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 "LambdaDelta-1/s/defs.ma".
20 \forall (k: K).(\forall (i: nat).(eq nat (s k (S i)) (S (s k i))))
22 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (S
23 i)) (S (s k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (s
24 (Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (s (Flat
28 \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j))
31 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
32 nat).(eq nat (s k0 (plus i j)) (plus (s k0 i) j))))) (\lambda (b: B).(\lambda
33 (i: nat).(\lambda (j: nat).(refl_equal nat (plus (s (Bind b) i) j)))))
34 (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (s
35 (Flat f) i) j))))) k).
38 \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j))
41 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
42 nat).(eq nat (s k0 (plus i j)) (plus i (s k0 j)))))) (\lambda (_: B).(\lambda
43 (i: nat).(\lambda (j: nat).(eq_ind_r nat (plus i (S j)) (\lambda (n: nat).(eq
44 nat n (plus i (S j)))) (refl_equal nat (plus i (S j))) (S (plus i j))
45 (plus_n_Sm i j))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j:
46 nat).(refl_equal nat (plus i (s (Flat f) j)))))) k).
49 \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le j i) \to (eq nat (s
50 k (minus i j)) (minus (s k i) j)))))
52 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
53 nat).((le j i) \to (eq nat (s k0 (minus i j)) (minus (s k0 i) j))))))
54 (\lambda (_: B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le j
55 i)).(eq_ind_r nat (minus (S i) j) (\lambda (n: nat).(eq nat n (minus (S i)
56 j))) (refl_equal nat (minus (S i) j)) (S (minus i j)) (minus_Sn_m i j H))))))
57 (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (_: (le j
58 i)).(refl_equal nat (minus (s (Flat f) i) j)))))) k).
61 \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (s k i) (s
64 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
65 nat).(eq nat (minus (s k0 i) (s k0 j)) (minus i j))))) (\lambda (_:
66 B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i j)))))
67 (\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i
71 \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le i j) \to (le (s k i)
74 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
75 nat).((le i j) \to (le (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i:
76 nat).(\lambda (j: nat).(\lambda (H: (le i j)).(le_n_S i j H))))) (\lambda (_:
77 F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le i j)).H)))) k).
80 \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt i j) \to (lt (s k i)
83 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
84 nat).((lt i j) \to (lt (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i:
85 nat).(\lambda (j: nat).(\lambda (H: (lt i j)).(le_n_S (S i) j H))))) (\lambda
86 (_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt i j)).H)))) k).
89 \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (s k i) (s k j))
92 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
93 nat).((eq nat (s k0 i) (s k0 j)) \to (eq nat i j))))) (\lambda (b:
94 B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (eq nat (s (Bind b) i) (s
95 (Bind b) j))).(eq_add_S i j H))))) (\lambda (f: F).(\lambda (i: nat).(\lambda
96 (j: nat).(\lambda (H: (eq nat (s (Flat f) i) (s (Flat f) j))).H)))) k).
99 \forall (k: K).(\forall (i: nat).(le i (s k i)))
101 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(le i (s k0 i))))
102 (\lambda (b: B).(\lambda (i: nat).(le_S_n i (s (Bind b) i) (le_S (S i) (s
103 (Bind b) i) (le_n (s (Bind b) i)))))) (\lambda (f: F).(\lambda (i: nat).(le_n
104 (s (Flat f) i)))) k).
107 \forall (k: K).(\forall (i: nat).(eq nat (minus (s k i) (s k O)) i))
109 \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (minus i O) (\lambda (n:
110 nat).(eq nat n i)) (eq_ind nat i (\lambda (n: nat).(eq nat n i)) (refl_equal
111 nat i) (minus i O) (minus_n_O i)) (minus (s k i) (s k O)) (minus_s_s k i O))).
114 \forall (b: B).(\forall (i: nat).(eq nat (minus (s (Bind b) i) (S O)) i))
116 \lambda (_: B).(\lambda (i: nat).(eq_ind nat i (\lambda (n: nat).(eq nat n
117 i)) (refl_equal nat i) (minus i O) (minus_n_O i))).