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 set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/leq/asucc".
19 include "leq/props.ma".
21 include "aplus/props.ma".
24 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g
25 (asucc g a1) (asucc g a2)))))
27 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
28 a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g (asucc g a) (asucc g
29 a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2:
30 nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g
31 (ASort h2 n2) k))).(nat_ind (\lambda (n: nat).((eq A (aplus g (ASort n n1) k)
32 (aplus g (ASort h2 n2) k)) \to (leq g (match n with [O \Rightarrow (ASort O
33 (next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow
34 (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) (\lambda (H1: (eq
35 A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda (n:
36 nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort n n2) k)) \to (leq g
37 (ASort O (next g n1)) (match n with [O \Rightarrow (ASort O (next g n2)) | (S
38 h) \Rightarrow (ASort h n2)])))) (\lambda (H2: (eq A (aplus g (ASort O n1) k)
39 (aplus g (ASort O n2) k))).(leq_sort g O O (next g n1) (next g n2) k (eq_ind
40 A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O
41 (next g n2)) k))) (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq
42 A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort O n2) k)
43 (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k))))
44 (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort O n1) k)
45 H2) (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)) (aplus g
46 (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))) (\lambda (h3:
47 nat).(\lambda (_: (((eq A (aplus g (ASort O n1) k) (aplus g (ASort h3 n2) k))
48 \to (leq g (ASort O (next g n1)) (match h3 with [O \Rightarrow (ASort O (next
49 g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H2: (eq A (aplus g
50 (ASort O n1) k) (aplus g (ASort (S h3) n2) k))).(leq_sort g O h3 (next g n1)
51 n2 k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g
52 (ASort h3 n2) k))) (eq_ind A (aplus g (ASort (S h3) n2) (S k)) (\lambda (a:
53 A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort (S h3)
54 n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort (S h3) n2)
55 k)))) (refl_equal A (asucc g (aplus g (ASort (S h3) n2) k))) (aplus g (ASort
56 O n1) k) H2) (aplus g (ASort h3 n2) k) (aplus_sort_S_S_simpl g n2 h3 k))
57 (aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))))) h2 H1))
58 (\lambda (h3: nat).(\lambda (IHh1: (((eq A (aplus g (ASort h3 n1) k) (aplus g
59 (ASort h2 n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next g
60 n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow (ASort
61 O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H1: (eq A
62 (aplus g (ASort (S h3) n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda
63 (n: nat).((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort n n2) k)) \to
64 ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort n n2) k)) \to (leq g
65 (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow
66 (ASort h n1)]) (match n with [O \Rightarrow (ASort O (next g n2)) | (S h)
67 \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match n with [O
68 \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))))
69 (\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort O n2)
70 k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort O n2) k))
71 \to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h)
72 \Rightarrow (ASort h n1)]) (ASort O (next g n2)))))).(leq_sort g h3 O n1
73 (next g n2) k (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A
74 (aplus g (ASort h3 n1) k) a)) (eq_ind A (aplus g (ASort (S h3) n1) (S k))
75 (\lambda (a: A).(eq A a (aplus g (ASort O n2) (S k)))) (eq_ind_r A (aplus g
76 (ASort O n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O
77 n2) k)))) (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort
78 (S h3) n1) k) H2) (aplus g (ASort h3 n1) k) (aplus_sort_S_S_simpl g n1 h3 k))
79 (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k))))) (\lambda
80 (h4: nat).(\lambda (_: (((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort
81 h4 n2) k)) \to ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort h4 n2) k))
82 \to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h)
83 \Rightarrow (ASort h n1)]) (match h4 with [O \Rightarrow (ASort O (next g
84 n2)) | (S h) \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match h4
85 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h
86 n2)])))))).(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort
87 (S h4) n2) k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g
88 (ASort (S h4) n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next
89 g n1)) | (S h) \Rightarrow (ASort h n1)]) (ASort h4 n2))))).(leq_sort g h3 h4
90 n1 n2 k (eq_ind A (aplus g (ASort (S h3) n1) (S k)) (\lambda (a: A).(eq A a
91 (aplus g (ASort h4 n2) k))) (eq_ind A (aplus g (ASort (S h4) n2) (S k))
92 (\lambda (a: A).(eq A (aplus g (ASort (S h3) n1) (S k)) a)) (eq_ind_r A
93 (aplus g (ASort (S h4) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g
94 (aplus g (ASort (S h4) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S
95 h4) n2) k))) (aplus g (ASort (S h3) n1) k) H2) (aplus g (ASort h4 n2) k)
96 (aplus_sort_S_S_simpl g n2 h4 k)) (aplus g (ASort h3 n1) k)
97 (aplus_sort_S_S_simpl g n1 h3 k))))))) h2 H1 IHh1)))) h1 H0))))))) (\lambda
98 (a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3 a4)).(\lambda (_: (leq g
99 (asucc g a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_:
100 (leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g
101 a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))).
104 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc
105 g a2)) \to (leq g a1 a2))))
107 \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
108 A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n:
109 nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g
110 (asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda
111 (n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0))
112 (asucc g (ASort n1 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort
113 n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))
114 (\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1
115 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g
116 (ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2)))) (\lambda (H1: (leq g
117 (asucc g (ASort O n0)) (asucc g (ASort O n2)))).(let H2 \def (match H1 in leq
118 return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a
119 (ASort O (next g n0))) \to ((eq A a0 (ASort O (next g n2))) \to (leq g (ASort
120 O n0) (ASort O n2))))))) with [(leq_sort h1 h2 n3 n4 k H2) \Rightarrow
121 (\lambda (H3: (eq A (ASort h1 n3) (ASort O (next g n0)))).(\lambda (H4: (eq A
122 (ASort h2 n4) (ASort O (next g n2)))).((let H5 \def (f_equal A nat (\lambda
123 (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n5)
124 \Rightarrow n5 | (AHead _ _) \Rightarrow n3])) (ASort h1 n3) (ASort O (next g
125 n0)) H3) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e in A return
126 (\lambda (_: A).nat) with [(ASort n5 _) \Rightarrow n5 | (AHead _ _)
127 \Rightarrow h1])) (ASort h1 n3) (ASort O (next g n0)) H3) in (eq_ind nat O
128 (\lambda (n5: nat).((eq nat n3 (next g n0)) \to ((eq A (ASort h2 n4) (ASort O
129 (next g n2))) \to ((eq A (aplus g (ASort n5 n3) k) (aplus g (ASort h2 n4) k))
130 \to (leq g (ASort O n0) (ASort O n2)))))) (\lambda (H7: (eq nat n3 (next g
131 n0))).(eq_ind nat (next g n0) (\lambda (n5: nat).((eq A (ASort h2 n4) (ASort
132 O (next g n2))) \to ((eq A (aplus g (ASort O n5) k) (aplus g (ASort h2 n4)
133 k)) \to (leq g (ASort O n0) (ASort O n2))))) (\lambda (H8: (eq A (ASort h2
134 n4) (ASort O (next g n2)))).(let H9 \def (f_equal A nat (\lambda (e:
135 A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n5) \Rightarrow
136 n5 | (AHead _ _) \Rightarrow n4])) (ASort h2 n4) (ASort O (next g n2)) H8) in
137 ((let H10 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda
138 (_: A).nat) with [(ASort n5 _) \Rightarrow n5 | (AHead _ _) \Rightarrow h2]))
139 (ASort h2 n4) (ASort O (next g n2)) H8) in (eq_ind nat O (\lambda (n5:
140 nat).((eq nat n4 (next g n2)) \to ((eq A (aplus g (ASort O (next g n0)) k)
141 (aplus g (ASort n5 n4) k)) \to (leq g (ASort O n0) (ASort O n2))))) (\lambda
142 (H11: (eq nat n4 (next g n2))).(eq_ind nat (next g n2) (\lambda (n5:
143 nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O n5) k)) \to
144 (leq g (ASort O n0) (ASort O n2)))) (\lambda (H12: (eq A (aplus g (ASort O
145 (next g n0)) k) (aplus g (ASort O (next g n2)) k))).(let H13 \def (eq_ind_r A
146 (aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a (aplus g (ASort O
147 (next g n2)) k))) H12 (aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g n0
148 k)) in (let H14 \def (eq_ind_r A (aplus g (ASort O (next g n2)) k) (\lambda
149 (a: A).(eq A (aplus g (ASort O n0) (S k)) a)) H13 (aplus g (ASort O n2) (S
150 k)) (aplus_sort_O_S_simpl g n2 k)) in (leq_sort g O O n0 n2 (S k) H14)))) n4
151 (sym_eq nat n4 (next g n2) H11))) h2 (sym_eq nat h2 O H10))) H9))) n3 (sym_eq
152 nat n3 (next g n0) H7))) h1 (sym_eq nat h1 O H6))) H5)) H4 H2))) | (leq_head
153 a0 a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A (AHead a0 a4) (ASort O
154 (next g n0)))).(\lambda (H5: (eq A (AHead a3 a5) (ASort O (next g
155 n2)))).((let H6 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match e in A
156 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
157 _) \Rightarrow True])) I (ASort O (next g n0)) H4) in (False_ind ((eq A
158 (AHead a3 a5) (ASort O (next g n2))) \to ((leq g a0 a3) \to ((leq g a4 a5)
159 \to (leq g (ASort O n0) (ASort O n2))))) H6)) H5 H2 H3)))]) in (H2
160 (refl_equal A (ASort O (next g n0))) (refl_equal A (ASort O (next g n2))))))
161 (\lambda (n3: nat).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g
162 (ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2))))).(\lambda (H1: (leq
163 g (asucc g (ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H2 \def (match H1
164 in leq return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a
165 a0)).((eq A a (ASort O (next g n0))) \to ((eq A a0 (ASort n3 n2)) \to (leq g
166 (ASort O n0) (ASort (S n3) n2))))))) with [(leq_sort h1 h2 n4 n5 k H2)
167 \Rightarrow (\lambda (H3: (eq A (ASort h1 n4) (ASort O (next g
168 n0)))).(\lambda (H4: (eq A (ASort h2 n5) (ASort n3 n2))).((let H5 \def
169 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
170 [(ASort _ n6) \Rightarrow n6 | (AHead _ _) \Rightarrow n4])) (ASort h1 n4)
171 (ASort O (next g n0)) H3) in ((let H6 \def (f_equal A nat (\lambda (e:
172 A).(match e in A return (\lambda (_: A).nat) with [(ASort n6 _) \Rightarrow
173 n6 | (AHead _ _) \Rightarrow h1])) (ASort h1 n4) (ASort O (next g n0)) H3) in
174 (eq_ind nat O (\lambda (n6: nat).((eq nat n4 (next g n0)) \to ((eq A (ASort
175 h2 n5) (ASort n3 n2)) \to ((eq A (aplus g (ASort n6 n4) k) (aplus g (ASort h2
176 n5) k)) \to (leq g (ASort O n0) (ASort (S n3) n2)))))) (\lambda (H7: (eq nat
177 n4 (next g n0))).(eq_ind nat (next g n0) (\lambda (n6: nat).((eq A (ASort h2
178 n5) (ASort n3 n2)) \to ((eq A (aplus g (ASort O n6) k) (aplus g (ASort h2 n5)
179 k)) \to (leq g (ASort O n0) (ASort (S n3) n2))))) (\lambda (H8: (eq A (ASort
180 h2 n5) (ASort n3 n2))).(let H9 \def (f_equal A nat (\lambda (e: A).(match e
181 in A return (\lambda (_: A).nat) with [(ASort _ n6) \Rightarrow n6 | (AHead _
182 _) \Rightarrow n5])) (ASort h2 n5) (ASort n3 n2) H8) in ((let H10 \def
183 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
184 [(ASort n6 _) \Rightarrow n6 | (AHead _ _) \Rightarrow h2])) (ASort h2 n5)
185 (ASort n3 n2) H8) in (eq_ind nat n3 (\lambda (n6: nat).((eq nat n5 n2) \to
186 ((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n6 n5) k)) \to (leq
187 g (ASort O n0) (ASort (S n3) n2))))) (\lambda (H11: (eq nat n5 n2)).(eq_ind
188 nat n2 (\lambda (n6: nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g
189 (ASort n3 n6) k)) \to (leq g (ASort O n0) (ASort (S n3) n2)))) (\lambda (H12:
190 (eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n3 n2) k))).(let H13
191 \def (eq_ind_r A (aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a
192 (aplus g (ASort n3 n2) k))) H12 (aplus g (ASort O n0) (S k))
193 (aplus_sort_O_S_simpl g n0 k)) in (let H14 \def (eq_ind_r A (aplus g (ASort
194 n3 n2) k) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S k)) a)) H13 (aplus g
195 (ASort (S n3) n2) (S k)) (aplus_sort_S_S_simpl g n2 n3 k)) in (leq_sort g O
196 (S n3) n0 n2 (S k) H14)))) n5 (sym_eq nat n5 n2 H11))) h2 (sym_eq nat h2 n3
197 H10))) H9))) n4 (sym_eq nat n4 (next g n0) H7))) h1 (sym_eq nat h1 O H6)))
198 H5)) H4 H2))) | (leq_head a0 a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A
199 (AHead a0 a4) (ASort O (next g n0)))).(\lambda (H5: (eq A (AHead a3 a5)
200 (ASort n3 n2))).((let H6 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match
201 e in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
202 (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H4) in (False_ind
203 ((eq A (AHead a3 a5) (ASort n3 n2)) \to ((leq g a0 a3) \to ((leq g a4 a5) \to
204 (leq g (ASort O n0) (ASort (S n3) n2))))) H6)) H5 H2 H3)))]) in (H2
205 (refl_equal A (ASort O (next g n0))) (refl_equal A (ASort n3 n2))))))) n1
206 H0)) (\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0))
207 (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda
208 (H0: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(nat_ind
209 (\lambda (n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4
210 n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq
211 g (ASort n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4
212 n2))))) (\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort O
213 n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort O n2)))
214 \to (leq g (ASort n3 n0) (ASort O n2))))).(let H2 \def (match H1 in leq
215 return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a
216 (ASort n3 n0)) \to ((eq A a0 (ASort O (next g n2))) \to (leq g (ASort (S n3)
217 n0) (ASort O n2))))))) with [(leq_sort h1 h2 n4 n5 k H2) \Rightarrow (\lambda
218 (H3: (eq A (ASort h1 n4) (ASort n3 n0))).(\lambda (H4: (eq A (ASort h2 n5)
219 (ASort O (next g n2)))).((let H5 \def (f_equal A nat (\lambda (e: A).(match e
220 in A return (\lambda (_: A).nat) with [(ASort _ n6) \Rightarrow n6 | (AHead _
221 _) \Rightarrow n4])) (ASort h1 n4) (ASort n3 n0) H3) in ((let H6 \def
222 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
223 [(ASort n6 _) \Rightarrow n6 | (AHead _ _) \Rightarrow h1])) (ASort h1 n4)
224 (ASort n3 n0) H3) in (eq_ind nat n3 (\lambda (n6: nat).((eq nat n4 n0) \to
225 ((eq A (ASort h2 n5) (ASort O (next g n2))) \to ((eq A (aplus g (ASort n6 n4)
226 k) (aplus g (ASort h2 n5) k)) \to (leq g (ASort (S n3) n0) (ASort O n2))))))
227 (\lambda (H7: (eq nat n4 n0)).(eq_ind nat n0 (\lambda (n6: nat).((eq A (ASort
228 h2 n5) (ASort O (next g n2))) \to ((eq A (aplus g (ASort n3 n6) k) (aplus g
229 (ASort h2 n5) k)) \to (leq g (ASort (S n3) n0) (ASort O n2))))) (\lambda (H8:
230 (eq A (ASort h2 n5) (ASort O (next g n2)))).(let H9 \def (f_equal A nat
231 (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n6)
232 \Rightarrow n6 | (AHead _ _) \Rightarrow n5])) (ASort h2 n5) (ASort O (next g
233 n2)) H8) in ((let H10 \def (f_equal A nat (\lambda (e: A).(match e in A
234 return (\lambda (_: A).nat) with [(ASort n6 _) \Rightarrow n6 | (AHead _ _)
235 \Rightarrow h2])) (ASort h2 n5) (ASort O (next g n2)) H8) in (eq_ind nat O
236 (\lambda (n6: nat).((eq nat n5 (next g n2)) \to ((eq A (aplus g (ASort n3 n0)
237 k) (aplus g (ASort n6 n5) k)) \to (leq g (ASort (S n3) n0) (ASort O n2)))))
238 (\lambda (H11: (eq nat n5 (next g n2))).(eq_ind nat (next g n2) (\lambda (n6:
239 nat).((eq A (aplus g (ASort n3 n0) k) (aplus g (ASort O n6) k)) \to (leq g
240 (ASort (S n3) n0) (ASort O n2)))) (\lambda (H12: (eq A (aplus g (ASort n3 n0)
241 k) (aplus g (ASort O (next g n2)) k))).(let H13 \def (eq_ind_r A (aplus g
242 (ASort n3 n0) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n2)) k)))
243 H12 (aplus g (ASort (S n3) n0) (S k)) (aplus_sort_S_S_simpl g n0 n3 k)) in
244 (let H14 \def (eq_ind_r A (aplus g (ASort O (next g n2)) k) (\lambda (a:
245 A).(eq A (aplus g (ASort (S n3) n0) (S k)) a)) H13 (aplus g (ASort O n2) (S
246 k)) (aplus_sort_O_S_simpl g n2 k)) in (leq_sort g (S n3) O n0 n2 (S k)
247 H14)))) n5 (sym_eq nat n5 (next g n2) H11))) h2 (sym_eq nat h2 O H10))) H9)))
248 n4 (sym_eq nat n4 n0 H7))) h1 (sym_eq nat h1 n3 H6))) H5)) H4 H2))) |
249 (leq_head a0 a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A (AHead a0 a4)
250 (ASort n3 n0))).(\lambda (H5: (eq A (AHead a3 a5) (ASort O (next g
251 n2)))).((let H6 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match e in A
252 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
253 _) \Rightarrow True])) I (ASort n3 n0) H4) in (False_ind ((eq A (AHead a3 a5)
254 (ASort O (next g n2))) \to ((leq g a0 a3) \to ((leq g a4 a5) \to (leq g
255 (ASort (S n3) n0) (ASort O n2))))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A
256 (ASort n3 n0)) (refl_equal A (ASort O (next g n2))))))) (\lambda (n4:
257 nat).(\lambda (_: (((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4
258 n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq
259 g (ASort n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4
260 n2)))))).(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S
261 n4) n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort (S
262 n4) n2))) \to (leq g (ASort n3 n0) (ASort (S n4) n2))))).(let H2 \def (match
263 H1 in leq return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a
264 a0)).((eq A a (ASort n3 n0)) \to ((eq A a0 (ASort n4 n2)) \to (leq g (ASort
265 (S n3) n0) (ASort (S n4) n2))))))) with [(leq_sort h1 h2 n5 n6 k H2)
266 \Rightarrow (\lambda (H3: (eq A (ASort h1 n5) (ASort n3 n0))).(\lambda (H4:
267 (eq A (ASort h2 n6) (ASort n4 n2))).((let H5 \def (f_equal A nat (\lambda (e:
268 A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n7) \Rightarrow
269 n7 | (AHead _ _) \Rightarrow n5])) (ASort h1 n5) (ASort n3 n0) H3) in ((let
270 H6 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_:
271 A).nat) with [(ASort n7 _) \Rightarrow n7 | (AHead _ _) \Rightarrow h1]))
272 (ASort h1 n5) (ASort n3 n0) H3) in (eq_ind nat n3 (\lambda (n7: nat).((eq nat
273 n5 n0) \to ((eq A (ASort h2 n6) (ASort n4 n2)) \to ((eq A (aplus g (ASort n7
274 n5) k) (aplus g (ASort h2 n6) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4)
275 n2)))))) (\lambda (H7: (eq nat n5 n0)).(eq_ind nat n0 (\lambda (n7: nat).((eq
276 A (ASort h2 n6) (ASort n4 n2)) \to ((eq A (aplus g (ASort n3 n7) k) (aplus g
277 (ASort h2 n6) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))) (\lambda
278 (H8: (eq A (ASort h2 n6) (ASort n4 n2))).(let H9 \def (f_equal A nat (\lambda
279 (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n7)
280 \Rightarrow n7 | (AHead _ _) \Rightarrow n6])) (ASort h2 n6) (ASort n4 n2)
281 H8) in ((let H10 \def (f_equal A nat (\lambda (e: A).(match e in A return
282 (\lambda (_: A).nat) with [(ASort n7 _) \Rightarrow n7 | (AHead _ _)
283 \Rightarrow h2])) (ASort h2 n6) (ASort n4 n2) H8) in (eq_ind nat n4 (\lambda
284 (n7: nat).((eq nat n6 n2) \to ((eq A (aplus g (ASort n3 n0) k) (aplus g
285 (ASort n7 n6) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))) (\lambda
286 (H11: (eq nat n6 n2)).(eq_ind nat n2 (\lambda (n7: nat).((eq A (aplus g
287 (ASort n3 n0) k) (aplus g (ASort n4 n7) k)) \to (leq g (ASort (S n3) n0)
288 (ASort (S n4) n2)))) (\lambda (H12: (eq A (aplus g (ASort n3 n0) k) (aplus g
289 (ASort n4 n2) k))).(let H13 \def (eq_ind_r A (aplus g (ASort n3 n0) k)
290 (\lambda (a: A).(eq A a (aplus g (ASort n4 n2) k))) H12 (aplus g (ASort (S
291 n3) n0) (S k)) (aplus_sort_S_S_simpl g n0 n3 k)) in (let H14 \def (eq_ind_r A
292 (aplus g (ASort n4 n2) k) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S
293 k)) a)) H13 (aplus g (ASort (S n4) n2) (S k)) (aplus_sort_S_S_simpl g n2 n4
294 k)) in (leq_sort g (S n3) (S n4) n0 n2 (S k) H14)))) n6 (sym_eq nat n6 n2
295 H11))) h2 (sym_eq nat h2 n4 H10))) H9))) n5 (sym_eq nat n5 n0 H7))) h1
296 (sym_eq nat h1 n3 H6))) H5)) H4 H2))) | (leq_head a0 a3 H2 a4 a5 H3)
297 \Rightarrow (\lambda (H4: (eq A (AHead a0 a4) (ASort n3 n0))).(\lambda (H5:
298 (eq A (AHead a3 a5) (ASort n4 n2))).((let H6 \def (eq_ind A (AHead a0 a4)
299 (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
300 \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n3 n0) H4) in
301 (False_ind ((eq A (AHead a3 a5) (ASort n4 n2)) \to ((leq g a0 a3) \to ((leq g
302 a4 a5) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))) H6)) H5 H2 H3)))])
303 in (H2 (refl_equal A (ASort n3 n0)) (refl_equal A (ASort n4 n2)))))))) n1 H0
304 IHn)))) n H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n n0))
305 (asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda (H0:
306 (((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0)
307 a0)))).(\lambda (H1: (leq g (asucc g (ASort n n0)) (asucc g (AHead a
308 a0)))).(nat_ind (\lambda (n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g
309 a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 n0))
310 (asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1
311 n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a a0))))))
312 (\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O
313 n0) a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq
314 g (ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g
315 (AHead a a0)))).(let H5 \def (match H4 in leq return (\lambda (a3:
316 A).(\lambda (a4: A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (ASort O (next g
317 n0))) \to ((eq A a4 (AHead a (asucc g a0))) \to (leq g (ASort O n0) (AHead a
318 a0))))))) with [(leq_sort h1 h2 n1 n2 k H5) \Rightarrow (\lambda (H6: (eq A
319 (ASort h1 n1) (ASort O (next g n0)))).(\lambda (H7: (eq A (ASort h2 n2)
320 (AHead a (asucc g a0)))).((let H8 \def (f_equal A nat (\lambda (e: A).(match
321 e in A return (\lambda (_: A).nat) with [(ASort _ n3) \Rightarrow n3 | (AHead
322 _ _) \Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H6) in ((let H9
323 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat)
324 with [(ASort n3 _) \Rightarrow n3 | (AHead _ _) \Rightarrow h1])) (ASort h1
325 n1) (ASort O (next g n0)) H6) in (eq_ind nat O (\lambda (n3: nat).((eq nat n1
326 (next g n0)) \to ((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A
327 (aplus g (ASort n3 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0)
328 (AHead a a0)))))) (\lambda (H10: (eq nat n1 (next g n0))).(eq_ind nat (next g
329 n0) (\lambda (n3: nat).((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq
330 A (aplus g (ASort O n3) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0)
331 (AHead a a0))))) (\lambda (H11: (eq A (ASort h2 n2) (AHead a (asucc g
332 a0)))).(let H12 \def (eq_ind A (ASort h2 n2) (\lambda (e: A).(match e in A
333 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
334 \Rightarrow False])) I (AHead a (asucc g a0)) H11) in (False_ind ((eq A
335 (aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n2) k)) \to (leq g
336 (ASort O n0) (AHead a a0))) H12))) n1 (sym_eq nat n1 (next g n0) H10))) h1
337 (sym_eq nat h1 O H9))) H8)) H7 H5))) | (leq_head a3 a4 H5 a5 a6 H6)
338 \Rightarrow (\lambda (H7: (eq A (AHead a3 a5) (ASort O (next g
339 n0)))).(\lambda (H8: (eq A (AHead a4 a6) (AHead a (asucc g a0)))).((let H9
340 \def (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda
341 (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
342 True])) I (ASort O (next g n0)) H7) in (False_ind ((eq A (AHead a4 a6) (AHead
343 a (asucc g a0))) \to ((leq g a3 a4) \to ((leq g a5 a6) \to (leq g (ASort O
344 n0) (AHead a a0))))) H9)) H8 H5 H6)))]) in (H5 (refl_equal A (ASort O (next g
345 n0))) (refl_equal A (AHead a (asucc g a0)))))))) (\lambda (n1: nat).(\lambda
346 (_: (((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0)
347 a))) \to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1
348 n0) a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to
349 (leq g (ASort n1 n0) (AHead a a0))))))).(\lambda (_: (((leq g (asucc g (ASort
350 (S n1) n0)) (asucc g a)) \to (leq g (ASort (S n1) n0) a)))).(\lambda (_:
351 (((leq g (asucc g (ASort (S n1) n0)) (asucc g a0)) \to (leq g (ASort (S n1)
352 n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort (S n1) n0)) (asucc g (AHead a
353 a0)))).(let H5 \def (match H4 in leq return (\lambda (a3: A).(\lambda (a4:
354 A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (ASort n1 n0)) \to ((eq A a4 (AHead
355 a (asucc g a0))) \to (leq g (ASort (S n1) n0) (AHead a a0))))))) with
356 [(leq_sort h1 h2 n2 n3 k H5) \Rightarrow (\lambda (H6: (eq A (ASort h1 n2)
357 (ASort n1 n0))).(\lambda (H7: (eq A (ASort h2 n3) (AHead a (asucc g
358 a0)))).((let H8 \def (f_equal A nat (\lambda (e: A).(match e in A return
359 (\lambda (_: A).nat) with [(ASort _ n4) \Rightarrow n4 | (AHead _ _)
360 \Rightarrow n2])) (ASort h1 n2) (ASort n1 n0) H6) in ((let H9 \def (f_equal A
361 nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort n4
362 _) \Rightarrow n4 | (AHead _ _) \Rightarrow h1])) (ASort h1 n2) (ASort n1 n0)
363 H6) in (eq_ind nat n1 (\lambda (n4: nat).((eq nat n2 n0) \to ((eq A (ASort h2
364 n3) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n4 n2) k) (aplus g
365 (ASort h2 n3) k)) \to (leq g (ASort (S n1) n0) (AHead a a0)))))) (\lambda
366 (H10: (eq nat n2 n0)).(eq_ind nat n0 (\lambda (n4: nat).((eq A (ASort h2 n3)
367 (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n1 n4) k) (aplus g (ASort
368 h2 n3) k)) \to (leq g (ASort (S n1) n0) (AHead a a0))))) (\lambda (H11: (eq A
369 (ASort h2 n3) (AHead a (asucc g a0)))).(let H12 \def (eq_ind A (ASort h2 n3)
370 (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
371 \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0))
372 H11) in (False_ind ((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n3)
373 k)) \to (leq g (ASort (S n1) n0) (AHead a a0))) H12))) n2 (sym_eq nat n2 n0
374 H10))) h1 (sym_eq nat h1 n1 H9))) H8)) H7 H5))) | (leq_head a3 a4 H5 a5 a6
375 H6) \Rightarrow (\lambda (H7: (eq A (AHead a3 a5) (ASort n1 n0))).(\lambda
376 (H8: (eq A (AHead a4 a6) (AHead a (asucc g a0)))).((let H9 \def (eq_ind A
377 (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with
378 [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1
379 n0) H7) in (False_ind ((eq A (AHead a4 a6) (AHead a (asucc g a0))) \to ((leq
380 g a3 a4) \to ((leq g a5 a6) \to (leq g (ASort (S n1) n0) (AHead a a0)))))
381 H9)) H8 H5 H6)))]) in (H5 (refl_equal A (ASort n1 n0)) (refl_equal A (AHead a
382 (asucc g a0)))))))))) n H H0 H1)))))) a2)))) (\lambda (a: A).(\lambda (_:
383 ((\forall (a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq g a
384 a2))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0)
385 (asucc g a2)) \to (leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3:
386 A).((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))
387 (\lambda (n: nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a
388 a0)) (asucc g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g
389 (AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1
390 n0)))) (\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O
391 n0)))).(let H3 \def (match H2 in leq return (\lambda (a3: A).(\lambda (a4:
392 A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (AHead a (asucc g a0))) \to ((eq A
393 a4 (ASort O (next g n0))) \to (leq g (AHead a a0) (ASort O n0))))))) with
394 [(leq_sort h1 h2 n1 n2 k H3) \Rightarrow (\lambda (H4: (eq A (ASort h1 n1)
395 (AHead a (asucc g a0)))).(\lambda (H5: (eq A (ASort h2 n2) (ASort O (next g
396 n0)))).((let H6 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A
397 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
398 \Rightarrow False])) I (AHead a (asucc g a0)) H4) in (False_ind ((eq A (ASort
399 h2 n2) (ASort O (next g n0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g
400 (ASort h2 n2) k)) \to (leq g (AHead a a0) (ASort O n0)))) H6)) H5 H3))) |
401 (leq_head a3 a4 H3 a5 a6 H4) \Rightarrow (\lambda (H5: (eq A (AHead a3 a5)
402 (AHead a (asucc g a0)))).(\lambda (H6: (eq A (AHead a4 a6) (ASort O (next g
403 n0)))).((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return
404 (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a7)
405 \Rightarrow a7])) (AHead a3 a5) (AHead a (asucc g a0)) H5) in ((let H8 \def
406 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
407 [(ASort _ _) \Rightarrow a3 | (AHead a7 _) \Rightarrow a7])) (AHead a3 a5)
408 (AHead a (asucc g a0)) H5) in (eq_ind A a (\lambda (a7: A).((eq A a5 (asucc g
409 a0)) \to ((eq A (AHead a4 a6) (ASort O (next g n0))) \to ((leq g a7 a4) \to
410 ((leq g a5 a6) \to (leq g (AHead a a0) (ASort O n0))))))) (\lambda (H9: (eq A
411 a5 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a7: A).((eq A (AHead a4
412 a6) (ASort O (next g n0))) \to ((leq g a a4) \to ((leq g a7 a6) \to (leq g
413 (AHead a a0) (ASort O n0)))))) (\lambda (H10: (eq A (AHead a4 a6) (ASort O
414 (next g n0)))).(let H11 \def (eq_ind A (AHead a4 a6) (\lambda (e: A).(match e
415 in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
416 (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H10) in (False_ind
417 ((leq g a a4) \to ((leq g (asucc g a0) a6) \to (leq g (AHead a a0) (ASort O
418 n0)))) H11))) a5 (sym_eq A a5 (asucc g a0) H9))) a3 (sym_eq A a3 a H8))) H7))
419 H6 H3 H4)))]) in (H3 (refl_equal A (AHead a (asucc g a0))) (refl_equal A
420 (ASort O (next g n0)))))) (\lambda (n1: nat).(\lambda (_: (((leq g (asucc g
421 (AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1
422 n0))))).(\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort (S n1)
423 n0)))).(let H3 \def (match H2 in leq return (\lambda (a3: A).(\lambda (a4:
424 A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (AHead a (asucc g a0))) \to ((eq A
425 a4 (ASort n1 n0)) \to (leq g (AHead a a0) (ASort (S n1) n0))))))) with
426 [(leq_sort h1 h2 n2 n3 k H3) \Rightarrow (\lambda (H4: (eq A (ASort h1 n2)
427 (AHead a (asucc g a0)))).(\lambda (H5: (eq A (ASort h2 n3) (ASort n1
428 n0))).((let H6 \def (eq_ind A (ASort h1 n2) (\lambda (e: A).(match e in A
429 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
430 \Rightarrow False])) I (AHead a (asucc g a0)) H4) in (False_ind ((eq A (ASort
431 h2 n3) (ASort n1 n0)) \to ((eq A (aplus g (ASort h1 n2) k) (aplus g (ASort h2
432 n3) k)) \to (leq g (AHead a a0) (ASort (S n1) n0)))) H6)) H5 H3))) |
433 (leq_head a3 a4 H3 a5 a6 H4) \Rightarrow (\lambda (H5: (eq A (AHead a3 a5)
434 (AHead a (asucc g a0)))).(\lambda (H6: (eq A (AHead a4 a6) (ASort n1
435 n0))).((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return
436 (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a7)
437 \Rightarrow a7])) (AHead a3 a5) (AHead a (asucc g a0)) H5) in ((let H8 \def
438 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
439 [(ASort _ _) \Rightarrow a3 | (AHead a7 _) \Rightarrow a7])) (AHead a3 a5)
440 (AHead a (asucc g a0)) H5) in (eq_ind A a (\lambda (a7: A).((eq A a5 (asucc g
441 a0)) \to ((eq A (AHead a4 a6) (ASort n1 n0)) \to ((leq g a7 a4) \to ((leq g
442 a5 a6) \to (leq g (AHead a a0) (ASort (S n1) n0))))))) (\lambda (H9: (eq A a5
443 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a7: A).((eq A (AHead a4 a6)
444 (ASort n1 n0)) \to ((leq g a a4) \to ((leq g a7 a6) \to (leq g (AHead a a0)
445 (ASort (S n1) n0)))))) (\lambda (H10: (eq A (AHead a4 a6) (ASort n1
446 n0))).(let H11 \def (eq_ind A (AHead a4 a6) (\lambda (e: A).(match e in A
447 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
448 _) \Rightarrow True])) I (ASort n1 n0) H10) in (False_ind ((leq g a a4) \to
449 ((leq g (asucc g a0) a6) \to (leq g (AHead a a0) (ASort (S n1) n0)))) H11)))
450 a5 (sym_eq A a5 (asucc g a0) H9))) a3 (sym_eq A a3 a H8))) H7)) H6 H3 H4)))])
451 in (H3 (refl_equal A (AHead a (asucc g a0))) (refl_equal A (ASort n1
452 n0))))))) n H1)))) (\lambda (a3: A).(\lambda (_: (((leq g (asucc g (AHead a
453 a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda (a4: A).(\lambda
454 (_: (((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g (AHead a a0)
455 a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g (AHead a3
456 a4)))).(let H4 \def (match H3 in leq return (\lambda (a5: A).(\lambda (a6:
457 A).(\lambda (_: (leq ? a5 a6)).((eq A a5 (AHead a (asucc g a0))) \to ((eq A
458 a6 (AHead a3 (asucc g a4))) \to (leq g (AHead a a0) (AHead a3 a4))))))) with
459 [(leq_sort h1 h2 n1 n2 k H4) \Rightarrow (\lambda (H5: (eq A (ASort h1 n1)
460 (AHead a (asucc g a0)))).(\lambda (H6: (eq A (ASort h2 n2) (AHead a3 (asucc g
461 a4)))).((let H7 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A
462 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
463 \Rightarrow False])) I (AHead a (asucc g a0)) H5) in (False_ind ((eq A (ASort
464 h2 n2) (AHead a3 (asucc g a4))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g
465 (ASort h2 n2) k)) \to (leq g (AHead a a0) (AHead a3 a4)))) H7)) H6 H4))) |
466 (leq_head a5 a6 H4 a7 a8 H5) \Rightarrow (\lambda (H6: (eq A (AHead a5 a7)
467 (AHead a (asucc g a0)))).(\lambda (H7: (eq A (AHead a6 a8) (AHead a3 (asucc g
468 a4)))).((let H8 \def (f_equal A A (\lambda (e: A).(match e in A return
469 (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 | (AHead _ a9)
470 \Rightarrow a9])) (AHead a5 a7) (AHead a (asucc g a0)) H6) in ((let H9 \def
471 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
472 [(ASort _ _) \Rightarrow a5 | (AHead a9 _) \Rightarrow a9])) (AHead a5 a7)
473 (AHead a (asucc g a0)) H6) in (eq_ind A a (\lambda (a9: A).((eq A a7 (asucc g
474 a0)) \to ((eq A (AHead a6 a8) (AHead a3 (asucc g a4))) \to ((leq g a9 a6) \to
475 ((leq g a7 a8) \to (leq g (AHead a a0) (AHead a3 a4))))))) (\lambda (H10: (eq
476 A a7 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a9: A).((eq A (AHead a6
477 a8) (AHead a3 (asucc g a4))) \to ((leq g a a6) \to ((leq g a9 a8) \to (leq g
478 (AHead a a0) (AHead a3 a4)))))) (\lambda (H11: (eq A (AHead a6 a8) (AHead a3
479 (asucc g a4)))).(let H12 \def (f_equal A A (\lambda (e: A).(match e in A
480 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a8 | (AHead _ a9)
481 \Rightarrow a9])) (AHead a6 a8) (AHead a3 (asucc g a4)) H11) in ((let H13
482 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
483 with [(ASort _ _) \Rightarrow a6 | (AHead a9 _) \Rightarrow a9])) (AHead a6
484 a8) (AHead a3 (asucc g a4)) H11) in (eq_ind A a3 (\lambda (a9: A).((eq A a8
485 (asucc g a4)) \to ((leq g a a9) \to ((leq g (asucc g a0) a8) \to (leq g
486 (AHead a a0) (AHead a3 a4)))))) (\lambda (H14: (eq A a8 (asucc g
487 a4))).(eq_ind A (asucc g a4) (\lambda (a9: A).((leq g a a3) \to ((leq g
488 (asucc g a0) a9) \to (leq g (AHead a a0) (AHead a3 a4))))) (\lambda (H15:
489 (leq g a a3)).(\lambda (H16: (leq g (asucc g a0) (asucc g a4))).(leq_head g a
490 a3 H15 a0 a4 (H0 a4 H16)))) a8 (sym_eq A a8 (asucc g a4) H14))) a6 (sym_eq A
491 a6 a3 H13))) H12))) a7 (sym_eq A a7 (asucc g a0) H10))) a5 (sym_eq A a5 a
492 H9))) H8)) H7 H4 H5)))]) in (H4 (refl_equal A (AHead a (asucc g a0)))
493 (refl_equal A (AHead a3 (asucc g a4)))))))))) a2)))))) a1)).
496 \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g
499 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(ex A (\lambda (a1:
500 A).(leq g a0 (asucc g a1))))) (\lambda (n: nat).(\lambda (n0: nat).(ex_intro
501 A (\lambda (a0: A).(leq g (ASort n n0) (asucc g a0))) (ASort (S n) n0)
502 (leq_refl g (ASort n n0))))) (\lambda (a0: A).(\lambda (_: (ex A (\lambda
503 (a1: A).(leq g a0 (asucc g a1))))).(\lambda (a1: A).(\lambda (H0: (ex A
504 (\lambda (a2: A).(leq g a1 (asucc g a2))))).(let H1 \def H0 in (ex_ind A
505 (\lambda (a2: A).(leq g a1 (asucc g a2))) (ex A (\lambda (a2: A).(leq g
506 (AHead a0 a1) (asucc g a2)))) (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc
507 g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2)))
508 (AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1))))))
511 theorem leq_ahead_asucc_false:
512 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2)
513 (asucc g a1)) \to (\forall (P: Prop).P))))
515 \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
516 A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda
517 (n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead
518 (ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
519 \Rightarrow (ASort h n0)]))).(\lambda (P: Prop).(nat_ind (\lambda (n1:
520 nat).((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O
521 (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P)) (\lambda (H0: (leq g
522 (AHead (ASort O n0) a2) (ASort O (next g n0)))).(let H1 \def (match H0 in leq
523 return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a
524 (AHead (ASort O n0) a2)) \to ((eq A a0 (ASort O (next g n0))) \to P))))) with
525 [(leq_sort h1 h2 n1 n2 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n1)
526 (AHead (ASort O n0) a2))).(\lambda (H3: (eq A (ASort h2 n2) (ASort O (next g
527 n0)))).((let H4 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A
528 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
529 \Rightarrow False])) I (AHead (ASort O n0) a2) H2) in (False_ind ((eq A
530 (ASort h2 n2) (ASort O (next g n0))) \to ((eq A (aplus g (ASort h1 n1) k)
531 (aplus g (ASort h2 n2) k)) \to P)) H4)) H3 H1))) | (leq_head a0 a3 H1 a4 a5
532 H2) \Rightarrow (\lambda (H3: (eq A (AHead a0 a4) (AHead (ASort O n0)
533 a2))).(\lambda (H4: (eq A (AHead a3 a5) (ASort O (next g n0)))).((let H5 \def
534 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
535 [(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a0 a4)
536 (AHead (ASort O n0) a2) H3) in ((let H6 \def (f_equal A A (\lambda (e:
537 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 |
538 (AHead a _) \Rightarrow a])) (AHead a0 a4) (AHead (ASort O n0) a2) H3) in
539 (eq_ind A (ASort O n0) (\lambda (a: A).((eq A a4 a2) \to ((eq A (AHead a3 a5)
540 (ASort O (next g n0))) \to ((leq g a a3) \to ((leq g a4 a5) \to P)))))
541 (\lambda (H7: (eq A a4 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a3 a5)
542 (ASort O (next g n0))) \to ((leq g (ASort O n0) a3) \to ((leq g a a5) \to
543 P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort O (next g n0)))).(let H9 \def
544 (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda (_:
545 A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
546 True])) I (ASort O (next g n0)) H8) in (False_ind ((leq g (ASort O n0) a3)
547 \to ((leq g a2 a5) \to P)) H9))) a4 (sym_eq A a4 a2 H7))) a0 (sym_eq A a0
548 (ASort O n0) H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead (ASort O
549 n0) a2)) (refl_equal A (ASort O (next g n0)))))) (\lambda (n1: nat).(\lambda
550 (_: (((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O
551 (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P))).(\lambda (H0: (leq
552 g (AHead (ASort (S n1) n0) a2) (ASort n1 n0))).(let H1 \def (match H0 in leq
553 return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a
554 (AHead (ASort (S n1) n0) a2)) \to ((eq A a0 (ASort n1 n0)) \to P))))) with
555 [(leq_sort h1 h2 n2 n3 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n2)
556 (AHead (ASort (S n1) n0) a2))).(\lambda (H3: (eq A (ASort h2 n3) (ASort n1
557 n0))).((let H4 \def (eq_ind A (ASort h1 n2) (\lambda (e: A).(match e in A
558 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
559 \Rightarrow False])) I (AHead (ASort (S n1) n0) a2) H2) in (False_ind ((eq A
560 (ASort h2 n3) (ASort n1 n0)) \to ((eq A (aplus g (ASort h1 n2) k) (aplus g
561 (ASort h2 n3) k)) \to P)) H4)) H3 H1))) | (leq_head a0 a3 H1 a4 a5 H2)
562 \Rightarrow (\lambda (H3: (eq A (AHead a0 a4) (AHead (ASort (S n1) n0)
563 a2))).(\lambda (H4: (eq A (AHead a3 a5) (ASort n1 n0))).((let H5 \def
564 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
565 [(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a0 a4)
566 (AHead (ASort (S n1) n0) a2) H3) in ((let H6 \def (f_equal A A (\lambda (e:
567 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 |
568 (AHead a _) \Rightarrow a])) (AHead a0 a4) (AHead (ASort (S n1) n0) a2) H3)
569 in (eq_ind A (ASort (S n1) n0) (\lambda (a: A).((eq A a4 a2) \to ((eq A
570 (AHead a3 a5) (ASort n1 n0)) \to ((leq g a a3) \to ((leq g a4 a5) \to P)))))
571 (\lambda (H7: (eq A a4 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a3 a5)
572 (ASort n1 n0)) \to ((leq g (ASort (S n1) n0) a3) \to ((leq g a a5) \to P))))
573 (\lambda (H8: (eq A (AHead a3 a5) (ASort n1 n0))).(let H9 \def (eq_ind A
574 (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with
575 [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1
576 n0) H8) in (False_ind ((leq g (ASort (S n1) n0) a3) \to ((leq g a2 a5) \to
577 P)) H9))) a4 (sym_eq A a4 a2 H7))) a0 (sym_eq A a0 (ASort (S n1) n0) H6)))
578 H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead (ASort (S n1) n0) a2))
579 (refl_equal A (ASort n1 n0))))))) n H)))))) (\lambda (a: A).(\lambda (_:
580 ((\forall (a2: A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P:
581 Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead
582 a0 a2) (asucc g a0)) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda
583 (H1: (leq g (AHead (AHead a a0) a2) (AHead a (asucc g a0)))).(\lambda (P:
584 Prop).(let H2 \def (match H1 in leq return (\lambda (a3: A).(\lambda (a4:
585 A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (AHead (AHead a a0) a2)) \to ((eq A
586 a4 (AHead a (asucc g a0))) \to P))))) with [(leq_sort h1 h2 n1 n2 k H2)
587 \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead (AHead a a0)
588 a2))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a (asucc g a0)))).((let H5
589 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A return (\lambda
590 (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
591 False])) I (AHead (AHead a a0) a2) H3) in (False_ind ((eq A (ASort h2 n2)
592 (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort
593 h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a3 a4 H2 a5 a6 H3) \Rightarrow
594 (\lambda (H4: (eq A (AHead a3 a5) (AHead (AHead a a0) a2))).(\lambda (H5: (eq
595 A (AHead a4 a6) (AHead a (asucc g a0)))).((let H6 \def (f_equal A A (\lambda
596 (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow
597 a5 | (AHead _ a7) \Rightarrow a7])) (AHead a3 a5) (AHead (AHead a a0) a2) H4)
598 in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda
599 (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a7 _) \Rightarrow a7]))
600 (AHead a3 a5) (AHead (AHead a a0) a2) H4) in (eq_ind A (AHead a a0) (\lambda
601 (a7: A).((eq A a5 a2) \to ((eq A (AHead a4 a6) (AHead a (asucc g a0))) \to
602 ((leq g a7 a4) \to ((leq g a5 a6) \to P))))) (\lambda (H8: (eq A a5
603 a2)).(eq_ind A a2 (\lambda (a7: A).((eq A (AHead a4 a6) (AHead a (asucc g
604 a0))) \to ((leq g (AHead a a0) a4) \to ((leq g a7 a6) \to P)))) (\lambda (H9:
605 (eq A (AHead a4 a6) (AHead a (asucc g a0)))).(let H10 \def (f_equal A A
606 (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
607 \Rightarrow a6 | (AHead _ a7) \Rightarrow a7])) (AHead a4 a6) (AHead a (asucc
608 g a0)) H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e in A
609 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead a7 _)
610 \Rightarrow a7])) (AHead a4 a6) (AHead a (asucc g a0)) H9) in (eq_ind A a
611 (\lambda (a7: A).((eq A a6 (asucc g a0)) \to ((leq g (AHead a a0) a7) \to
612 ((leq g a2 a6) \to P)))) (\lambda (H12: (eq A a6 (asucc g a0))).(eq_ind A
613 (asucc g a0) (\lambda (a7: A).((leq g (AHead a a0) a) \to ((leq g a2 a7) \to
614 P))) (\lambda (H13: (leq g (AHead a a0) a)).(\lambda (_: (leq g a2 (asucc g
615 a0))).(leq_ahead_false_1 g a a0 H13 P))) a6 (sym_eq A a6 (asucc g a0) H12)))
616 a4 (sym_eq A a4 a H11))) H10))) a5 (sym_eq A a5 a2 H8))) a3 (sym_eq A a3
617 (AHead a a0) H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead (AHead a
618 a0) a2)) (refl_equal A (AHead a (asucc g a0)))))))))))) a1)).
620 theorem leq_asucc_false:
621 \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P:
624 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0)
625 a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda
626 (H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
627 \Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).(nat_ind
628 (\lambda (n1: nat).((leq g (match n1 with [O \Rightarrow (ASort O (next g
629 n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P)) (\lambda (H0:
630 (leq g (ASort O (next g n0)) (ASort O n0))).(let H1 \def (match H0 in leq
631 return (\lambda (a0: A).(\lambda (a1: A).(\lambda (_: (leq ? a0 a1)).((eq A
632 a0 (ASort O (next g n0))) \to ((eq A a1 (ASort O n0)) \to P))))) with
633 [(leq_sort h1 h2 n1 n2 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n1)
634 (ASort O (next g n0)))).(\lambda (H3: (eq A (ASort h2 n2) (ASort O
635 n0))).((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return
636 (\lambda (_: A).nat) with [(ASort _ n3) \Rightarrow n3 | (AHead _ _)
637 \Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H2) in ((let H5 \def
638 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
639 [(ASort n3 _) \Rightarrow n3 | (AHead _ _) \Rightarrow h1])) (ASort h1 n1)
640 (ASort O (next g n0)) H2) in (eq_ind nat O (\lambda (n3: nat).((eq nat n1
641 (next g n0)) \to ((eq A (ASort h2 n2) (ASort O n0)) \to ((eq A (aplus g
642 (ASort n3 n1) k) (aplus g (ASort h2 n2) k)) \to P)))) (\lambda (H6: (eq nat
643 n1 (next g n0))).(eq_ind nat (next g n0) (\lambda (n3: nat).((eq A (ASort h2
644 n2) (ASort O n0)) \to ((eq A (aplus g (ASort O n3) k) (aplus g (ASort h2 n2)
645 k)) \to P))) (\lambda (H7: (eq A (ASort h2 n2) (ASort O n0))).(let H8 \def
646 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
647 [(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow n2])) (ASort h2 n2)
648 (ASort O n0) H7) in ((let H9 \def (f_equal A nat (\lambda (e: A).(match e in
649 A return (\lambda (_: A).nat) with [(ASort n3 _) \Rightarrow n3 | (AHead _ _)
650 \Rightarrow h2])) (ASort h2 n2) (ASort O n0) H7) in (eq_ind nat O (\lambda
651 (n3: nat).((eq nat n2 n0) \to ((eq A (aplus g (ASort O (next g n0)) k) (aplus
652 g (ASort n3 n2) k)) \to P))) (\lambda (H10: (eq nat n2 n0)).(eq_ind nat n0
653 (\lambda (n3: nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O
654 n3) k)) \to P)) (\lambda (H11: (eq A (aplus g (ASort O (next g n0)) k) (aplus
655 g (ASort O n0) k))).(let H12 \def (eq_ind_r A (aplus g (ASort O (next g n0))
656 k) (\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) k))) H11 (aplus g (ASort O
657 n0) (S k)) (aplus_sort_O_S_simpl g n0 k)) in (let H_y \def (aplus_inj g (S k)
658 k (ASort O n0) H12) in (le_Sx_x k (eq_ind_r nat k (\lambda (n3: nat).(le n3
659 k)) (le_n k) (S k) H_y) P)))) n2 (sym_eq nat n2 n0 H10))) h2 (sym_eq nat h2 O
660 H9))) H8))) n1 (sym_eq nat n1 (next g n0) H6))) h1 (sym_eq nat h1 O H5)))
661 H4)) H3 H1))) | (leq_head a1 a2 H1 a3 a4 H2) \Rightarrow (\lambda (H3: (eq A
662 (AHead a1 a3) (ASort O (next g n0)))).(\lambda (H4: (eq A (AHead a2 a4)
663 (ASort O n0))).((let H5 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e
664 in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
665 (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H3) in (False_ind
666 ((eq A (AHead a2 a4) (ASort O n0)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to
667 P))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (ASort O (next g n0)))
668 (refl_equal A (ASort O n0))))) (\lambda (n1: nat).(\lambda (_: (((leq g
669 (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow
670 (ASort h n0)]) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (ASort n1 n0)
671 (ASort (S n1) n0))).(let H1 \def (match H0 in leq return (\lambda (a0:
672 A).(\lambda (a1: A).(\lambda (_: (leq ? a0 a1)).((eq A a0 (ASort n1 n0)) \to
673 ((eq A a1 (ASort (S n1) n0)) \to P))))) with [(leq_sort h1 h2 n2 n3 k H1)
674 \Rightarrow (\lambda (H2: (eq A (ASort h1 n2) (ASort n1 n0))).(\lambda (H3:
675 (eq A (ASort h2 n3) (ASort (S n1) n0))).((let H4 \def (f_equal A nat (\lambda
676 (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n4)
677 \Rightarrow n4 | (AHead _ _) \Rightarrow n2])) (ASort h1 n2) (ASort n1 n0)
678 H2) in ((let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return
679 (\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _)
680 \Rightarrow h1])) (ASort h1 n2) (ASort n1 n0) H2) in (eq_ind nat n1 (\lambda
681 (n4: nat).((eq nat n2 n0) \to ((eq A (ASort h2 n3) (ASort (S n1) n0)) \to
682 ((eq A (aplus g (ASort n4 n2) k) (aplus g (ASort h2 n3) k)) \to P))))
683 (\lambda (H6: (eq nat n2 n0)).(eq_ind nat n0 (\lambda (n4: nat).((eq A (ASort
684 h2 n3) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort n1 n4) k) (aplus g
685 (ASort h2 n3) k)) \to P))) (\lambda (H7: (eq A (ASort h2 n3) (ASort (S n1)
686 n0))).(let H8 \def (f_equal A nat (\lambda (e: A).(match e in A return
687 (\lambda (_: A).nat) with [(ASort _ n4) \Rightarrow n4 | (AHead _ _)
688 \Rightarrow n3])) (ASort h2 n3) (ASort (S n1) n0) H7) in ((let H9 \def
689 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
690 [(ASort n4 _) \Rightarrow n4 | (AHead _ _) \Rightarrow h2])) (ASort h2 n3)
691 (ASort (S n1) n0) H7) in (eq_ind nat (S n1) (\lambda (n4: nat).((eq nat n3
692 n0) \to ((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort n4 n3) k)) \to P)))
693 (\lambda (H10: (eq nat n3 n0)).(eq_ind nat n0 (\lambda (n4: nat).((eq A
694 (aplus g (ASort n1 n0) k) (aplus g (ASort (S n1) n4) k)) \to P)) (\lambda
695 (H11: (eq A (aplus g (ASort n1 n0) k) (aplus g (ASort (S n1) n0) k))).(let
696 H12 \def (eq_ind_r A (aplus g (ASort n1 n0) k) (\lambda (a0: A).(eq A a0
697 (aplus g (ASort (S n1) n0) k))) H11 (aplus g (ASort (S n1) n0) (S k))
698 (aplus_sort_S_S_simpl g n0 n1 k)) in (let H_y \def (aplus_inj g (S k) k
699 (ASort (S n1) n0) H12) in (le_Sx_x k (eq_ind_r nat k (\lambda (n4: nat).(le
700 n4 k)) (le_n k) (S k) H_y) P)))) n3 (sym_eq nat n3 n0 H10))) h2 (sym_eq nat
701 h2 (S n1) H9))) H8))) n2 (sym_eq nat n2 n0 H6))) h1 (sym_eq nat h1 n1 H5)))
702 H4)) H3 H1))) | (leq_head a1 a2 H1 a3 a4 H2) \Rightarrow (\lambda (H3: (eq A
703 (AHead a1 a3) (ASort n1 n0))).(\lambda (H4: (eq A (AHead a2 a4) (ASort (S n1)
704 n0))).((let H5 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e in A
705 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
706 _) \Rightarrow True])) I (ASort n1 n0) H3) in (False_ind ((eq A (AHead a2 a4)
707 (ASort (S n1) n0)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to P))) H5)) H4 H1
708 H2)))]) in (H1 (refl_equal A (ASort n1 n0)) (refl_equal A (ASort (S n1)
709 n0))))))) n H))))) (\lambda (a0: A).(\lambda (_: (((leq g (asucc g a0) a0)
710 \to (\forall (P: Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g
711 a1) a1) \to (\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g
712 a1)) (AHead a0 a1))).(\lambda (P: Prop).(let H2 \def (match H1 in leq return
713 (\lambda (a2: A).(\lambda (a3: A).(\lambda (_: (leq ? a2 a3)).((eq A a2
714 (AHead a0 (asucc g a1))) \to ((eq A a3 (AHead a0 a1)) \to P))))) with
715 [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1)
716 (AHead a0 (asucc g a1)))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a0
717 a1))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A
718 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
719 \Rightarrow False])) I (AHead a0 (asucc g a1)) H3) in (False_ind ((eq A
720 (ASort h2 n2) (AHead a0 a1)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g
721 (ASort h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a2 a3 H2 a4 a5 H3)
722 \Rightarrow (\lambda (H4: (eq A (AHead a2 a4) (AHead a0 (asucc g
723 a1)))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a0 a1))).((let H6 \def
724 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
725 [(ASort _ _) \Rightarrow a4 | (AHead _ a6) \Rightarrow a6])) (AHead a2 a4)
726 (AHead a0 (asucc g a1)) H4) in ((let H7 \def (f_equal A A (\lambda (e:
727 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a2 |
728 (AHead a6 _) \Rightarrow a6])) (AHead a2 a4) (AHead a0 (asucc g a1)) H4) in
729 (eq_ind A a0 (\lambda (a6: A).((eq A a4 (asucc g a1)) \to ((eq A (AHead a3
730 a5) (AHead a0 a1)) \to ((leq g a6 a3) \to ((leq g a4 a5) \to P))))) (\lambda
731 (H8: (eq A a4 (asucc g a1))).(eq_ind A (asucc g a1) (\lambda (a6: A).((eq A
732 (AHead a3 a5) (AHead a0 a1)) \to ((leq g a0 a3) \to ((leq g a6 a5) \to P))))
733 (\lambda (H9: (eq A (AHead a3 a5) (AHead a0 a1))).(let H10 \def (f_equal A A
734 (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
735 \Rightarrow a5 | (AHead _ a6) \Rightarrow a6])) (AHead a3 a5) (AHead a0 a1)
736 H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e in A return
737 (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a6 _)
738 \Rightarrow a6])) (AHead a3 a5) (AHead a0 a1) H9) in (eq_ind A a0 (\lambda
739 (a6: A).((eq A a5 a1) \to ((leq g a0 a6) \to ((leq g (asucc g a1) a5) \to
740 P)))) (\lambda (H12: (eq A a5 a1)).(eq_ind A a1 (\lambda (a6: A).((leq g a0
741 a0) \to ((leq g (asucc g a1) a6) \to P))) (\lambda (_: (leq g a0
742 a0)).(\lambda (H14: (leq g (asucc g a1) a1)).(H0 H14 P))) a5 (sym_eq A a5 a1
743 H12))) a3 (sym_eq A a3 a0 H11))) H10))) a4 (sym_eq A a4 (asucc g a1) H8))) a2
744 (sym_eq A a2 a0 H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead a0
745 (asucc g a1))) (refl_equal A (AHead a0 a1)))))))))) a)).
projects / helm.git / blob