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).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (r k0 (S
25 i)) (S (r k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (r
26 (Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (r (Flat
33 \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j))
36 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
37 nat).(eq nat (r k0 (plus i j)) (plus (r k0 i) j))))) (\lambda (b: B).(\lambda
38 (i: nat).(\lambda (j: nat).(refl_equal nat (plus (r (Bind b) i) j)))))
39 (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (r
40 (Flat f) i) j))))) k).
46 \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j))
49 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
50 nat).(eq nat (r k0 (plus i j)) (plus i (r k0 j)))))) (\lambda (_: B).(\lambda
51 (i: nat).(\lambda (j: nat).(refl_equal nat (plus i j))))) (\lambda (_:
52 F).(\lambda (i: nat).(\lambda (j: nat).(plus_n_Sm i j)))) k).
58 \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (k: K).(eq nat
59 (minus (r k i) (S n)) (r k (minus i (S n)))))))
61 \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (k:
62 K).(K_ind (\lambda (k0: K).(eq nat (minus (r k0 i) (S n)) (r k0 (minus i (S
63 n))))) (\lambda (_: B).(refl_equal nat (minus i (S n)))) (\lambda (_:
64 F).(minus_x_Sy i n H)) k)))).
70 \forall (k: K).(\forall (P: Prop).(((((\forall (i: nat).(eq nat (r k i) i)))
71 \to P)) \to (((((\forall (i: nat).(eq nat (r k i) (S i)))) \to P)) \to P)))
73 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (P: Prop).(((((\forall (i:
74 nat).(eq nat (r k0 i) i))) \to P)) \to (((((\forall (i: nat).(eq nat (r k0 i)
75 (S i)))) \to P)) \to P)))) (\lambda (b: B).(\lambda (P: Prop).(\lambda (H:
76 ((((\forall (i: nat).(eq nat (r (Bind b) i) i))) \to P))).(\lambda (_:
77 ((((\forall (i: nat).(eq nat (r (Bind b) i) (S i)))) \to P))).(H (\lambda (i:
78 nat).(refl_equal nat i))))))) (\lambda (f: F).(\lambda (P: Prop).(\lambda (_:
79 ((((\forall (i: nat).(eq nat (r (Flat f) i) i))) \to P))).(\lambda (H0:
80 ((((\forall (i: nat).(eq nat (r (Flat f) i) (S i)))) \to P))).(H0 (\lambda
81 (i: nat).(refl_equal nat (S i)))))))) k).
87 \forall (k: K).(\forall (i: nat).(eq nat (s k (r k i)) (S i)))
89 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (r k0
90 i)) (S i)))) (\lambda (_: B).(\lambda (i: nat).(refl_equal nat (S i))))
91 (\lambda (_: F).(\lambda (i: nat).(refl_equal nat (S i)))) k).
97 \forall (k: K).(\forall (i: nat).(eq nat (minus (r k (S i)) (S O)) (r k i)))
99 \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (S (r k i)) (\lambda (n:
100 nat).(eq nat (minus n (S O)) (r k i))) (eq_ind_r nat (r k i) (\lambda (n:
101 nat).(eq nat n (r k i))) (refl_equal nat (r k i)) (minus (S (r k i)) (S O))
102 (minus_Sx_SO (r k i))) (r k (S i)) (r_S k i))).
108 \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k (S
109 i)) (S j)) (minus (r k i) j))))
111 \lambda (k: K).(\lambda (i: nat).(\lambda (j: nat).(eq_ind_r nat (S (r k i))
112 (\lambda (n: nat).(eq nat (minus n (S j)) (minus (r k i) j))) (refl_equal nat
113 (minus (r k i) j)) (r k (S i)) (r_S k i)))).