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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/defs.ma".
19 include "aplus/props.ma".
21 theorem ahead_inj_snd:
22 \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall
23 (a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4))))))
25 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda
26 (a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let H0 \def (match
27 H in leq return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a
28 a0)).((eq A a (AHead a1 a2)) \to ((eq A a0 (AHead a3 a4)) \to (leq g a2
29 a4)))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A
30 (ASort h1 n1) (AHead a1 a2))).(\lambda (H2: (eq A (ASort h2 n2) (AHead a3
31 a4))).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A
32 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
33 \Rightarrow False])) I (AHead a1 a2) H1) in (False_ind ((eq A (ASort h2 n2)
34 (AHead a3 a4)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
35 k)) \to (leq g a2 a4))) H3)) H2 H0))) | (leq_head a0 a5 H0 a6 a7 H1)
36 \Rightarrow (\lambda (H2: (eq A (AHead a0 a6) (AHead a1 a2))).(\lambda (H3:
37 (eq A (AHead a5 a7) (AHead a3 a4))).((let H4 \def (f_equal A A (\lambda (e:
38 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a6 |
39 (AHead _ a) \Rightarrow a])) (AHead a0 a6) (AHead a1 a2) H2) in ((let H5 \def
40 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
41 [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a6)
42 (AHead a1 a2) H2) in (eq_ind A a1 (\lambda (a: A).((eq A a6 a2) \to ((eq A
43 (AHead a5 a7) (AHead a3 a4)) \to ((leq g a a5) \to ((leq g a6 a7) \to (leq g
44 a2 a4)))))) (\lambda (H6: (eq A a6 a2)).(eq_ind A a2 (\lambda (a: A).((eq A
45 (AHead a5 a7) (AHead a3 a4)) \to ((leq g a1 a5) \to ((leq g a a7) \to (leq g
46 a2 a4))))) (\lambda (H7: (eq A (AHead a5 a7) (AHead a3 a4))).(let H8 \def
47 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
48 [(ASort _ _) \Rightarrow a7 | (AHead _ a) \Rightarrow a])) (AHead a5 a7)
49 (AHead a3 a4) H7) in ((let H9 \def (f_equal A A (\lambda (e: A).(match e in A
50 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead a _)
51 \Rightarrow a])) (AHead a5 a7) (AHead a3 a4) H7) in (eq_ind A a3 (\lambda (a:
52 A).((eq A a7 a4) \to ((leq g a1 a) \to ((leq g a2 a7) \to (leq g a2 a4)))))
53 (\lambda (H10: (eq A a7 a4)).(eq_ind A a4 (\lambda (a: A).((leq g a1 a3) \to
54 ((leq g a2 a) \to (leq g a2 a4)))) (\lambda (_: (leq g a1 a3)).(\lambda (H12:
55 (leq g a2 a4)).H12)) a7 (sym_eq A a7 a4 H10))) a5 (sym_eq A a5 a3 H9))) H8)))
56 a6 (sym_eq A a6 a2 H6))) a0 (sym_eq A a0 a1 H5))) H4)) H3 H0 H1)))]) in (H0
57 (refl_equal A (AHead a1 a2)) (refl_equal A (AHead a3 a4))))))))).
60 \forall (g: G).(\forall (a: A).(leq g a a))
62 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(leq g a0 a0))
63 (\lambda (n: nat).(\lambda (n0: nat).(leq_sort g n n n0 n0 O (refl_equal A
64 (aplus g (ASort n n0) O))))) (\lambda (a0: A).(\lambda (H: (leq g a0
65 a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1
69 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1
72 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1
73 a2)).(eq_ind_r A a2 (\lambda (a: A).(leq g a a2)) (leq_refl g a2) a1 H)))).
76 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g
79 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
80 a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g a0 a))) (\lambda (h1:
81 nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k:
82 nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
83 k))).(leq_sort g h2 h1 n2 n1 k (sym_eq A (aplus g (ASort h1 n1) k) (aplus g
84 (ASort h2 n2) k) H0)))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_:
85 (leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5: A).(\lambda (a6:
86 A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3
87 H1 a6 a5 H3))))))))) a1 a2 H)))).
90 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
91 (a3: A).((leq g a2 a3) \to (leq g a1 a3))))))
93 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
94 a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0
95 a3) \to (leq g a a3))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1:
96 nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort
97 h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (a3: A).(\lambda (H1: (leq g
98 (ASort h2 n2) a3)).(let H2 \def (match H1 in leq return (\lambda (a:
99 A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort h2 n2)) \to
100 ((eq A a0 a3) \to (leq g (ASort h1 n1) a3)))))) with [(leq_sort h0 h3 n0 n3
101 k0 H2) \Rightarrow (\lambda (H3: (eq A (ASort h0 n0) (ASort h2 n2))).(\lambda
102 (H4: (eq A (ASort h3 n3) a3)).((let H5 \def (f_equal A nat (\lambda (e:
103 A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n
104 | (AHead _ _) \Rightarrow n0])) (ASort h0 n0) (ASort h2 n2) H3) in ((let H6
105 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat)
106 with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0)
107 (ASort h2 n2) H3) in (eq_ind nat h2 (\lambda (n: nat).((eq nat n0 n2) \to
108 ((eq A (ASort h3 n3) a3) \to ((eq A (aplus g (ASort n n0) k0) (aplus g (ASort
109 h3 n3) k0)) \to (leq g (ASort h1 n1) a3))))) (\lambda (H7: (eq nat n0
110 n2)).(eq_ind nat n2 (\lambda (n: nat).((eq A (ASort h3 n3) a3) \to ((eq A
111 (aplus g (ASort h2 n) k0) (aplus g (ASort h3 n3) k0)) \to (leq g (ASort h1
112 n1) a3)))) (\lambda (H8: (eq A (ASort h3 n3) a3)).(eq_ind A (ASort h3 n3)
113 (\lambda (a: A).((eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0))
114 \to (leq g (ASort h1 n1) a))) (\lambda (H9: (eq A (aplus g (ASort h2 n2) k0)
115 (aplus g (ASort h3 n3) k0))).(lt_le_e k k0 (leq g (ASort h1 n1) (ASort h3
116 n3)) (\lambda (H10: (lt k k0)).(let H_y \def (aplus_reg_r g (ASort h1 n1)
117 (ASort h2 n2) k k H0 (minus k0 k)) in (let H11 \def (eq_ind_r nat (plus
118 (minus k0 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n) (aplus g
119 (ASort h2 n2) n))) H_y k0 (le_plus_minus_sym k k0 (le_trans k (S k) k0 (le_S
120 k k (le_n k)) H10))) in (leq_sort g h1 h3 n1 n3 k0 (trans_eq A (aplus g
121 (ASort h1 n1) k0) (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0) H11
122 H9))))) (\lambda (H10: (le k0 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2)
123 (ASort h3 n3) k0 k0 H9 (minus k k0)) in (let H11 \def (eq_ind_r nat (plus
124 (minus k k0) k0) (\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g
125 (ASort h3 n3) n))) H_y k (le_plus_minus_sym k0 k H10)) in (leq_sort g h1 h3
126 n1 n3 k (trans_eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)
127 (aplus g (ASort h3 n3) k) H0 H11))))))) a3 H8)) n0 (sym_eq nat n0 n2 H7))) h0
128 (sym_eq nat h0 h2 H6))) H5)) H4 H2))) | (leq_head a0 a4 H2 a5 a6 H3)
129 \Rightarrow (\lambda (H4: (eq A (AHead a0 a5) (ASort h2 n2))).(\lambda (H5:
130 (eq A (AHead a4 a6) a3)).((let H6 \def (eq_ind A (AHead a0 a5) (\lambda (e:
131 A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow
132 False | (AHead _ _) \Rightarrow True])) I (ASort h2 n2) H4) in (False_ind
133 ((eq A (AHead a4 a6) a3) \to ((leq g a0 a4) \to ((leq g a5 a6) \to (leq g
134 (ASort h1 n1) a3)))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (ASort h2 n2))
135 (refl_equal A a3))))))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_:
136 (leq g a3 a4)).(\lambda (H1: ((\forall (a5: A).((leq g a4 a5) \to (leq g a3
137 a5))))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: (leq g a5
138 a6)).(\lambda (H3: ((\forall (a7: A).((leq g a6 a7) \to (leq g a5
139 a7))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6) a0)).(let H5 \def
140 (match H4 in leq return (\lambda (a: A).(\lambda (a7: A).(\lambda (_: (leq ?
141 a a7)).((eq A a (AHead a4 a6)) \to ((eq A a7 a0) \to (leq g (AHead a3 a5)
142 a0)))))) with [(leq_sort h1 h2 n1 n2 k H5) \Rightarrow (\lambda (H6: (eq A
143 (ASort h1 n1) (AHead a4 a6))).(\lambda (H7: (eq A (ASort h2 n2) a0)).((let H8
144 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A return (\lambda
145 (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
146 False])) I (AHead a4 a6) H6) in (False_ind ((eq A (ASort h2 n2) a0) \to ((eq
147 A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (AHead a3
148 a5) a0))) H8)) H7 H5))) | (leq_head a7 a8 H5 a9 a10 H6) \Rightarrow (\lambda
149 (H7: (eq A (AHead a7 a9) (AHead a4 a6))).(\lambda (H8: (eq A (AHead a8 a10)
150 a0)).((let H9 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda
151 (_: A).A) with [(ASort _ _) \Rightarrow a9 | (AHead _ a) \Rightarrow a]))
152 (AHead a7 a9) (AHead a4 a6) H7) in ((let H10 \def (f_equal A A (\lambda (e:
153 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 |
154 (AHead a _) \Rightarrow a])) (AHead a7 a9) (AHead a4 a6) H7) in (eq_ind A a4
155 (\lambda (a: A).((eq A a9 a6) \to ((eq A (AHead a8 a10) a0) \to ((leq g a a8)
156 \to ((leq g a9 a10) \to (leq g (AHead a3 a5) a0)))))) (\lambda (H11: (eq A a9
157 a6)).(eq_ind A a6 (\lambda (a: A).((eq A (AHead a8 a10) a0) \to ((leq g a4
158 a8) \to ((leq g a a10) \to (leq g (AHead a3 a5) a0))))) (\lambda (H12: (eq A
159 (AHead a8 a10) a0)).(eq_ind A (AHead a8 a10) (\lambda (a: A).((leq g a4 a8)
160 \to ((leq g a6 a10) \to (leq g (AHead a3 a5) a)))) (\lambda (H13: (leq g a4
161 a8)).(\lambda (H14: (leq g a6 a10)).(leq_head g a3 a8 (H1 a8 H13) a5 a10 (H3
162 a10 H14)))) a0 H12)) a9 (sym_eq A a9 a6 H11))) a7 (sym_eq A a7 a4 H10))) H9))
163 H8 H5 H6)))]) in (H5 (refl_equal A (AHead a4 a6)) (refl_equal A
164 a0))))))))))))) a1 a2 H)))).
166 theorem leq_ahead_false_1:
167 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1)
168 \to (\forall (P: Prop).P))))
170 \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
171 A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n:
172 nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n
173 n0) a2) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g
174 (AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead
175 (ASort O n0) a2) (ASort O n0))).(let H1 \def (match H0 in leq return (\lambda
176 (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (AHead (ASort O
177 n0) a2)) \to ((eq A a0 (ASort O n0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k
178 H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n1) (AHead (ASort O n0)
179 a2))).(\lambda (H3: (eq A (ASort h2 n2) (ASort O n0))).((let H4 \def (eq_ind
180 A (ASort h1 n1) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop)
181 with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I
182 (AHead (ASort O n0) a2) H2) in (False_ind ((eq A (ASort h2 n2) (ASort O n0))
183 \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H4))
184 H3 H1))) | (leq_head a0 a3 H1 a4 a5 H2) \Rightarrow (\lambda (H3: (eq A
185 (AHead a0 a4) (AHead (ASort O n0) a2))).(\lambda (H4: (eq A (AHead a3 a5)
186 (ASort O n0))).((let H5 \def (f_equal A A (\lambda (e: A).(match e in A
187 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a)
188 \Rightarrow a])) (AHead a0 a4) (AHead (ASort O n0) a2) H3) in ((let H6 \def
189 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
190 [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a4)
191 (AHead (ASort O n0) a2) H3) in (eq_ind A (ASort O n0) (\lambda (a: A).((eq A
192 a4 a2) \to ((eq A (AHead a3 a5) (ASort O n0)) \to ((leq g a a3) \to ((leq g
193 a4 a5) \to P))))) (\lambda (H7: (eq A a4 a2)).(eq_ind A a2 (\lambda (a:
194 A).((eq A (AHead a3 a5) (ASort O n0)) \to ((leq g (ASort O n0) a3) \to ((leq
195 g a a5) \to P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort O n0))).(let H9
196 \def (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda
197 (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
198 True])) I (ASort O n0) H8) in (False_ind ((leq g (ASort O n0) a3) \to ((leq g
199 a2 a5) \to P)) H9))) a4 (sym_eq A a4 a2 H7))) a0 (sym_eq A a0 (ASort O n0)
200 H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead (ASort O n0) a2))
201 (refl_equal A (ASort O n0))))) (\lambda (n1: nat).(\lambda (_: (((leq g
202 (AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead
203 (ASort (S n1) n0) a2) (ASort (S n1) n0))).(let H1 \def (match H0 in leq
204 return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a
205 (AHead (ASort (S n1) n0) a2)) \to ((eq A a0 (ASort (S n1) n0)) \to P)))))
206 with [(leq_sort h1 h2 n2 n3 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1
207 n2) (AHead (ASort (S n1) n0) a2))).(\lambda (H3: (eq A (ASort h2 n3) (ASort
208 (S n1) n0))).((let H4 \def (eq_ind A (ASort h1 n2) (\lambda (e: A).(match e
209 in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead
210 _ _) \Rightarrow False])) I (AHead (ASort (S n1) n0) a2) H2) in (False_ind
211 ((eq A (ASort h2 n3) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort h1 n2) k)
212 (aplus g (ASort h2 n3) k)) \to P)) H4)) H3 H1))) | (leq_head a0 a3 H1 a4 a5
213 H2) \Rightarrow (\lambda (H3: (eq A (AHead a0 a4) (AHead (ASort (S n1) n0)
214 a2))).(\lambda (H4: (eq A (AHead a3 a5) (ASort (S n1) n0))).((let H5 \def
215 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
216 [(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a0 a4)
217 (AHead (ASort (S n1) n0) a2) H3) in ((let H6 \def (f_equal A A (\lambda (e:
218 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 |
219 (AHead a _) \Rightarrow a])) (AHead a0 a4) (AHead (ASort (S n1) n0) a2) H3)
220 in (eq_ind A (ASort (S n1) n0) (\lambda (a: A).((eq A a4 a2) \to ((eq A
221 (AHead a3 a5) (ASort (S n1) n0)) \to ((leq g a a3) \to ((leq g a4 a5) \to
222 P))))) (\lambda (H7: (eq A a4 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead
223 a3 a5) (ASort (S n1) n0)) \to ((leq g (ASort (S n1) n0) a3) \to ((leq g a a5)
224 \to P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort (S n1) n0))).(let H9 \def
225 (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda (_:
226 A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
227 True])) I (ASort (S n1) n0) H8) in (False_ind ((leq g (ASort (S n1) n0) a3)
228 \to ((leq g a2 a5) \to P)) H9))) a4 (sym_eq A a4 a2 H7))) a0 (sym_eq A a0
229 (ASort (S n1) n0) H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead (ASort
230 (S n1) n0) a2)) (refl_equal A (ASort (S n1) n0))))))) n H)))))) (\lambda (a:
231 A).(\lambda (H: ((\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P:
232 Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead
233 a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq
234 g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P: Prop).(let H2 \def
235 (match H1 in leq return (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: (leq ?
236 a3 a4)).((eq A a3 (AHead (AHead a a0) a2)) \to ((eq A a4 (AHead a a0)) \to
237 P))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A
238 (ASort h1 n1) (AHead (AHead a a0) a2))).(\lambda (H4: (eq A (ASort h2 n2)
239 (AHead a a0))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e
240 in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead
241 _ _) \Rightarrow False])) I (AHead (AHead a a0) a2) H3) in (False_ind ((eq A
242 (ASort h2 n2) (AHead a a0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g
243 (ASort h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a3 a4 H2 a5 a6 H3)
244 \Rightarrow (\lambda (H4: (eq A (AHead a3 a5) (AHead (AHead a a0)
245 a2))).(\lambda (H5: (eq A (AHead a4 a6) (AHead a a0))).((let H6 \def (f_equal
246 A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
247 \Rightarrow a5 | (AHead _ a7) \Rightarrow a7])) (AHead a3 a5) (AHead (AHead a
248 a0) a2) H4) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A
249 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a7 _)
250 \Rightarrow a7])) (AHead a3 a5) (AHead (AHead a a0) a2) H4) in (eq_ind A
251 (AHead a a0) (\lambda (a7: A).((eq A a5 a2) \to ((eq A (AHead a4 a6) (AHead a
252 a0)) \to ((leq g a7 a4) \to ((leq g a5 a6) \to P))))) (\lambda (H8: (eq A a5
253 a2)).(eq_ind A a2 (\lambda (a7: A).((eq A (AHead a4 a6) (AHead a a0)) \to
254 ((leq g (AHead a a0) a4) \to ((leq g a7 a6) \to P)))) (\lambda (H9: (eq A
255 (AHead a4 a6) (AHead a a0))).(let H10 \def (f_equal A A (\lambda (e:
256 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a6 |
257 (AHead _ a7) \Rightarrow a7])) (AHead a4 a6) (AHead a a0) H9) in ((let H11
258 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
259 with [(ASort _ _) \Rightarrow a4 | (AHead a7 _) \Rightarrow a7])) (AHead a4
260 a6) (AHead a a0) H9) in (eq_ind A a (\lambda (a7: A).((eq A a6 a0) \to ((leq
261 g (AHead a a0) a7) \to ((leq g a2 a6) \to P)))) (\lambda (H12: (eq A a6
262 a0)).(eq_ind A a0 (\lambda (a7: A).((leq g (AHead a a0) a) \to ((leq g a2 a7)
263 \to P))) (\lambda (H13: (leq g (AHead a a0) a)).(\lambda (_: (leq g a2
264 a0)).(H a0 H13 P))) a6 (sym_eq A a6 a0 H12))) a4 (sym_eq A a4 a H11))) H10)))
265 a5 (sym_eq A a5 a2 H8))) a3 (sym_eq A a3 (AHead a a0) H7))) H6)) H5 H2
266 H3)))]) in (H2 (refl_equal A (AHead (AHead a a0) a2)) (refl_equal A (AHead a
269 theorem leq_ahead_false_2:
270 \forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2)
271 \to (\forall (P: Prop).P))))
273 \lambda (g: G).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (a1:
274 A).((leq g (AHead a1 a) a) \to (\forall (P: Prop).P)))) (\lambda (n:
275 nat).(\lambda (n0: nat).(\lambda (a1: A).(\lambda (H: (leq g (AHead a1 (ASort
276 n n0)) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g
277 (AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead
278 a1 (ASort O n0)) (ASort O n0))).(let H1 \def (match H0 in leq return (\lambda
279 (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (AHead a1 (ASort
280 O n0))) \to ((eq A a0 (ASort O n0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k
281 H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n1) (AHead a1 (ASort O
282 n0)))).(\lambda (H3: (eq A (ASort h2 n2) (ASort O n0))).((let H4 \def (eq_ind
283 A (ASort h1 n1) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop)
284 with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I
285 (AHead a1 (ASort O n0)) H2) in (False_ind ((eq A (ASort h2 n2) (ASort O n0))
286 \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H4))
287 H3 H1))) | (leq_head a0 a3 H1 a4 a5 H2) \Rightarrow (\lambda (H3: (eq A
288 (AHead a0 a4) (AHead a1 (ASort O n0)))).(\lambda (H4: (eq A (AHead a3 a5)
289 (ASort O n0))).((let H5 \def (f_equal A A (\lambda (e: A).(match e in A
290 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a)
291 \Rightarrow a])) (AHead a0 a4) (AHead a1 (ASort O n0)) H3) in ((let H6 \def
292 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
293 [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a4)
294 (AHead a1 (ASort O n0)) H3) in (eq_ind A a1 (\lambda (a: A).((eq A a4 (ASort
295 O n0)) \to ((eq A (AHead a3 a5) (ASort O n0)) \to ((leq g a a3) \to ((leq g
296 a4 a5) \to P))))) (\lambda (H7: (eq A a4 (ASort O n0))).(eq_ind A (ASort O
297 n0) (\lambda (a: A).((eq A (AHead a3 a5) (ASort O n0)) \to ((leq g a1 a3) \to
298 ((leq g a a5) \to P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort O n0))).(let
299 H9 \def (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda
300 (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
301 True])) I (ASort O n0) H8) in (False_ind ((leq g a1 a3) \to ((leq g (ASort O
302 n0) a5) \to P)) H9))) a4 (sym_eq A a4 (ASort O n0) H7))) a0 (sym_eq A a0 a1
303 H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead a1 (ASort O n0)))
304 (refl_equal A (ASort O n0))))) (\lambda (n1: nat).(\lambda (_: (((leq g
305 (AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead
306 a1 (ASort (S n1) n0)) (ASort (S n1) n0))).(let H1 \def (match H0 in leq
307 return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a
308 (AHead a1 (ASort (S n1) n0))) \to ((eq A a0 (ASort (S n1) n0)) \to P)))))
309 with [(leq_sort h1 h2 n2 n3 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1
310 n2) (AHead a1 (ASort (S n1) n0)))).(\lambda (H3: (eq A (ASort h2 n3) (ASort
311 (S n1) n0))).((let H4 \def (eq_ind A (ASort h1 n2) (\lambda (e: A).(match e
312 in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead
313 _ _) \Rightarrow False])) I (AHead a1 (ASort (S n1) n0)) H2) in (False_ind
314 ((eq A (ASort h2 n3) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort h1 n2) k)
315 (aplus g (ASort h2 n3) k)) \to P)) H4)) H3 H1))) | (leq_head a0 a3 H1 a4 a5
316 H2) \Rightarrow (\lambda (H3: (eq A (AHead a0 a4) (AHead a1 (ASort (S n1)
317 n0)))).(\lambda (H4: (eq A (AHead a3 a5) (ASort (S n1) n0))).((let H5 \def
318 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
319 [(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a0 a4)
320 (AHead a1 (ASort (S n1) n0)) H3) in ((let H6 \def (f_equal A A (\lambda (e:
321 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 |
322 (AHead a _) \Rightarrow a])) (AHead a0 a4) (AHead a1 (ASort (S n1) n0)) H3)
323 in (eq_ind A a1 (\lambda (a: A).((eq A a4 (ASort (S n1) n0)) \to ((eq A
324 (AHead a3 a5) (ASort (S n1) n0)) \to ((leq g a a3) \to ((leq g a4 a5) \to
325 P))))) (\lambda (H7: (eq A a4 (ASort (S n1) n0))).(eq_ind A (ASort (S n1) n0)
326 (\lambda (a: A).((eq A (AHead a3 a5) (ASort (S n1) n0)) \to ((leq g a1 a3)
327 \to ((leq g a a5) \to P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort (S n1)
328 n0))).(let H9 \def (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A
329 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
330 _) \Rightarrow True])) I (ASort (S n1) n0) H8) in (False_ind ((leq g a1 a3)
331 \to ((leq g (ASort (S n1) n0) a5) \to P)) H9))) a4 (sym_eq A a4 (ASort (S n1)
332 n0) H7))) a0 (sym_eq A a0 a1 H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A
333 (AHead a1 (ASort (S n1) n0))) (refl_equal A (ASort (S n1) n0))))))) n H))))))
334 (\lambda (a: A).(\lambda (_: ((\forall (a1: A).((leq g (AHead a1 a) a) \to
335 (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a1:
336 A).((leq g (AHead a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1:
337 A).(\lambda (H1: (leq g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P:
338 Prop).(let H2 \def (match H1 in leq return (\lambda (a3: A).(\lambda (a4:
339 A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (AHead a1 (AHead a a0))) \to ((eq A
340 a4 (AHead a a0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow
341 (\lambda (H3: (eq A (ASort h1 n1) (AHead a1 (AHead a a0)))).(\lambda (H4: (eq
342 A (ASort h2 n2) (AHead a a0))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda
343 (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
344 \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a1 (AHead a a0))
345 H3) in (False_ind ((eq A (ASort h2 n2) (AHead a a0)) \to ((eq A (aplus g
346 (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head
347 a3 a4 H2 a5 a6 H3) \Rightarrow (\lambda (H4: (eq A (AHead a3 a5) (AHead a1
348 (AHead a a0)))).(\lambda (H5: (eq A (AHead a4 a6) (AHead a a0))).((let H6
349 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
350 with [(ASort _ _) \Rightarrow a5 | (AHead _ a7) \Rightarrow a7])) (AHead a3
351 a5) (AHead a1 (AHead a a0)) H4) in ((let H7 \def (f_equal A A (\lambda (e:
352 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 |
353 (AHead a7 _) \Rightarrow a7])) (AHead a3 a5) (AHead a1 (AHead a a0)) H4) in
354 (eq_ind A a1 (\lambda (a7: A).((eq A a5 (AHead a a0)) \to ((eq A (AHead a4
355 a6) (AHead a a0)) \to ((leq g a7 a4) \to ((leq g a5 a6) \to P))))) (\lambda
356 (H8: (eq A a5 (AHead a a0))).(eq_ind A (AHead a a0) (\lambda (a7: A).((eq A
357 (AHead a4 a6) (AHead a a0)) \to ((leq g a1 a4) \to ((leq g a7 a6) \to P))))
358 (\lambda (H9: (eq A (AHead a4 a6) (AHead a a0))).(let H10 \def (f_equal A A
359 (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
360 \Rightarrow a6 | (AHead _ a7) \Rightarrow a7])) (AHead a4 a6) (AHead a a0)
361 H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e in A return
362 (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead a7 _)
363 \Rightarrow a7])) (AHead a4 a6) (AHead a a0) H9) in (eq_ind A a (\lambda (a7:
364 A).((eq A a6 a0) \to ((leq g a1 a7) \to ((leq g (AHead a a0) a6) \to P))))
365 (\lambda (H12: (eq A a6 a0)).(eq_ind A a0 (\lambda (a7: A).((leq g a1 a) \to
366 ((leq g (AHead a a0) a7) \to P))) (\lambda (_: (leq g a1 a)).(\lambda (H14:
367 (leq g (AHead a a0) a0)).(H0 a H14 P))) a6 (sym_eq A a6 a0 H12))) a4 (sym_eq
368 A a4 a H11))) H10))) a5 (sym_eq A a5 (AHead a a0) H8))) a3 (sym_eq A a3 a1
369 H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead a1 (AHead a a0)))
370 (refl_equal A (AHead a a0))))))))))) a2)).