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/r/defs.ma".
19 include "basic_1/s/defs.ma".
22 \forall (k: K).(\forall (i: nat).(eq nat (r k (S i)) (S (r k i))))
24 \lambda (k: K).(let TMP_5 \def (\lambda (k0: K).(\forall (i: nat).(let TMP_1
25 \def (S i) in (let TMP_2 \def (r k0 TMP_1) in (let TMP_3 \def (r k0 i) in
26 (let TMP_4 \def (S TMP_3) in (eq nat TMP_2 TMP_4))))))) in (let TMP_9 \def
27 (\lambda (b: B).(\lambda (i: nat).(let TMP_6 \def (Bind b) in (let TMP_7 \def
28 (r TMP_6 i) in (let TMP_8 \def (S TMP_7) in (refl_equal nat TMP_8)))))) in
29 (let TMP_13 \def (\lambda (f: F).(\lambda (i: nat).(let TMP_10 \def (Flat f)
30 in (let TMP_11 \def (r TMP_10 i) in (let TMP_12 \def (S TMP_11) in
31 (refl_equal nat TMP_12)))))) in (K_ind TMP_5 TMP_9 TMP_13 k)))).
34 \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j))
37 \lambda (k: K).(let TMP_5 \def (\lambda (k0: K).(\forall (i: nat).(\forall
38 (j: nat).(let TMP_1 \def (plus i j) in (let TMP_2 \def (r k0 TMP_1) in (let
39 TMP_3 \def (r k0 i) in (let TMP_4 \def (plus TMP_3 j) in (eq nat TMP_2
40 TMP_4)))))))) in (let TMP_9 \def (\lambda (b: B).(\lambda (i: nat).(\lambda
41 (j: nat).(let TMP_6 \def (Bind b) in (let TMP_7 \def (r TMP_6 i) in (let
42 TMP_8 \def (plus TMP_7 j) in (refl_equal nat TMP_8))))))) in (let TMP_13 \def
43 (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(let TMP_10 \def (Flat f)
44 in (let TMP_11 \def (r TMP_10 i) in (let TMP_12 \def (plus TMP_11 j) in
45 (refl_equal nat TMP_12))))))) in (K_ind TMP_5 TMP_9 TMP_13 k)))).
48 \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j))
51 \lambda (k: K).(let TMP_5 \def (\lambda (k0: K).(\forall (i: nat).(\forall
52 (j: nat).(let TMP_1 \def (plus i j) in (let TMP_2 \def (r k0 TMP_1) in (let
53 TMP_3 \def (r k0 j) in (let TMP_4 \def (plus i TMP_3) in (eq nat TMP_2
54 TMP_4)))))))) in (let TMP_7 \def (\lambda (_: B).(\lambda (i: nat).(\lambda
55 (j: nat).(let TMP_6 \def (plus i j) in (refl_equal nat TMP_6))))) in (let
56 TMP_8 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(plus_n_Sm i
57 j)))) in (K_ind TMP_5 TMP_7 TMP_8 k)))).
60 \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (k: K).(eq nat
61 (minus (r k i) (S n)) (r k (minus i (S n)))))))
63 \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (k:
64 K).(let TMP_7 \def (\lambda (k0: K).(let TMP_1 \def (r k0 i) in (let TMP_2
65 \def (S n) in (let TMP_3 \def (minus TMP_1 TMP_2) in (let TMP_4 \def (S n) in
66 (let TMP_5 \def (minus i TMP_4) in (let TMP_6 \def (r k0 TMP_5) in (eq nat
67 TMP_3 TMP_6)))))))) in (let TMP_10 \def (\lambda (_: B).(let TMP_8 \def (S n)
68 in (let TMP_9 \def (minus i TMP_8) in (refl_equal nat TMP_9)))) in (let
69 TMP_11 \def (\lambda (_: F).(minus_x_Sy i n H)) in (K_ind TMP_7 TMP_10 TMP_11
73 \forall (k: K).(\forall (P: Prop).(((((\forall (i: nat).(eq nat (r k i) i)))
74 \to P)) \to (((((\forall (i: nat).(eq nat (r k i) (S i)))) \to P)) \to P)))
76 \lambda (k: K).(let TMP_1 \def (\lambda (k0: K).(\forall (P:
77 Prop).(((((\forall (i: nat).(eq nat (r k0 i) i))) \to P)) \to (((((\forall
78 (i: nat).(eq nat (r k0 i) (S i)))) \to P)) \to P)))) in (let TMP_3 \def
79 (\lambda (b: B).(\lambda (P: Prop).(\lambda (H: ((((\forall (i: nat).(eq nat
80 (r (Bind b) i) i))) \to P))).(\lambda (_: ((((\forall (i: nat).(eq nat (r
81 (Bind b) i) (S i)))) \to P))).(let TMP_2 \def (\lambda (i: nat).(refl_equal
82 nat i)) in (H TMP_2)))))) in (let TMP_6 \def (\lambda (f: F).(\lambda (P:
83 Prop).(\lambda (_: ((((\forall (i: nat).(eq nat (r (Flat f) i) i))) \to
84 P))).(\lambda (H0: ((((\forall (i: nat).(eq nat (r (Flat f) i) (S i)))) \to
85 P))).(let TMP_5 \def (\lambda (i: nat).(let TMP_4 \def (S i) in (refl_equal
86 nat TMP_4))) in (H0 TMP_5)))))) in (K_ind TMP_1 TMP_3 TMP_6 k)))).
89 \forall (k: K).(\forall (i: nat).(eq nat (s k (r k i)) (S i)))
91 \lambda (k: K).(let TMP_4 \def (\lambda (k0: K).(\forall (i: nat).(let TMP_1
92 \def (r k0 i) in (let TMP_2 \def (s k0 TMP_1) in (let TMP_3 \def (S i) in (eq
93 nat TMP_2 TMP_3)))))) in (let TMP_6 \def (\lambda (_: B).(\lambda (i:
94 nat).(let TMP_5 \def (S i) in (refl_equal nat TMP_5)))) in (let TMP_8 \def
95 (\lambda (_: F).(\lambda (i: nat).(let TMP_7 \def (S i) in (refl_equal nat
96 TMP_7)))) in (K_ind TMP_4 TMP_6 TMP_8 k)))).
99 \forall (k: K).(\forall (i: nat).(eq nat (minus (r k (S i)) (S O)) (r k i)))
101 \lambda (k: K).(\lambda (i: nat).(let TMP_1 \def (r k i) in (let TMP_2 \def
102 (S TMP_1) in (let TMP_6 \def (\lambda (n: nat).(let TMP_3 \def (S O) in (let
103 TMP_4 \def (minus n TMP_3) in (let TMP_5 \def (r k i) in (eq nat TMP_4
104 TMP_5))))) in (let TMP_7 \def (r k i) in (let TMP_9 \def (\lambda (n:
105 nat).(let TMP_8 \def (r k i) in (eq nat n TMP_8))) in (let TMP_10 \def (r k
106 i) in (let TMP_11 \def (refl_equal nat TMP_10) in (let TMP_12 \def (r k i) in
107 (let TMP_13 \def (S TMP_12) in (let TMP_14 \def (S O) in (let TMP_15 \def
108 (minus TMP_13 TMP_14) in (let TMP_16 \def (r k i) in (let TMP_17 \def
109 (minus_Sx_SO TMP_16) in (let TMP_18 \def (eq_ind_r nat TMP_7 TMP_9 TMP_11
110 TMP_15 TMP_17) in (let TMP_19 \def (S i) in (let TMP_20 \def (r k TMP_19) in
111 (let TMP_21 \def (r_S k i) in (eq_ind_r nat TMP_2 TMP_6 TMP_18 TMP_20
112 TMP_21))))))))))))))))))).
115 \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k (S
116 i)) (S j)) (minus (r k i) j))))
118 \lambda (k: K).(\lambda (i: nat).(\lambda (j: nat).(let TMP_1 \def (r k i)
119 in (let TMP_2 \def (S TMP_1) in (let TMP_7 \def (\lambda (n: nat).(let TMP_3
120 \def (S j) in (let TMP_4 \def (minus n TMP_3) in (let TMP_5 \def (r k i) in
121 (let TMP_6 \def (minus TMP_5 j) in (eq nat TMP_4 TMP_6)))))) in (let TMP_8
122 \def (r k i) in (let TMP_9 \def (minus TMP_8 j) in (let TMP_10 \def
123 (refl_equal nat TMP_9) in (let TMP_11 \def (S i) in (let TMP_12 \def (r k
124 TMP_11) in (let TMP_13 \def (r_S k i) in (eq_ind_r nat TMP_2 TMP_7 TMP_10
125 TMP_12 TMP_13)))))))))))).
128 \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le (S i) (s k j)) \to
131 \lambda (k: K).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(\forall
132 (j: nat).((le (S i) (s k0 j)) \to (let TMP_1 \def (r k0 i) in (le TMP_1
133 j)))))) in (let TMP_3 \def (\lambda (_: B).(\lambda (i: nat).(\lambda (j:
134 nat).(\lambda (H: (le (S i) (S j))).(let H_y \def (le_S_n i j H) in H_y)))))
135 in (let TMP_4 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j:
136 nat).(\lambda (H: (le (S i) j)).H)))) in (K_ind TMP_2 TMP_3 TMP_4 k)))).
139 \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le (s k j) (S i)) \to
142 \lambda (k: K).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(\forall
143 (j: nat).((le (s k0 j) (S i)) \to (let TMP_1 \def (r k0 i) in (le j
144 TMP_1)))))) in (let TMP_3 \def (\lambda (_: B).(\lambda (i: nat).(\lambda (j:
145 nat).(\lambda (H: (le (S j) (S i))).(let H_y \def (le_S_n j i H) in H_y)))))
146 in (let TMP_4 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j:
147 nat).(\lambda (H: (le j (S i))).H)))) in (K_ind TMP_2 TMP_3 TMP_4 k)))).
150 \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (S i) (s k
151 j)) (minus (r k i) j))))
153 \lambda (k: K).(let TMP_6 \def (\lambda (k0: K).(\forall (i: nat).(\forall
154 (j: nat).(let TMP_1 \def (S i) in (let TMP_2 \def (s k0 j) in (let TMP_3 \def
155 (minus TMP_1 TMP_2) in (let TMP_4 \def (r k0 i) in (let TMP_5 \def (minus
156 TMP_4 j) in (eq nat TMP_3 TMP_5))))))))) in (let TMP_10 \def (\lambda (b:
157 B).(\lambda (i: nat).(\lambda (j: nat).(let TMP_7 \def (Bind b) in (let TMP_8
158 \def (r TMP_7 i) in (let TMP_9 \def (minus TMP_8 j) in (refl_equal nat
159 TMP_9))))))) in (let TMP_14 \def (\lambda (f: F).(\lambda (i: nat).(\lambda
160 (j: nat).(let TMP_11 \def (Flat f) in (let TMP_12 \def (r TMP_11 i) in (let
161 TMP_13 \def (minus TMP_12 j) in (refl_equal nat TMP_13))))))) in (K_ind TMP_6
165 \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt (s k j) (S i)) \to
168 \lambda (k: K).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(\forall
169 (j: nat).((lt (s k0 j) (S i)) \to (let TMP_1 \def (r k0 i) in (lt j
170 TMP_1)))))) in (let TMP_3 \def (\lambda (_: B).(\lambda (i: nat).(\lambda (j:
171 nat).(\lambda (H: (lt (S j) (S i))).(lt_S_n j i H))))) in (let TMP_4 \def
172 (\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt j (S
173 i))).H)))) in (K_ind TMP_2 TMP_3 TMP_4 k)))).
176 \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k i) (S
177 j)) (minus i (s k j)))))
179 \lambda (k: K).(let TMP_6 \def (\lambda (k0: K).(\forall (i: nat).(\forall
180 (j: nat).(let TMP_1 \def (r k0 i) in (let TMP_2 \def (S j) in (let TMP_3 \def
181 (minus TMP_1 TMP_2) in (let TMP_4 \def (s k0 j) in (let TMP_5 \def (minus i
182 TMP_4) in (eq nat TMP_3 TMP_5))))))))) in (let TMP_10 \def (\lambda (b:
183 B).(\lambda (i: nat).(\lambda (j: nat).(let TMP_7 \def (Bind b) in (let TMP_8
184 \def (s TMP_7 j) in (let TMP_9 \def (minus i TMP_8) in (refl_equal nat
185 TMP_9))))))) in (let TMP_14 \def (\lambda (f: F).(\lambda (i: nat).(\lambda
186 (j: nat).(let TMP_11 \def (Flat f) in (let TMP_12 \def (s TMP_11 j) in (let
187 TMP_13 \def (minus i TMP_12) in (refl_equal nat TMP_13))))))) in (K_ind TMP_6
191 \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (S i) (s k j))
192 \to (eq nat (r k i) j))))
194 \lambda (k: K).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(\forall
195 (j: nat).((eq nat (S i) (s k0 j)) \to (let TMP_1 \def (r k0 i) in (eq nat
196 TMP_1 j)))))) in (let TMP_3 \def (\lambda (_: B).(\lambda (i: nat).(\lambda
197 (j: nat).(\lambda (H: (eq nat (S i) (S j))).(eq_add_S i j H))))) in (let
198 TMP_4 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H:
199 (eq nat (S i) j)).H)))) in (K_ind TMP_2 TMP_3 TMP_4 k)))).