leq g (match n1 with [O \Rightarrow (ASort O (next g
627 n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P)) (\lambda (H0:
628 (leq g (ASort O (next g n0)) (ASort O n0))).(let H1 \def (match H0 in leq
629 return (\lambda (a0: A).(\lambda (a1: A).(\lambda (_: (leq ? a0 a1)).((eq A
630 a0 (ASort O (next g n0))) \to ((eq A a1 (ASort O n0)) \to P))))) with
631 [(leq_sort h1 h2 n1 n2 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n1)
632 (ASort O (next g n0)))).(\lambda (H3: (eq A (ASort h2 n2) (ASort O
633 n0))).((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return
634 (\lambda (_: A).nat) with [(ASort _ n3) \Rightarrow n3 | (AHead _ _)
635 \Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H2) in ((let H5 \def
636 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
637 [(ASort n3 _) \Rightarrow n3 | (AHead _ _) \Rightarrow h1])) (ASort h1 n1)
638 (ASort O (next g n0)) H2) in (eq_ind nat O (\lambda (n3: nat).((eq nat n1
639 (next g n0)) \to ((eq A (ASort h2 n2) (ASort O n0)) \to ((eq A (aplus g
640 (ASort n3 n1) k) (aplus g (ASort h2 n2) k)) \to P)))) (\lambda (H6: (eq nat
641 n1 (next g n0))).(eq_ind nat (next g n0) (\lambda (n3: nat).((eq A (ASort h2
642 n2) (ASort O n0)) \to ((eq A (aplus g (ASort O n3) k) (aplus g (ASort h2 n2)
643 k)) \to P))) (\lambda (H7: (eq A (ASort h2 n2) (ASort O n0))).(let H8 \def
644 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
645 [(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow n2])) (ASort h2 n2)
646 (ASort O n0) H7) in ((let H9 \def (f_equal A nat (\lambda (e: A).(match e in
647 A return (\lambda (_: A).nat) with [(ASort n3 _) \Rightarrow n3 | (AHead _ _)
648 \Rightarrow h2])) (ASort h2 n2) (ASort O n0) H7) in (eq_ind nat O (\lambda
649 (n3: nat).((eq nat n2 n0) \to ((eq A (aplus g (ASort O (next g n0)) k) (aplus
650 g (ASort n3 n2) k)) \to P))) (\lambda (H10: (eq nat n2 n0)).(eq_ind nat n0
651 (\lambda (n3: nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O
652 n3) k)) \to P)) (\lambda (H11: (eq A (aplus g (ASort O (next g n0)) k) (aplus
653 g (ASort O n0) k))).(let H12 \def (eq_ind_r A (aplus g (ASort O (next g n0))
654 k) (\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) k))) H11 (aplus g (ASort O
655 n0) (S k)) (aplus_sort_O_S_simpl g n0 k)) in (let H_y \def (aplus_inj g (S k)
656 k (ASort O n0) H12) in (le_Sx_x k (eq_ind_r nat k (\lambda (n3: nat).(le n3
657 k)) (le_n k) (S k) H_y) P)))) n2 (sym_eq nat n2 n0 H10))) h2 (sym_eq nat h2 O
658 H9))) H8))) n1 (sym_eq nat n1 (next g n0) H6))) h1 (sym_eq nat h1 O H5)))
659 H4)) H3 H1))) | (leq_head a1 a2 H1 a3 a4 H2) \Rightarrow (\lambda (H3: (eq A
660 (AHead a1 a3) (ASort O (next g n0)))).(\lambda (H4: (eq A (AHead a2 a4)
661 (ASort O n0))).((let H5 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e
662 in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
663 (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H3) in (False_ind
664 ((eq A (AHead a2 a4) (ASort O n0)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to
665 P))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (ASort O (next g n0)))
666 (refl_equal A (ASort O n0))))) (\lambda (n1: nat).(\lambda (_: (((leq g
667 (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow
668 (ASort h n0)]) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (ASort n1 n0)
669 (ASort (S n1) n0))).(let H1 \def (match H0 in leq return (\lambda (a0:
670 A).(\lambda (a1: A).(\lambda (_: (leq ? a0 a1)).((eq A a0 (ASort n1 n0)) \to
671 ((eq A a1 (ASort (S n1) n0)) \to P))))) with [(leq_sort h1 h2 n2 n3 k H1)
672 \Rightarrow (\lambda (H2: (eq A (ASort h1 n2) (ASort n1 n0))).(\lambda (H3:
673 (eq A (ASort h2 n3) (ASort (S n1) n0))).((let H4 \def (f_equal A nat (\lambda
674 (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n4)
675 \Rightarrow n4 | (AHead _ _) \Rightarrow n2])) (ASort h1 n2) (ASort n1 n0)
676 H2) in ((let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return
677 (\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _)
678 \Rightarrow h1])) (ASort h1 n2) (ASort n1 n0) H2) in (eq_ind nat n1 (\lambda
679 (n4: nat).((eq nat n2 n0) \to ((eq A (ASort h2 n3) (ASort (S n1) n0)) \to
680 ((eq A (aplus g (ASort n4 n2) k) (aplus g (ASort h2 n3) k)) \to P))))
681 (\lambda (H6: (eq nat n2 n0)).(eq_ind nat n0 (\lambda (n4: nat).((eq A (ASort
682 h2 n3) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort n1 n4) k) (aplus g
683 (ASort h2 n3) k)) \to P))) (\lambda (H7: (eq A (ASort h2 n3) (ASort (S n1)
684 n0))).(let H8 \def (f_equal A nat (\lambda (e: A).(match e in A return
685 (\lambda (_: A).nat) with [(ASort _ n4) \Rightarrow n4 | (AHead _ _)
686 \Rightarrow n3])) (ASort h2 n3) (ASort (S n1) n0) H7) in ((let H9 \def
687 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
688 [(ASort n4 _) \Rightarrow n4 | (AHead _ _) \Rightarrow h2])) (ASort h2 n3)
689 (ASort (S n1) n0) H7) in (eq_ind nat (S n1) (\lambda (n4: nat).((eq nat n3
690 n0) \to ((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort n4 n3) k)) \to P)))
691 (\lambda (H10: (eq nat n3 n0)).(eq_ind nat n0 (\lambda (n4: nat).((eq A
692 (aplus g (ASort n1 n0) k) (aplus g (ASort (S n1) n4) k)) \to P)) (\lambda
693 (H11: (eq A (aplus g (ASort n1 n0) k) (aplus g (ASort (S n1) n0) k))).(let
694 H12 \def (eq_ind_r A (aplus g (ASort n1 n0) k) (\lambda (a0: A).(eq A a0
695 (aplus g (ASort (S n1) n0) k))) H11 (aplus g (ASort (S n1) n0) (S k))
696 (aplus_sort_S_S_simpl g n0 n1 k)) in (let H_y \def (aplus_inj g (S k) k
697 (ASort (S n1) n0) H12) in (le_Sx_x k (eq_ind_r nat k (\lambda (n4: nat).(le
698 n4 k)) (le_n k) (S k) H_y) P)))) n3 (sym_eq nat n3 n0 H10))) h2 (sym_eq nat
699 h2 (S n1) H9))) H8))) n2 (sym_eq nat n2 n0 H6))) h1 (sym_eq nat h1 n1 H5)))
700 H4)) H3 H1))) | (leq_head a1 a2 H1 a3 a4 H2) \Rightarrow (\lambda (H3: (eq A
701 (AHead a1 a3) (ASort n1 n0))).(\lambda (H4: (eq A (AHead a2 a4) (ASort (S n1)
702 n0))).((let H5 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e in A
703 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
704 _) \Rightarrow True])) I (ASort n1 n0) H3) in (False_ind ((eq A (AHead a2 a4)
705 (ASort (S n1) n0)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to P))) H5)) H4 H1
706 H2)))]) in (H1 (refl_equal A (ASort n1 n0)) (refl_equal A (ASort (S n1)
707 n0))))))) n H))))) (\lambda (a0: A).(\lambda (_: (((leq g (asucc g a0) a0)
708 \to (\forall (P: Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g
709 a1) a1) \to (\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g
710 a1)) (AHead a0 a1))).(\lambda (P: Prop).(let H2 \def (match H1 in leq return
711 (\lambda (a2: A).(\lambda (a3: A).(\lambda (_: (leq ? a2 a3)).((eq A a2
712 (AHead a0 (asucc g a1))) \to ((eq A a3 (AHead a0 a1)) \to P))))) with
713 [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1)
714 (AHead a0 (asucc g a1)))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a0
715 a1))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A
716 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
717 \Rightarrow False])) I (AHead a0 (asucc g a1)) H3) in (False_ind ((eq A
718 (ASort h2 n2) (AHead a0 a1)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g
719 (ASort h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a2 a3 H2 a4 a5 H3)
720 \Rightarrow (\lambda (H4: (eq A (AHead a2 a4) (AHead a0 (asucc g
721 a1)))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a0 a1))).((let H6 \def
722 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
723 [(ASort _ _) \Rightarrow a4 | (AHead _ a6) \Rightarrow a6])) (AHead a2 a4)
724 (AHead a0 (asucc g a1)) H4) in ((let H7 \def (f_equal A A (\lambda (e:
725 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a2 |
726 (AHead a6 _) \Rightarrow a6])) (AHead a2 a4) (AHead a0 (asucc g a1)) H4) in
727 (eq_ind A a0 (\lambda (a6: A).((eq A a4 (asucc g a1)) \to ((eq A (AHead a3
728 a5) (AHead a0 a1)) \to ((leq g a6 a3) \to ((leq g a4 a5) \to P))))) (\lambda
729 (H8: (eq A a4 (asucc g a1))).(eq_ind A (asucc g a1) (\lambda (a6: A).((eq A
730 (AHead a3 a5) (AHead a0 a1)) \to ((leq g a0 a3) \to ((leq g a6 a5) \to P))))
731 (\lambda (H9: (eq A (AHead a3 a5) (AHead a0 a1))).(let H10 \def (f_equal A A
732 (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
733 \Rightarrow a5 | (AHead _ a6) \Rightarrow a6])) (AHead a3 a5) (AHead a0 a1)
734 H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e in A return
735 (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a6 _)
736 \Rightarrow a6])) (AHead a3 a5) (AHead a0 a1) H9) in (eq_ind A a0 (\lambda
737 (a6: A).((eq A a5 a1) \to ((leq g a0 a6) \to ((leq g (asucc g a1) a5) \to
738 P)))) (\lambda (H12: (eq A a5 a1)).(eq_ind A a1 (\lambda (a6: A).((leq g a0
739 a0) \to ((leq g (asucc g a1) a6) \to P))) (\lambda (_: (leq g a0
740 a0)).(\lambda (H14: (leq g (asucc g a1) a1)).(H0 H14 P))) a5 (sym_eq A a5 a1
741 H12))) a3 (sym_eq A a3 a0 H11))) H10))) a4 (sym_eq A a4 (asucc g a1) H8))) a2
742 (sym_eq A a2 a0 H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead a0
743 (asucc g a1))) (refl_equal A (AHead a0 a1)))))))))) a)).
projects / helm.git / blob