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6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/leq/props".
19 include "leq/defs.ma".
21 include "aplus/props.ma".
23 theorem ahead_inj_snd:
24 \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall
25 (a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4))))))
27 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda
28 (a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let H0 \def (match
29 H in leq return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a
30 a0)).((eq A a (AHead a1 a2)) \to ((eq A a0 (AHead a3 a4)) \to (leq g a2
31 a4)))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A
32 (ASort h1 n1) (AHead a1 a2))).(\lambda (H2: (eq A (ASort h2 n2) (AHead a3
33 a4))).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A
34 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
35 \Rightarrow False])) I (AHead a1 a2) H1) in (False_ind ((eq A (ASort h2 n2)
36 (AHead a3 a4)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
37 k)) \to (leq g a2 a4))) H3)) H2 H0))) | (leq_head a0 a5 H0 a6 a7 H1)
38 \Rightarrow (\lambda (H2: (eq A (AHead a0 a6) (AHead a1 a2))).(\lambda (H3:
39 (eq A (AHead a5 a7) (AHead a3 a4))).((let H4 \def (f_equal A A (\lambda (e:
40 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a6 |
41 (AHead _ a) \Rightarrow a])) (AHead a0 a6) (AHead a1 a2) H2) in ((let H5 \def
42 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
43 [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a6)
44 (AHead a1 a2) H2) in (eq_ind A a1 (\lambda (a: A).((eq A a6 a2) \to ((eq A
45 (AHead a5 a7) (AHead a3 a4)) \to ((leq g a a5) \to ((leq g a6 a7) \to (leq g
46 a2 a4)))))) (\lambda (H6: (eq A a6 a2)).(eq_ind A a2 (\lambda (a: A).((eq A
47 (AHead a5 a7) (AHead a3 a4)) \to ((leq g a1 a5) \to ((leq g a a7) \to (leq g
48 a2 a4))))) (\lambda (H7: (eq A (AHead a5 a7) (AHead a3 a4))).(let H8 \def
49 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
50 [(ASort _ _) \Rightarrow a7 | (AHead _ a) \Rightarrow a])) (AHead a5 a7)
51 (AHead a3 a4) H7) in ((let H9 \def (f_equal A A (\lambda (e: A).(match e in A
52 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead a _)
53 \Rightarrow a])) (AHead a5 a7) (AHead a3 a4) H7) in (eq_ind A a3 (\lambda (a:
54 A).((eq A a7 a4) \to ((leq g a1 a) \to ((leq g a2 a7) \to (leq g a2 a4)))))
55 (\lambda (H10: (eq A a7 a4)).(eq_ind A a4 (\lambda (a: A).((leq g a1 a3) \to
56 ((leq g a2 a) \to (leq g a2 a4)))) (\lambda (_: (leq g a1 a3)).(\lambda (H12:
57 (leq g a2 a4)).H12)) a7 (sym_eq A a7 a4 H10))) a5 (sym_eq A a5 a3 H9))) H8)))
58 a6 (sym_eq A a6 a2 H6))) a0 (sym_eq A a0 a1 H5))) H4)) H3 H0 H1)))]) in (H0
59 (refl_equal A (AHead a1 a2)) (refl_equal A (AHead a3 a4))))))))).
62 \forall (g: G).(\forall (a: A).(leq g a a))
64 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(leq g a0 a0))
65 (\lambda (n: nat).(\lambda (n0: nat).(leq_sort g n n n0 n0 O (refl_equal A
66 (aplus g (ASort n n0) O))))) (\lambda (a0: A).(\lambda (H: (leq g a0
67 a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1
71 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1
74 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1
75 a2)).(eq_ind_r A a2 (\lambda (a: A).(leq g a a2)) (leq_refl g a2) a1 H)))).
78 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g
81 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
82 a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g a0 a))) (\lambda (h1:
83 nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k:
84 nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
85 k))).(leq_sort g h2 h1 n2 n1 k (sym_eq A (aplus g (ASort h1 n1) k) (aplus g
86 (ASort h2 n2) k) H0)))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_:
87 (leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5: A).(\lambda (a6:
88 A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3
89 H1 a6 a5 H3))))))))) a1 a2 H)))).
92 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
93 (a3: A).((leq g a2 a3) \to (leq g a1 a3))))))
95 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
96 a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0
97 a3) \to (leq g a a3))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1:
98 nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort
99 h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (a3: A).(\lambda (H1: (leq g
100 (ASort h2 n2) a3)).(let H2 \def (match H1 in leq return (\lambda (a:
101 A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort h2 n2)) \to
102 ((eq A a0 a3) \to (leq g (ASort h1 n1) a3)))))) with [(leq_sort h0 h3 n0 n3
103 k0 H2) \Rightarrow (\lambda (H3: (eq A (ASort h0 n0) (ASort h2 n2))).(\lambda
104 (H4: (eq A (ASort h3 n3) a3)).((let H5 \def (f_equal A nat (\lambda (e:
105 A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n
106 | (AHead _ _) \Rightarrow n0])) (ASort h0 n0) (ASort h2 n2) H3) in ((let H6
107 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat)
108 with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0)
109 (ASort h2 n2) H3) in (eq_ind nat h2 (\lambda (n: nat).((eq nat n0 n2) \to
110 ((eq A (ASort h3 n3) a3) \to ((eq A (aplus g (ASort n n0) k0) (aplus g (ASort
111 h3 n3) k0)) \to (leq g (ASort h1 n1) a3))))) (\lambda (H7: (eq nat n0
112 n2)).(eq_ind nat n2 (\lambda (n: nat).((eq A (ASort h3 n3) a3) \to ((eq A
113 (aplus g (ASort h2 n) k0) (aplus g (ASort h3 n3) k0)) \to (leq g (ASort h1
114 n1) a3)))) (\lambda (H8: (eq A (ASort h3 n3) a3)).(eq_ind A (ASort h3 n3)
115 (\lambda (a: A).((eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0))
116 \to (leq g (ASort h1 n1) a))) (\lambda (H9: (eq A (aplus g (ASort h2 n2) k0)
117 (aplus g (ASort h3 n3) k0))).(lt_le_e k k0 (leq g (ASort h1 n1) (ASort h3
118 n3)) (\lambda (H10: (lt k k0)).(let H_y \def (aplus_reg_r g (ASort h1 n1)
119 (ASort h2 n2) k k H0 (minus k0 k)) in (let H11 \def (eq_ind_r nat (plus
120 (minus k0 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n) (aplus g
121 (ASort h2 n2) n))) H_y k0 (le_plus_minus_sym k k0 (le_trans k (S k) k0 (le_S
122 k k (le_n k)) H10))) in (leq_sort g h1 h3 n1 n3 k0 (trans_eq A (aplus g
123 (ASort h1 n1) k0) (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0) H11
124 H9))))) (\lambda (H10: (le k0 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2)
125 (ASort h3 n3) k0 k0 H9 (minus k k0)) in (let H11 \def (eq_ind_r nat (plus
126 (minus k k0) k0) (\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g
127 (ASort h3 n3) n))) H_y k (le_plus_minus_sym k0 k H10)) in (leq_sort g h1 h3
128 n1 n3 k (trans_eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)
129 (aplus g (ASort h3 n3) k) H0 H11))))))) a3 H8)) n0 (sym_eq nat n0 n2 H7))) h0
130 (sym_eq nat h0 h2 H6))) H5)) H4 H2))) | (leq_head a0 a4 H2 a5 a6 H3)
131 \Rightarrow (\lambda (H4: (eq A (AHead a0 a5) (ASort h2 n2))).(\lambda (H5:
132 (eq A (AHead a4 a6) a3)).((let H6 \def (eq_ind A (AHead a0 a5) (\lambda (e:
133 A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow
134 False | (AHead _ _) \Rightarrow True])) I (ASort h2 n2) H4) in (False_ind
135 ((eq A (AHead a4 a6) a3) \to ((leq g a0 a4) \to ((leq g a5 a6) \to (leq g
136 (ASort h1 n1) a3)))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (ASort h2 n2))
137 (refl_equal A a3))))))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_:
138 (leq g a3 a4)).(\lambda (H1: ((\forall (a5: A).((leq g a4 a5) \to (leq g a3
139 a5))))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: (leq g a5
140 a6)).(\lambda (H3: ((\forall (a7: A).((leq g a6 a7) \to (leq g a5
141 a7))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6) a0)).(let H5 \def
142 (match H4 in leq return (\lambda (a: A).(\lambda (a7: A).(\lambda (_: (leq ?
143 a a7)).((eq A a (AHead a4 a6)) \to ((eq A a7 a0) \to (leq g (AHead a3 a5)
144 a0)))))) with [(leq_sort h1 h2 n1 n2 k H5) \Rightarrow (\lambda (H6: (eq A
145 (ASort h1 n1) (AHead a4 a6))).(\lambda (H7: (eq A (ASort h2 n2) a0)).((let H8
146 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A return (\lambda
147 (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
148 False])) I (AHead a4 a6) H6) in (False_ind ((eq A (ASort h2 n2) a0) \to ((eq
149 A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (AHead a3
150 a5) a0))) H8)) H7 H5))) | (leq_head a7 a8 H5 a9 a10 H6) \Rightarrow (\lambda
151 (H7: (eq A (AHead a7 a9) (AHead a4 a6))).(\lambda (H8: (eq A (AHead a8 a10)
152 a0)).((let H9 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda
153 (_: A).A) with [(ASort _ _) \Rightarrow a9 | (AHead _ a) \Rightarrow a]))
154 (AHead a7 a9) (AHead a4 a6) H7) in ((let H10 \def (f_equal A A (\lambda (e:
155 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 |
156 (AHead a _) \Rightarrow a])) (AHead a7 a9) (AHead a4 a6) H7) in (eq_ind A a4
157 (\lambda (a: A).((eq A a9 a6) \to ((eq A (AHead a8 a10) a0) \to ((leq g a a8)
158 \to ((leq g a9 a10) \to (leq g (AHead a3 a5) a0)))))) (\lambda (H11: (eq A a9
159 a6)).(eq_ind A a6 (\lambda (a: A).((eq A (AHead a8 a10) a0) \to ((leq g a4
160 a8) \to ((leq g a a10) \to (leq g (AHead a3 a5) a0))))) (\lambda (H12: (eq A
161 (AHead a8 a10) a0)).(eq_ind A (AHead a8 a10) (\lambda (a: A).((leq g a4 a8)
162 \to ((leq g a6 a10) \to (leq g (AHead a3 a5) a)))) (\lambda (H13: (leq g a4
163 a8)).(\lambda (H14: (leq g a6 a10)).(leq_head g a3 a8 (H1 a8 H13) a5 a10 (H3
164 a10 H14)))) a0 H12)) a9 (sym_eq A a9 a6 H11))) a7 (sym_eq A a7 a4 H10))) H9))
165 H8 H5 H6)))]) in (H5 (refl_equal A (AHead a4 a6)) (refl_equal A
166 a0))))))))))))) a1 a2 H)))).
168 theorem leq_ahead_false_1:
169 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1)
170 \to (\forall (P: Prop).P))))
172 \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
173 A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n:
174 nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n
175 n0) a2) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g
176 (AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead
177 (ASort O n0) a2) (ASort O n0))).(let H1 \def (match H0 in leq return (\lambda
178 (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (AHead (ASort O
179 n0) a2)) \to ((eq A a0 (ASort O n0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k
180 H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n1) (AHead (ASort O n0)
181 a2))).(\lambda (H3: (eq A (ASort h2 n2) (ASort O n0))).((let H4 \def (eq_ind
182 A (ASort h1 n1) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop)
183 with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I
184 (AHead (ASort O n0) a2) H2) in (False_ind ((eq A (ASort h2 n2) (ASort O n0))
185 \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H4))
186 H3 H1))) | (leq_head a0 a3 H1 a4 a5 H2) \Rightarrow (\lambda (H3: (eq A
187 (AHead a0 a4) (AHead (ASort O n0) a2))).(\lambda (H4: (eq A (AHead a3 a5)
188 (ASort O n0))).((let H5 \def (f_equal A A (\lambda (e: A).(match e in A
189 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a)
190 \Rightarrow a])) (AHead a0 a4) (AHead (ASort O n0) a2) H3) in ((let H6 \def
191 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
192 [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a4)
193 (AHead (ASort O n0) a2) H3) in (eq_ind A (ASort O n0) (\lambda (a: A).((eq A
194 a4 a2) \to ((eq A (AHead a3 a5) (ASort O n0)) \to ((leq g a a3) \to ((leq g
195 a4 a5) \to P))))) (\lambda (H7: (eq A a4 a2)).(eq_ind A a2 (\lambda (a:
196 A).((eq A (AHead a3 a5) (ASort O n0)) \to ((leq g (ASort O n0) a3) \to ((leq
197 g a a5) \to P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort O n0))).(let H9
198 \def (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda
199 (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
200 True])) I (ASort O n0) H8) in (False_ind ((leq g (ASort O n0) a3) \to ((leq g
201 a2 a5) \to P)) H9))) a4 (sym_eq A a4 a2 H7))) a0 (sym_eq A a0 (ASort O n0)
202 H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead (ASort O n0) a2))
203 (refl_equal A (ASort O n0))))) (\lambda (n1: nat).(\lambda (_: (((leq g
204 (AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead
205 (ASort (S n1) n0) a2) (ASort (S n1) n0))).(let H1 \def (match H0 in leq
206 return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a
207 (AHead (ASort (S n1) n0) a2)) \to ((eq A a0 (ASort (S n1) n0)) \to P)))))
208 with [(leq_sort h1 h2 n2 n3 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1
209 n2) (AHead (ASort (S n1) n0) a2))).(\lambda (H3: (eq A (ASort h2 n3) (ASort
210 (S n1) n0))).((let H4 \def (eq_ind A (ASort h1 n2) (\lambda (e: A).(match e
211 in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead
212 _ _) \Rightarrow False])) I (AHead (ASort (S n1) n0) a2) H2) in (False_ind
213 ((eq A (ASort h2 n3) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort h1 n2) k)
214 (aplus g (ASort h2 n3) k)) \to P)) H4)) H3 H1))) | (leq_head a0 a3 H1 a4 a5
215 H2) \Rightarrow (\lambda (H3: (eq A (AHead a0 a4) (AHead (ASort (S n1) n0)
216 a2))).(\lambda (H4: (eq A (AHead a3 a5) (ASort (S n1) n0))).((let H5 \def
217 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
218 [(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a0 a4)
219 (AHead (ASort (S n1) n0) a2) H3) in ((let H6 \def (f_equal A A (\lambda (e:
220 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 |
221 (AHead a _) \Rightarrow a])) (AHead a0 a4) (AHead (ASort (S n1) n0) a2) H3)
222 in (eq_ind A (ASort (S n1) n0) (\lambda (a: A).((eq A a4 a2) \to ((eq A
223 (AHead a3 a5) (ASort (S n1) n0)) \to ((leq g a a3) \to ((leq g a4 a5) \to
224 P))))) (\lambda (H7: (eq A a4 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead
225 a3 a5) (ASort (S n1) n0)) \to ((leq g (ASort (S n1) n0) a3) \to ((leq g a a5)
226 \to P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort (S n1) n0))).(let H9 \def
227 (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda (_:
228 A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
229 True])) I (ASort (S n1) n0) H8) in (False_ind ((leq g (ASort (S n1) n0) a3)
230 \to ((leq g a2 a5) \to P)) H9))) a4 (sym_eq A a4 a2 H7))) a0 (sym_eq A a0
231 (ASort (S n1) n0) H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead (ASort
232 (S n1) n0) a2)) (refl_equal A (ASort (S n1) n0))))))) n H)))))) (\lambda (a:
233 A).(\lambda (H: ((\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P:
234 Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead
235 a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq
236 g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P: Prop).(let H2 \def
237 (match H1 in leq return (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: (leq ?
238 a3 a4)).((eq A a3 (AHead (AHead a a0) a2)) \to ((eq A a4 (AHead a a0)) \to
239 P))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A
240 (ASort h1 n1) (AHead (AHead a a0) a2))).(\lambda (H4: (eq A (ASort h2 n2)
241 (AHead a a0))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e
242 in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead
243 _ _) \Rightarrow False])) I (AHead (AHead a a0) a2) H3) in (False_ind ((eq A
244 (ASort h2 n2) (AHead a a0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g
245 (ASort h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a3 a4 H2 a5 a6 H3)
246 \Rightarrow (\lambda (H4: (eq A (AHead a3 a5) (AHead (AHead a a0)
247 a2))).(\lambda (H5: (eq A (AHead a4 a6) (AHead a a0))).((let H6 \def (f_equal
248 A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
249 \Rightarrow a5 | (AHead _ a7) \Rightarrow a7])) (AHead a3 a5) (AHead (AHead a
250 a0) a2) H4) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A
251 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a7 _)
252 \Rightarrow a7])) (AHead a3 a5) (AHead (AHead a a0) a2) H4) in (eq_ind A
253 (AHead a a0) (\lambda (a7: A).((eq A a5 a2) \to ((eq A (AHead a4 a6) (AHead a
254 a0)) \to ((leq g a7 a4) \to ((leq g a5 a6) \to P))))) (\lambda (H8: (eq A a5
255 a2)).(eq_ind A a2 (\lambda (a7: A).((eq A (AHead a4 a6) (AHead a a0)) \to
256 ((leq g (AHead a a0) a4) \to ((leq g a7 a6) \to P)))) (\lambda (H9: (eq A
257 (AHead a4 a6) (AHead a a0))).(let H10 \def (f_equal A A (\lambda (e:
258 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a6 |
259 (AHead _ a7) \Rightarrow a7])) (AHead a4 a6) (AHead a a0) H9) in ((let H11
260 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
261 with [(ASort _ _) \Rightarrow a4 | (AHead a7 _) \Rightarrow a7])) (AHead a4
262 a6) (AHead a a0) H9) in (eq_ind A a (\lambda (a7: A).((eq A a6 a0) \to ((leq
263 g (AHead a a0) a7) \to ((leq g a2 a6) \to P)))) (\lambda (H12: (eq A a6
264 a0)).(eq_ind A a0 (\lambda (a7: A).((leq g (AHead a a0) a) \to ((leq g a2 a7)
265 \to P))) (\lambda (H13: (leq g (AHead a a0) a)).(\lambda (_: (leq g a2
266 a0)).(H a0 H13 P))) a6 (sym_eq A a6 a0 H12))) a4 (sym_eq A a4 a H11))) H10)))
267 a5 (sym_eq A a5 a2 H8))) a3 (sym_eq A a3 (AHead a a0) H7))) H6)) H5 H2
268 H3)))]) in (H2 (refl_equal A (AHead (AHead a a0) a2)) (refl_equal A (AHead a
271 theorem leq_ahead_false_2:
272 \forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2)
273 \to (\forall (P: Prop).P))))
275 \lambda (g: G).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (a1:
276 A).((leq g (AHead a1 a) a) \to (\forall (P: Prop).P)))) (\lambda (n:
277 nat).(\lambda (n0: nat).(\lambda (a1: A).(\lambda (H: (leq g (AHead a1 (ASort
278 n n0)) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g
279 (AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead
280 a1 (ASort O n0)) (ASort O n0))).(let H1 \def (match H0 in leq return (\lambda
281 (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (AHead a1 (ASort
282 O n0))) \to ((eq A a0 (ASort O n0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k
283 H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n1) (AHead a1 (ASort O
284 n0)))).(\lambda (H3: (eq A (ASort h2 n2) (ASort O n0))).((let H4 \def (eq_ind
285 A (ASort h1 n1) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop)
286 with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I
287 (AHead a1 (ASort O n0)) H2) in (False_ind ((eq A (ASort h2 n2) (ASort O n0))
288 \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H4))
289 H3 H1))) | (leq_head a0 a3 H1 a4 a5 H2) \Rightarrow (\lambda (H3: (eq A
290 (AHead a0 a4) (AHead a1 (ASort O n0)))).(\lambda (H4: (eq A (AHead a3 a5)
291 (ASort O n0))).((let H5 \def (f_equal A A (\lambda (e: A).(match e in A
292 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a)
293 \Rightarrow a])) (AHead a0 a4) (AHead a1 (ASort O n0)) H3) in ((let H6 \def
294 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
295 [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a4)
296 (AHead a1 (ASort O n0)) H3) in (eq_ind A a1 (\lambda (a: A).((eq A a4 (ASort
297 O n0)) \to ((eq A (AHead a3 a5) (ASort O n0)) \to ((leq g a a3) \to ((leq g
298 a4 a5) \to P))))) (\lambda (H7: (eq A a4 (ASort O n0))).(eq_ind A (ASort O
299 n0) (\lambda (a: A).((eq A (AHead a3 a5) (ASort O n0)) \to ((leq g a1 a3) \to
300 ((leq g a a5) \to P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort O n0))).(let
301 H9 \def (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda
302 (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
303 True])) I (ASort O n0) H8) in (False_ind ((leq g a1 a3) \to ((leq g (ASort O
304 n0) a5) \to P)) H9))) a4 (sym_eq A a4 (ASort O n0) H7))) a0 (sym_eq A a0 a1
305 H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead a1 (ASort O n0)))
306 (refl_equal A (ASort O n0))))) (\lambda (n1: nat).(\lambda (_: (((leq g
307 (AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead
308 a1 (ASort (S n1) n0)) (ASort (S n1) n0))).(let H1 \def (match H0 in leq
309 return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a
310 (AHead a1 (ASort (S n1) n0))) \to ((eq A a0 (ASort (S n1) n0)) \to P)))))
311 with [(leq_sort h1 h2 n2 n3 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1
312 n2) (AHead a1 (ASort (S n1) n0)))).(\lambda (H3: (eq A (ASort h2 n3) (ASort
313 (S n1) n0))).((let H4 \def (eq_ind A (ASort h1 n2) (\lambda (e: A).(match e
314 in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead
315 _ _) \Rightarrow False])) I (AHead a1 (ASort (S n1) n0)) H2) in (False_ind
316 ((eq A (ASort h2 n3) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort h1 n2) k)
317 (aplus g (ASort h2 n3) k)) \to P)) H4)) H3 H1))) | (leq_head a0 a3 H1 a4 a5
318 H2) \Rightarrow (\lambda (H3: (eq A (AHead a0 a4) (AHead a1 (ASort (S n1)
319 n0)))).(\lambda (H4: (eq A (AHead a3 a5) (ASort (S n1) n0))).((let H5 \def
320 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
321 [(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a0 a4)
322 (AHead a1 (ASort (S n1) n0)) H3) in ((let H6 \def (f_equal A A (\lambda (e:
323 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 |
324 (AHead a _) \Rightarrow a])) (AHead a0 a4) (AHead a1 (ASort (S n1) n0)) H3)
325 in (eq_ind A a1 (\lambda (a: A).((eq A a4 (ASort (S n1) n0)) \to ((eq A
326 (AHead a3 a5) (ASort (S n1) n0)) \to ((leq g a a3) \to ((leq g a4 a5) \to
327 P))))) (\lambda (H7: (eq A a4 (ASort (S n1) n0))).(eq_ind A (ASort (S n1) n0)
328 (\lambda (a: A).((eq A (AHead a3 a5) (ASort (S n1) n0)) \to ((leq g a1 a3)
329 \to ((leq g a a5) \to P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort (S n1)
330 n0))).(let H9 \def (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A
331 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
332 _) \Rightarrow True])) I (ASort (S n1) n0) H8) in (False_ind ((leq g a1 a3)
333 \to ((leq g (ASort (S n1) n0) a5) \to P)) H9))) a4 (sym_eq A a4 (ASort (S n1)
334 n0) H7))) a0 (sym_eq A a0 a1 H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A
335 (AHead a1 (ASort (S n1) n0))) (refl_equal A (ASort (S n1) n0))))))) n H))))))
336 (\lambda (a: A).(\lambda (_: ((\forall (a1: A).((leq g (AHead a1 a) a) \to
337 (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a1:
338 A).((leq g (AHead a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1:
339 A).(\lambda (H1: (leq g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P:
340 Prop).(let H2 \def (match H1 in leq return (\lambda (a3: A).(\lambda (a4:
341 A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (AHead a1 (AHead a a0))) \to ((eq A
342 a4 (AHead a a0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow
343 (\lambda (H3: (eq A (ASort h1 n1) (AHead a1 (AHead a a0)))).(\lambda (H4: (eq
344 A (ASort h2 n2) (AHead a a0))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda
345 (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
346 \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a1 (AHead a a0))
347 H3) in (False_ind ((eq A (ASort h2 n2) (AHead a a0)) \to ((eq A (aplus g
348 (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head
349 a3 a4 H2 a5 a6 H3) \Rightarrow (\lambda (H4: (eq A (AHead a3 a5) (AHead a1
350 (AHead a a0)))).(\lambda (H5: (eq A (AHead a4 a6) (AHead a a0))).((let H6
351 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
352 with [(ASort _ _) \Rightarrow a5 | (AHead _ a7) \Rightarrow a7])) (AHead a3
353 a5) (AHead a1 (AHead a a0)) H4) in ((let H7 \def (f_equal A A (\lambda (e:
354 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 |
355 (AHead a7 _) \Rightarrow a7])) (AHead a3 a5) (AHead a1 (AHead a a0)) H4) in
356 (eq_ind A a1 (\lambda (a7: A).((eq A a5 (AHead a a0)) \to ((eq A (AHead a4
357 a6) (AHead a a0)) \to ((leq g a7 a4) \to ((leq g a5 a6) \to P))))) (\lambda
358 (H8: (eq A a5 (AHead a a0))).(eq_ind A (AHead a a0) (\lambda (a7: A).((eq A
359 (AHead a4 a6) (AHead a a0)) \to ((leq g a1 a4) \to ((leq g a7 a6) \to P))))
360 (\lambda (H9: (eq A (AHead a4 a6) (AHead a a0))).(let H10 \def (f_equal A A
361 (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
362 \Rightarrow a6 | (AHead _ a7) \Rightarrow a7])) (AHead a4 a6) (AHead a a0)
363 H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e in A return
364 (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead a7 _)
365 \Rightarrow a7])) (AHead a4 a6) (AHead a a0) H9) in (eq_ind A a (\lambda (a7:
366 A).((eq A a6 a0) \to ((leq g a1 a7) \to ((leq g (AHead a a0) a6) \to P))))
367 (\lambda (H12: (eq A a6 a0)).(eq_ind A a0 (\lambda (a7: A).((leq g a1 a) \to
368 ((leq g (AHead a a0) a7) \to P))) (\lambda (_: (leq g a1 a)).(\lambda (H14:
369 (leq g (AHead a a0) a0)).(H0 a H14 P))) a6 (sym_eq A a6 a0 H12))) a4 (sym_eq
370 A a4 a H11))) H10))) a5 (sym_eq A a5 (AHead a a0) H8))) a3 (sym_eq A a3 a1
371 H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead a1 (AHead a a0)))
372 (refl_equal A (AHead a a0))))))))))) a2)).