1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 include "basic_1/leq/props.ma".
20 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g
21 (asucc g a1) (asucc g a2)))))
23 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
24 a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g (asucc g a) (asucc g
25 a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2:
26 nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g
27 (ASort h2 n2) k))).(nat_ind (\lambda (n: nat).((eq A (aplus g (ASort n n1) k)
28 (aplus g (ASort h2 n2) k)) \to (leq g (match n with [O \Rightarrow (ASort O
29 (next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow
30 (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) (\lambda (H1: (eq
31 A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda (n:
32 nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort n n2) k)) \to (leq g
33 (ASort O (next g n1)) (match n with [O \Rightarrow (ASort O (next g n2)) | (S
34 h) \Rightarrow (ASort h n2)])))) (\lambda (H2: (eq A (aplus g (ASort O n1) k)
35 (aplus g (ASort O n2) k))).(leq_sort g O O (next g n1) (next g n2) k (eq_ind
36 A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O
37 (next g n2)) k))) (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq
38 A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort O n2) k)
39 (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k))))
40 (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort O n1) k)
41 H2) (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)) (aplus g
42 (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))) (\lambda (h3:
43 nat).(\lambda (_: (((eq A (aplus g (ASort O n1) k) (aplus g (ASort h3 n2) k))
44 \to (leq g (ASort O (next g n1)) (match h3 with [O \Rightarrow (ASort O (next
45 g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H2: (eq A (aplus g
46 (ASort O n1) k) (aplus g (ASort (S h3) n2) k))).(leq_sort g O h3 (next g n1)
47 n2 k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g
48 (ASort h3 n2) k))) (eq_ind A (aplus g (ASort (S h3) n2) (S k)) (\lambda (a:
49 A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort (S h3)
50 n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort (S h3) n2)
51 k)))) (refl_equal A (asucc g (aplus g (ASort (S h3) n2) k))) (aplus g (ASort
52 O n1) k) H2) (aplus g (ASort h3 n2) k) (aplus_sort_S_S_simpl g n2 h3 k))
53 (aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))))) h2 H1))
54 (\lambda (h3: nat).(\lambda (IHh1: (((eq A (aplus g (ASort h3 n1) k) (aplus g
55 (ASort h2 n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next g
56 n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow (ASort
57 O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H1: (eq A
58 (aplus g (ASort (S h3) n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda
59 (n: nat).((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort n n2) k)) \to
60 ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort n n2) k)) \to (leq g
61 (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow
62 (ASort h n1)]) (match n with [O \Rightarrow (ASort O (next g n2)) | (S h)
63 \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match n with [O
64 \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))))
65 (\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort O n2)
66 k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort O n2) k))
67 \to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h)
68 \Rightarrow (ASort h n1)]) (ASort O (next g n2)))))).(leq_sort g h3 O n1
69 (next g n2) k (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A
70 (aplus g (ASort h3 n1) k) a)) (eq_ind A (aplus g (ASort (S h3) n1) (S k))
71 (\lambda (a: A).(eq A a (aplus g (ASort O n2) (S k)))) (eq_ind_r A (aplus g
72 (ASort O n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O
73 n2) k)))) (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort
74 (S h3) n1) k) H2) (aplus g (ASort h3 n1) k) (aplus_sort_S_S_simpl g n1 h3 k))
75 (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k))))) (\lambda
76 (h4: nat).(\lambda (_: (((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort
77 h4 n2) k)) \to ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort h4 n2) k))
78 \to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h)
79 \Rightarrow (ASort h n1)]) (match h4 with [O \Rightarrow (ASort O (next g
80 n2)) | (S h) \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match h4
81 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h
82 n2)])))))).(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort
83 (S h4) n2) k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g
84 (ASort (S h4) n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next
85 g n1)) | (S h) \Rightarrow (ASort h n1)]) (ASort h4 n2))))).(leq_sort g h3 h4
86 n1 n2 k (eq_ind A (aplus g (ASort (S h3) n1) (S k)) (\lambda (a: A).(eq A a
87 (aplus g (ASort h4 n2) k))) (eq_ind A (aplus g (ASort (S h4) n2) (S k))
88 (\lambda (a: A).(eq A (aplus g (ASort (S h3) n1) (S k)) a)) (eq_ind_r A
89 (aplus g (ASort (S h4) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g
90 (aplus g (ASort (S h4) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S
91 h4) n2) k))) (aplus g (ASort (S h3) n1) k) H2) (aplus g (ASort h4 n2) k)
92 (aplus_sort_S_S_simpl g n2 h4 k)) (aplus g (ASort h3 n1) k)
93 (aplus_sort_S_S_simpl g n1 h3 k))))))) h2 H1 IHh1)))) h1 H0))))))) (\lambda
94 (a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3 a4)).(\lambda (_: (leq g
95 (asucc g a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_:
96 (leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g
97 a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))).
100 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc
101 g a2)) \to (leq g a1 a2))))
103 \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
104 A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n:
105 nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g
106 (asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda
107 (n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0))
108 (asucc g (ASort n1 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort
109 n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))
110 (\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1
111 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g
112 (ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2)))) (\lambda (H1: (leq g
113 (asucc g (ASort O n0)) (asucc g (ASort O n2)))).(let H_x \def (leq_gen_sort1
114 g O (next g n0) (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind
115 nat nat nat (\lambda (n3: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A
116 (aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n3) k))))) (\lambda (n3:
117 nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort
118 h2 n3))))) (leq g (ASort O n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1:
119 nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0))
120 x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2))
121 (ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with
122 [(ASort n3 _) \Rightarrow n3 | (AHead _ _) \Rightarrow O])) (ASort O (next g
123 n2)) (ASort x1 x0) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match
124 e with [(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow ((match g with
125 [(mk_G next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0)
126 H4) in (\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda
127 (n3: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 x0)
128 x2))) H3 O H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n3: nat).(eq A
129 (aplus g (ASort O (next g n0)) x2) (aplus g (ASort O n3) x2))) H8 (next g n2)
130 H6) in (let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) (\lambda
131 (a: A).(eq A a (aplus g (ASort O (next g n2)) x2))) H9 (aplus g (ASort O n0)
132 (S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g
133 (ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2))
134 a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in
135 (leq_sort g O O n0 n2 (S x2) H11))))))) H5))))))) H2)))) (\lambda (n3:
136 nat).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2)))
137 \to (leq g (ASort O n0) (ASort n3 n2))))).(\lambda (H1: (leq g (asucc g
138 (ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H_x \def (leq_gen_sort1 g O
139 (next g n0) (ASort n3 n2) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat
140 (\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
141 O (next g n0)) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda
142 (h2: nat).(\lambda (_: nat).(eq A (ASort n3 n2) (ASort h2 n4))))) (leq g
143 (ASort O n0) (ASort (S n3) n2)) (\lambda (x0: nat).(\lambda (x1:
144 nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0))
145 x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n3 n2) (ASort x1
146 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort n4 _)
147 \Rightarrow n4 | (AHead _ _) \Rightarrow n3])) (ASort n3 n2) (ASort x1 x0)
148 H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort _
149 n4) \Rightarrow n4 | (AHead _ _) \Rightarrow n2])) (ASort n3 n2) (ASort x1
150 x0) H4) in (\lambda (H7: (eq nat n3 x1)).(let H8 \def (eq_ind_r nat x1
151 (\lambda (n4: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort
152 n4 x0) x2))) H3 n3 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n4:
153 nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 n4) x2))) H8
154 n2 H6) in (let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2)
155 (\lambda (a: A).(eq A a (aplus g (ASort n3 n2) x2))) H9 (aplus g (ASort O n0)
156 (S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g
157 (ASort n3 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) a)) H10
158 (aplus g (ASort (S n3) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n3 x2)) in
159 (leq_sort g O (S n3) n0 n2 (S x2) H11))))))) H5))))))) H2)))))) n1 H0))
160 (\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0)) (asucc g
161 (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda (H0: (leq
162 g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(nat_ind (\lambda
163 (n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to
164 ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort
165 n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 n2)))))
166 (\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort O
167 n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort O n2)))
168 \to (leq g (ASort n3 n0) (ASort O n2))))).(let H_x \def (leq_gen_sort1 g n3
169 n0 (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat
170 (\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
171 n3 n0) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda (h2:
172 nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort h2 n4))))) (leq g
173 (ASort (S n3) n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1:
174 nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort n3 n0) x2) (aplus
175 g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) (ASort x1
176 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort n4 _)
177 \Rightarrow n4 | (AHead _ _) \Rightarrow O])) (ASort O (next g n2)) (ASort x1
178 x0) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort
179 _ n4) \Rightarrow n4 | (AHead _ _) \Rightarrow ((match g with [(mk_G next _)
180 \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in (\lambda
181 (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n4: nat).(eq A
182 (aplus g (ASort n3 n0) x2) (aplus g (ASort n4 x0) x2))) H3 O H7) in (let H9
183 \def (eq_ind_r nat x0 (\lambda (n4: nat).(eq A (aplus g (ASort n3 n0) x2)
184 (aplus g (ASort O n4) x2))) H8 (next g n2) H6) in (let H10 \def (eq_ind_r A
185 (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g (ASort O (next g
186 n2)) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) (aplus_sort_S_S_simpl g n0
187 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g (ASort O (next g n2)) x2)
188 (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S x2)) a)) H10 (aplus g
189 (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in (leq_sort g (S n3) O
190 n0 n2 (S x2) H11))))))) H5))))))) H2))))) (\lambda (n4: nat).(\lambda (_:
191 (((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to ((((leq g
192 (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort n3 n0)
193 (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 n2)))))).(\lambda
194 (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S n4)
195 n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort (S n4)
196 n2))) \to (leq g (ASort n3 n0) (ASort (S n4) n2))))).(let H_x \def
197 (leq_gen_sort1 g n3 n0 (ASort n4 n2) H1) in (let H2 \def H_x in (ex2_3_ind
198 nat nat nat (\lambda (n5: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A
199 (aplus g (ASort n3 n0) k) (aplus g (ASort h2 n5) k))))) (\lambda (n5:
200 nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort n4 n2) (ASort h2
201 n5))))) (leq g (ASort (S n3) n0) (ASort (S n4) n2)) (\lambda (x0:
202 nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g
203 (ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n4
204 n2) (ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with
205 [(ASort n5 _) \Rightarrow n5 | (AHead _ _) \Rightarrow n4])) (ASort n4 n2)
206 (ASort x1 x0) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e
207 with [(ASort _ n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2])) (ASort n4
208 n2) (ASort x1 x0) H4) in (\lambda (H7: (eq nat n4 x1)).(let H8 \def (eq_ind_r
209 nat x1 (\lambda (n5: nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n5
210 x0) x2))) H3 n4 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n5: nat).(eq A
211 (aplus g (ASort n3 n0) x2) (aplus g (ASort n4 n5) x2))) H8 n2 H6) in (let H10
212 \def (eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g
213 (ASort n4 n2) x2))) H9 (aplus g (ASort (S n3) n0) (S x2))
214 (aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g
215 (ASort n4 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S x2))
216 a)) H10 (aplus g (ASort (S n4) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n4 x2))
217 in (leq_sort g (S n3) (S n4) n0 n2 (S x2) H11))))))) H5))))))) H2))))))) n1
218 H0 IHn)))) n H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n
219 n0)) (asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda
220 (H0: (((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0)
221 a0)))).(\lambda (H1: (leq g (asucc g (ASort n n0)) (asucc g (AHead a
222 a0)))).(nat_ind (\lambda (n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g
223 a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 n0))
224 (asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1
225 n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a a0))))))
226 (\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O
227 n0) a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq
228 g (ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g
229 (AHead a a0)))).(let H_x \def (leq_gen_sort1 g O (next g n0) (AHead a (asucc
230 g a0)) H4) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2:
231 nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort O (next g
232 n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2:
233 nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0)) (ASort h2 n2))))) (leq g
234 (ASort O n0) (AHead a a0)) (\lambda (x0: nat).(\lambda (x1: nat).(\lambda
235 (x2: nat).(\lambda (_: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g
236 (ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1
237 x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda (ee: A).(match
238 ee with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I
239 (ASort x1 x0) H7) in (False_ind (leq g (ASort O n0) (AHead a a0)) H8)))))))
240 H5)))))) (\lambda (n1: nat).(\lambda (_: (((((leq g (asucc g (ASort n1 n0))
241 (asucc g a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1
242 n0)) (asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort
243 n1 n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a
244 a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1) n0)) (asucc g a)) \to
245 (leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g (asucc g (ASort (S n1)
246 n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0) a0)))).(\lambda (H4: (leq g
247 (asucc g (ASort (S n1) n0)) (asucc g (AHead a a0)))).(let H_x \def
248 (leq_gen_sort1 g n1 n0 (AHead a (asucc g a0)) H4) in (let H5 \def H_x in
249 (ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k:
250 nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda
251 (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0))
252 (ASort h2 n2))))) (leq g (ASort (S n1) n0) (AHead a a0)) (\lambda (x0:
253 nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A (aplus g (ASort
254 n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g
255 a0)) (ASort x1 x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda
256 (ee: A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _)
257 \Rightarrow True])) I (ASort x1 x0) H7) in (False_ind (leq g (ASort (S n1)
258 n0) (AHead a a0)) H8))))))) H5)))))))) n H H0 H1)))))) a2)))) (\lambda (a:
259 A).(\lambda (_: ((\forall (a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq
260 g a a2))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g
261 a0) (asucc g a2)) \to (leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3:
262 A).((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))
263 (\lambda (n: nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a
264 a0)) (asucc g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g
265 (AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1
266 n0)))) (\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O
267 n0)))).(let H_x \def (leq_gen_head1 g a (asucc g a0) (ASort O (next g n0))
268 H2) in (let H3 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_:
269 A).(leq g a a3))) (\lambda (_: A).(\lambda (a4: A).(leq g (asucc g a0) a4)))
270 (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O (next g n0)) (AHead a3
271 a4)))) (leq g (AHead a a0) (ASort O n0)) (\lambda (x0: A).(\lambda (x1:
272 A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda
273 (H6: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H7 \def (eq_ind A
274 (ASort O (next g n0)) (\lambda (ee: A).(match ee with [(ASort _ _)
275 \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in
276 (False_ind (leq g (AHead a a0) (ASort O n0)) H7))))))) H3)))) (\lambda (n1:
277 nat).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0)))
278 \to (leq g (AHead a a0) (ASort n1 n0))))).(\lambda (H2: (leq g (asucc g
279 (AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H_x \def (leq_gen_head1 g a
280 (asucc g a0) (ASort n1 n0) H2) in (let H3 \def H_x in (ex3_2_ind A A (\lambda
281 (a3: A).(\lambda (_: A).(leq g a a3))) (\lambda (_: A).(\lambda (a4: A).(leq
282 g (asucc g a0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0)
283 (AHead a3 a4)))) (leq g (AHead a a0) (ASort (S n1) n0)) (\lambda (x0:
284 A).(\lambda (x1: A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g
285 a0) x1)).(\lambda (H6: (eq A (ASort n1 n0) (AHead x0 x1))).(let H7 \def
286 (eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee with [(ASort _ _)
287 \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in
288 (False_ind (leq g (AHead a a0) (ASort (S n1) n0)) H7))))))) H3)))))) n H1))))
289 (\lambda (a3: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a3))
290 \to (leq g (AHead a a0) a3)))).(\lambda (a4: A).(\lambda (_: (((leq g (asucc
291 g (AHead a a0)) (asucc g a4)) \to (leq g (AHead a a0) a4)))).(\lambda (H3:
292 (leq g (asucc g (AHead a a0)) (asucc g (AHead a3 a4)))).(let H_x \def
293 (leq_gen_head1 g a (asucc g a0) (AHead a3 (asucc g a4)) H3) in (let H4 \def
294 H_x in (ex3_2_ind A A (\lambda (a5: A).(\lambda (_: A).(leq g a a5)))
295 (\lambda (_: A).(\lambda (a6: A).(leq g (asucc g a0) a6))) (\lambda (a5:
296 A).(\lambda (a6: A).(eq A (AHead a3 (asucc g a4)) (AHead a5 a6)))) (leq g
297 (AHead a a0) (AHead a3 a4)) (\lambda (x0: A).(\lambda (x1: A).(\lambda (H5:
298 (leq g a x0)).(\lambda (H6: (leq g (asucc g a0) x1)).(\lambda (H7: (eq A
299 (AHead a3 (asucc g a4)) (AHead x0 x1))).(let H8 \def (f_equal A A (\lambda
300 (e: A).(match e with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow
301 a5])) (AHead a3 (asucc g a4)) (AHead x0 x1) H7) in ((let H9 \def (f_equal A A
302 (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow (asucc g a4) | (AHead
303 _ a5) \Rightarrow a5])) (AHead a3 (asucc g a4)) (AHead x0 x1) H7) in (\lambda
304 (H10: (eq A a3 x0)).(let H11 \def (eq_ind_r A x1 (\lambda (a5: A).(leq g
305 (asucc g a0) a5)) H6 (asucc g a4) H9) in (let H12 \def (eq_ind_r A x0
306 (\lambda (a5: A).(leq g a a5)) H5 a3 H10) in (leq_head g a a3 H12 a0 a4 (H0
307 a4 H11)))))) H8))))))) H4)))))))) a2)))))) a1)).
310 \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g
313 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(ex A (\lambda (a1:
314 A).(leq g a0 (asucc g a1))))) (\lambda (n: nat).(\lambda (n0: nat).(ex_intro
315 A (\lambda (a0: A).(leq g (ASort n n0) (asucc g a0))) (ASort (S n) n0)
316 (leq_refl g (ASort n n0))))) (\lambda (a0: A).(\lambda (_: (ex A (\lambda
317 (a1: A).(leq g a0 (asucc g a1))))).(\lambda (a1: A).(\lambda (H0: (ex A
318 (\lambda (a2: A).(leq g a1 (asucc g a2))))).(let H1 \def H0 in (ex_ind A
319 (\lambda (a2: A).(leq g a1 (asucc g a2))) (ex A (\lambda (a2: A).(leq g
320 (AHead a0 a1) (asucc g a2)))) (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc
321 g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2)))
322 (AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1))))))
325 theorem leq_ahead_asucc_false:
326 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2)
327 (asucc g a1)) \to (\forall (P: Prop).P))))
329 \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
330 A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda
331 (n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead
332 (ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
333 \Rightarrow (ASort h n0)]))).(\lambda (P: Prop).(nat_ind (\lambda (n1:
334 nat).((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O
335 (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P)) (\lambda (H0: (leq g
336 (AHead (ASort O n0) a2) (ASort O (next g n0)))).(let H_x \def (leq_gen_head1
337 g (ASort O n0) a2 (ASort O (next g n0)) H0) in (let H1 \def H_x in (ex3_2_ind
338 A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_:
339 A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A
340 (ASort O (next g n0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1:
341 A).(\lambda (_: (leq g (ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda
342 (H4: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H5 \def (eq_ind A
343 (ASort O (next g n0)) (\lambda (ee: A).(match ee with [(ASort _ _)
344 \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in
345 (False_ind P H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead
346 (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h)
347 \Rightarrow (ASort h n0)])) \to P))).(\lambda (H0: (leq g (AHead (ASort (S
348 n1) n0) a2) (ASort n1 n0))).(let H_x \def (leq_gen_head1 g (ASort (S n1) n0)
349 a2 (ASort n1 n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3:
350 A).(\lambda (_: A).(leq g (ASort (S n1) n0) a3))) (\lambda (_: A).(\lambda
351 (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0)
352 (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g
353 (ASort (S n1) n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort
354 n1 n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort n1 n0) (\lambda (ee:
355 A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
356 False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n H))))))
357 (\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a a2) (asucc g
358 a)) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall
359 (a2: A).((leq g (AHead a0 a2) (asucc g a0)) \to (\forall (P:
360 Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead (AHead a a0) a2)
361 (AHead a (asucc g a0)))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g
362 (AHead a a0) a2 (AHead a (asucc g a0)) H1) in (let H2 \def H_x in (ex3_2_ind
363 A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) (\lambda (_:
364 A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A
365 (AHead a (asucc g a0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1:
366 A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2
367 x1)).(\lambda (H5: (eq A (AHead a (asucc g a0)) (AHead x0 x1))).(let H6 \def
368 (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a |
369 (AHead a3 _) \Rightarrow a3])) (AHead a (asucc g a0)) (AHead x0 x1) H5) in
370 ((let H7 \def (f_equal A A (\lambda (e: A).(match e with [(ASort _ _)
371 \Rightarrow (asucc g a0) | (AHead _ a3) \Rightarrow a3])) (AHead a (asucc g
372 a0)) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def (eq_ind_r A
373 x1 (\lambda (a3: A).(leq g a2 a3)) H4 (asucc g a0) H7) in (let H10 \def
374 (eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a a0) a3)) H3 a H8) in
375 (leq_ahead_false_1 g a a0 H10 P))))) H6))))))) H2)))))))))) a1)).
377 theorem leq_asucc_false:
378 \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P:
381 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0)
382 a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda
383 (H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
384 \Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).(nat_ind
385 (\lambda (n1: nat).((leq g (match n1 with [O \Rightarrow (ASort O (next g
386 n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P)) (\lambda (H0:
387 (leq g (ASort O (next g n0)) (ASort O n0))).(let H_x \def (leq_gen_sort1 g O
388 (next g n0) (ASort O n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat
389 (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
390 O (next g n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda
391 (h2: nat).(\lambda (_: nat).(eq A (ASort O n0) (ASort h2 n2))))) P (\lambda
392 (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g
393 (ASort O (next g n0)) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A
394 (ASort O n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e:
395 A).(match e with [(ASort n1 _) \Rightarrow n1 | (AHead _ _) \Rightarrow O]))
396 (ASort O n0) (ASort x1 x0) H3) in ((let H5 \def (f_equal A nat (\lambda (e:
397 A).(match e with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) \Rightarrow n0]))
398 (ASort O n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat O x1)).(let H7 \def
399 (eq_ind_r nat x1 (\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2)
400 (aplus g (ASort n1 x0) x2))) H2 O H6) in (let H8 \def (eq_ind_r nat x0
401 (\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort O
402 n1) x2))) H7 n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort O (next g
403 n0)) x2) (\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) x2))) H8 (aplus g
404 (ASort O n0) (S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H_y \def
405 (aplus_inj g (S x2) x2 (ASort O n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2
406 (\lambda (n1: nat).(le n1 x2)) (le_n x2) (S x2) H_y) P))))))) H4)))))))
407 H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow
408 (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to
409 P))).(\lambda (H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let H_x \def
410 (leq_gen_sort1 g n1 n0 (ASort (S n1) n0) H0) in (let H1 \def H_x in
411 (ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k:
412 nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda
413 (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort (S n1) n0) (ASort
414 h2 n2))))) P (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2:
415 nat).(\lambda (H2: (eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0)
416 x2))).(\lambda (H3: (eq A (ASort (S n1) n0) (ASort x1 x0))).(let H4 \def
417 (f_equal A nat (\lambda (e: A).(match e with [(ASort n2 _) \Rightarrow n2 |
418 (AHead _ _) \Rightarrow (S n1)])) (ASort (S n1) n0) (ASort x1 x0) H3) in
419 ((let H5 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort _ n2)
420 \Rightarrow n2 | (AHead _ _) \Rightarrow n0])) (ASort (S n1) n0) (ASort x1
421 x0) H3) in (\lambda (H6: (eq nat (S n1) x1)).(let H7 \def (eq_ind_r nat x1
422 (\lambda (n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort n2 x0)
423 x2))) H2 (S n1) H6) in (let H8 \def (eq_ind_r nat x0 (\lambda (n2: nat).(eq A
424 (aplus g (ASort n1 n0) x2) (aplus g (ASort (S n1) n2) x2))) H7 n0 H5) in (let
425 H9 \def (eq_ind_r A (aplus g (ASort n1 n0) x2) (\lambda (a0: A).(eq A a0
426 (aplus g (ASort (S n1) n0) x2))) H8 (aplus g (ASort (S n1) n0) (S x2))
427 (aplus_sort_S_S_simpl g n0 n1 x2)) in (let H_y \def (aplus_inj g (S x2) x2
428 (ASort (S n1) n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n2: nat).(le
429 n2 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))))) n H))))) (\lambda
430 (a0: A).(\lambda (_: (((leq g (asucc g a0) a0) \to (\forall (P:
431 Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g a1) a1) \to
432 (\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g a1)) (AHead
433 a0 a1))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g a0 (asucc g a1)
434 (AHead a0 a1) H1) in (let H2 \def H_x in (ex3_2_ind A A (\lambda (a3:
435 A).(\lambda (_: A).(leq g a0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g
436 (asucc g a1) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a0 a1)
437 (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (H3: (leq g a0
438 x0)).(\lambda (H4: (leq g (asucc g a1) x1)).(\lambda (H5: (eq A (AHead a0 a1)
439 (AHead x0 x1))).(let H6 \def (f_equal A A (\lambda (e: A).(match e with
440 [(ASort _ _) \Rightarrow a0 | (AHead a2 _) \Rightarrow a2])) (AHead a0 a1)
441 (AHead x0 x1) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e with
442 [(ASort _ _) \Rightarrow a1 | (AHead _ a2) \Rightarrow a2])) (AHead a0 a1)
443 (AHead x0 x1) H5) in (\lambda (H8: (eq A a0 x0)).(let H9 \def (eq_ind_r A x1
444 (\lambda (a2: A).(leq g (asucc g a1) a2)) H4 a1 H7) in (let H10 \def
445 (eq_ind_r A x0 (\lambda (a2: A).(leq g a0 a2)) H3 a0 H8) in (H0 H9 P)))))
446 H6))))))) H2))))))))) a)).