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 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/s/props".
22 \forall (k: K).(\forall (i: nat).(eq nat (s k (S i)) (S (s k i))))
24 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (S
25 i)) (S (s k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (s
26 (Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (s (Flat
30 \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j))
33 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
34 nat).(eq nat (s k0 (plus i j)) (plus (s k0 i) j))))) (\lambda (b: B).(\lambda
35 (i: nat).(\lambda (j: nat).(refl_equal nat (plus (s (Bind b) i) j)))))
36 (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (s
37 (Flat f) i) j))))) k).
40 \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j))
43 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
44 nat).(eq nat (s k0 (plus i j)) (plus i (s k0 j)))))) (\lambda (_: B).(\lambda
45 (i: nat).(\lambda (j: nat).(eq_ind_r nat (plus i (S j)) (\lambda (n: nat).(eq
46 nat n (plus i (S j)))) (refl_equal nat (plus i (S j))) (S (plus i j))
47 (plus_n_Sm i j))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j:
48 nat).(refl_equal nat (plus i (s (Flat f) j)))))) k).
51 \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le j i) \to (eq nat (s
52 k (minus i j)) (minus (s k i) j)))))
54 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
55 nat).((le j i) \to (eq nat (s k0 (minus i j)) (minus (s k0 i) j))))))
56 (\lambda (_: B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le j
57 i)).(eq_ind_r nat (minus (S i) j) (\lambda (n: nat).(eq nat n (minus (S i)
58 j))) (refl_equal nat (minus (S i) j)) (S (minus i j)) (minus_Sn_m i j H))))))
59 (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (_: (le j
60 i)).(refl_equal nat (minus (s (Flat f) i) j)))))) k).
63 \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (s k i) (s
66 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
67 nat).(eq nat (minus (s k0 i) (s k0 j)) (minus i j))))) (\lambda (_:
68 B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i j)))))
69 (\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i
73 \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le i j) \to (le (s k i)
76 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
77 nat).((le i j) \to (le (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i:
78 nat).(\lambda (j: nat).(\lambda (H: (le i j)).(le_S_n (S i) (S j) (lt_le_S (S
79 i) (S (S j)) (lt_n_S i (S j) (le_lt_n_Sm i j H)))))))) (\lambda (_:
80 F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le i j)).H)))) k).
83 \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt i j) \to (lt (s k i)
86 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
87 nat).((lt i j) \to (lt (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i:
88 nat).(\lambda (j: nat).(\lambda (H: (lt i j)).(le_S_n (S (S i)) (S j) (le_n_S
89 (S (S i)) (S j) (le_n_S (S i) j H))))))) (\lambda (_: F).(\lambda (i:
90 nat).(\lambda (j: nat).(\lambda (H: (lt i j)).H)))) k).
93 \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (s k i) (s k j))
96 \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
97 nat).((eq nat (s k0 i) (s k0 j)) \to (eq nat i j))))) (\lambda (b:
98 B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (eq nat (s (Bind b) i) (s
99 (Bind b) j))).(eq_add_S i j H))))) (\lambda (f: F).(\lambda (i: nat).(\lambda
100 (j: nat).(\lambda (H: (eq nat (s (Flat f) i) (s (Flat f) j))).H)))) k).
103 \forall (k: K).(\forall (i: nat).(eq nat (minus (s k i) (s k O)) i))
105 \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (minus i O) (\lambda (n:
106 nat).(eq nat n i)) (eq_ind nat i (\lambda (n: nat).(eq nat n i)) (refl_equal
107 nat i) (minus i O) (minus_n_O i)) (minus (s k i) (s k O)) (minus_s_s k i O))).
110 \forall (b: B).(\forall (i: nat).(eq nat (minus (s (Bind b) i) (S O)) i))
112 \lambda (_: B).(\lambda (i: nat).(eq_ind nat i (\lambda (n: nat).(eq nat n
113 i)) (refl_equal nat i) (minus i O) (minus_n_O i))).