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_1A/leq/fwd.ma".
19 include "basic_1A/aplus/props.ma".
22 \forall (g: G).(\forall (a: A).(leq g a a))
24 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(leq g a0 a0))
25 (\lambda (n: nat).(\lambda (n0: nat).(leq_sort g n n n0 n0 O (refl_equal A
26 (aplus g (ASort n n0) O))))) (\lambda (a0: A).(\lambda (H: (leq g a0
27 a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1
31 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1
34 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1
35 a2)).(eq_ind A a1 (\lambda (a: A).(leq g a1 a)) (leq_refl g a1) a2 H)))).
38 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g
41 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
42 a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g a0 a))) (\lambda (h1:
43 nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k:
44 nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
45 k))).(leq_sort g h2 h1 n2 n1 k (sym_eq A (aplus g (ASort h1 n1) k) (aplus g
46 (ASort h2 n2) k) H0)))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_:
47 (leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5: A).(\lambda (a6:
48 A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3
49 H1 a6 a5 H3))))))))) a1 a2 H)))).
52 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
53 (a3: A).((leq g a2 a3) \to (leq g a1 a3))))))
55 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
56 a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0
57 a3) \to (leq g a a3))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1:
58 nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort
59 h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (a3: A).(\lambda (H1: (leq g
60 (ASort h2 n2) a3)).(let H_x \def (leq_gen_sort1 g h2 n2 a3 H1) in (let H2
61 \def H_x in (ex2_3_ind nat nat nat (\lambda (n3: nat).(\lambda (h3:
62 nat).(\lambda (k0: nat).(eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3
63 n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a3
64 (ASort h3 n3))))) (leq g (ASort h1 n1) a3) (\lambda (x0: nat).(\lambda (x1:
65 nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort h2 n2) x2) (aplus
66 g (ASort x1 x0) x2))).(\lambda (H4: (eq A a3 (ASort x1 x0))).(let H5 \def
67 (f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H4) in (eq_ind_r A (ASort x1
68 x0) (\lambda (a: A).(leq g (ASort h1 n1) a)) (lt_le_e k x2 (leq g (ASort h1
69 n1) (ASort x1 x0)) (\lambda (H6: (lt k x2)).(let H_y \def (aplus_reg_r g
70 (ASort h1 n1) (ASort h2 n2) k k H0 (minus x2 k)) in (let H7 \def (eq_ind_r
71 nat (plus (minus x2 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n)
72 (aplus g (ASort h2 n2) n))) H_y x2 (le_plus_minus_sym k x2 (le_trans k (S k)
73 x2 (le_S k k (le_n k)) H6))) in (leq_sort g h1 x1 n1 x0 x2 (trans_eq A (aplus
74 g (ASort h1 n1) x2) (aplus g (ASort h2 n2) x2) (aplus g (ASort x1 x0) x2) H7
75 H3))))) (\lambda (H6: (le x2 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2)
76 (ASort x1 x0) x2 x2 H3 (minus k x2)) in (let H7 \def (eq_ind_r nat (plus
77 (minus k x2) x2) (\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g
78 (ASort x1 x0) n))) H_y k (le_plus_minus_sym x2 k H6)) in (leq_sort g h1 x1 n1
79 x0 k (trans_eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) (aplus g
80 (ASort x1 x0) k) H0 H7)))))) a3 H5))))))) H2))))))))))) (\lambda (a3:
81 A).(\lambda (a4: A).(\lambda (_: (leq g a3 a4)).(\lambda (H1: ((\forall (a5:
82 A).((leq g a4 a5) \to (leq g a3 a5))))).(\lambda (a5: A).(\lambda (a6:
83 A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: ((\forall (a7: A).((leq g a6 a7)
84 \to (leq g a5 a7))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6)
85 a0)).(let H_x \def (leq_gen_head1 g a4 a6 a0 H4) in (let H5 \def H_x in
86 (ex3_2_ind A A (\lambda (a7: A).(\lambda (_: A).(leq g a4 a7))) (\lambda (_:
87 A).(\lambda (a8: A).(leq g a6 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A
88 a0 (AHead a7 a8)))) (leq g (AHead a3 a5) a0) (\lambda (x0: A).(\lambda (x1:
89 A).(\lambda (H6: (leq g a4 x0)).(\lambda (H7: (leq g a6 x1)).(\lambda (H8:
90 (eq A a0 (AHead x0 x1))).(let H9 \def (f_equal A A (\lambda (e: A).e) a0
91 (AHead x0 x1) H8) in (eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead
92 a3 a5) a)) (leq_head g a3 x0 (H1 x0 H6) a5 x1 (H3 x1 H7)) a0 H9)))))))
93 H5))))))))))))) a1 a2 H)))).
95 lemma leq_ahead_false_1:
96 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1)
97 \to (\forall (P: Prop).P))))
99 \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
100 A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n:
101 nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n
102 n0) a2) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g
103 (AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead
104 (ASort O n0) a2) (ASort O n0))).(let H_x \def (leq_gen_head1 g (ASort O n0)
105 a2 (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3:
106 A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: A).(\lambda (a4:
107 A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0)
108 (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g
109 (ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort O
110 n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee:
111 A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
112 False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))) (\lambda (n1:
113 nat).(\lambda (_: (((leq g (AHead (ASort n1 n0) a2) (ASort n1 n0)) \to
114 P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort (S n1)
115 n0))).(let H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 (ASort (S n1) n0)
116 H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_:
117 A).(leq g (ASort (S n1) n0) a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2
118 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1) n0) (AHead a3
119 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g (ASort (S n1)
120 n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort (S n1) n0)
121 (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda (ee:
122 A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
123 False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n H))))))
124 (\lambda (a: A).(\lambda (H: ((\forall (a2: A).((leq g (AHead a a2) a) \to
125 (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2:
126 A).((leq g (AHead a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2:
127 A).(\lambda (H1: (leq g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P:
128 Prop).(let H_x \def (leq_gen_head1 g (AHead a a0) a2 (AHead a a0) H1) in (let
129 H2 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead
130 a a0) a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3:
131 A).(\lambda (a4: A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0:
132 A).(\lambda (x1: A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq
133 g a2 x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def
134 (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a |
135 (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) in ((let H7
136 \def (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a0 |
137 (AHead _ a3) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) in (\lambda
138 (H8: (eq A a x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3))
139 H4 a0 H7) in (let H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a
140 a0) a3)) H3 a H8) in (H a0 H10 P))))) H6))))))) H2)))))))))) a1)).
142 lemma leq_ahead_false_2:
143 \forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2)
144 \to (\forall (P: Prop).P))))
146 \lambda (g: G).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (a1:
147 A).((leq g (AHead a1 a) a) \to (\forall (P: Prop).P)))) (\lambda (n:
148 nat).(\lambda (n0: nat).(\lambda (a1: A).(\lambda (H: (leq g (AHead a1 (ASort
149 n n0)) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g
150 (AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead
151 a1 (ASort O n0)) (ASort O n0))).(let H_x \def (leq_gen_head1 g a1 (ASort O
152 n0) (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3:
153 A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g
154 (ASort O n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0)
155 (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1
156 x0)).(\lambda (_: (leq g (ASort O n0) x1)).(\lambda (H4: (eq A (ASort O n0)
157 (AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: A).(match
158 ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I
159 (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))) (\lambda (n1:
160 nat).(\lambda (_: (((leq g (AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to
161 P))).(\lambda (H0: (leq g (AHead a1 (ASort (S n1) n0)) (ASort (S n1)
162 n0))).(let H_x \def (leq_gen_head1 g a1 (ASort (S n1) n0) (ASort (S n1) n0)
163 H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_:
164 A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g (ASort (S n1) n0)
165 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1) n0) (AHead a3
166 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1
167 x0)).(\lambda (_: (leq g (ASort (S n1) n0) x1)).(\lambda (H4: (eq A (ASort (S
168 n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda
169 (ee: A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _)
170 \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n
171 H)))))) (\lambda (a: A).(\lambda (_: ((\forall (a1: A).((leq g (AHead a1 a)
172 a) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall
173 (a1: A).((leq g (AHead a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1:
174 A).(\lambda (H1: (leq g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P:
175 Prop).(let H_x \def (leq_gen_head1 g a1 (AHead a a0) (AHead a a0) H1) in (let
176 H2 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a1
177 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g (AHead a a0) a4))) (\lambda
178 (a3: A).(\lambda (a4: A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0:
179 A).(\lambda (x1: A).(\lambda (H3: (leq g a1 x0)).(\lambda (H4: (leq g (AHead
180 a a0) x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def
181 (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a |
182 (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) in ((let H7
183 \def (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a0 |
184 (AHead _ a3) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) in (\lambda
185 (H8: (eq A a x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a3: A).(leq g (AHead
186 a a0) a3)) H4 a0 H7) in (let H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g
187 a1 a3)) H3 a H8) in (H0 a H9 P))))) H6))))))) H2)))))))))) a2)).