--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/s/defs.ma".
+
+theorem s_S:
+ \forall (k: K).(\forall (i: nat).(eq nat (s k (S i)) (S (s k i))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (S
+i)) (S (s k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (s
+(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (s (Flat
+f) i))))) k).
+
+theorem s_plus:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j))
+(plus (s k i) j))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).(eq nat (s k0 (plus i j)) (plus (s k0 i) j))))) (\lambda (b: B).(\lambda
+(i: nat).(\lambda (j: nat).(refl_equal nat (plus (s (Bind b) i) j)))))
+(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (s
+(Flat f) i) j))))) k).
+
+theorem s_plus_sym:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j))
+(plus i (s k j)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).(eq nat (s k0 (plus i j)) (plus i (s k0 j)))))) (\lambda (_: B).(\lambda
+(i: nat).(\lambda (j: nat).(eq_ind_r nat (plus i (S j)) (\lambda (n: nat).(eq
+nat n (plus i (S j)))) (refl_equal nat (plus i (S j))) (S (plus i j))
+(plus_n_Sm i j))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j:
+nat).(refl_equal nat (plus i (s (Flat f) j)))))) k).
+
+theorem s_minus:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le j i) \to (eq nat (s
+k (minus i j)) (minus (s k i) j)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).((le j i) \to (eq nat (s k0 (minus i j)) (minus (s k0 i) j))))))
+(\lambda (_: B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le j
+i)).(eq_ind_r nat (minus (S i) j) (\lambda (n: nat).(eq nat n (minus (S i)
+j))) (refl_equal nat (minus (S i) j)) (S (minus i j)) (minus_Sn_m i j H))))))
+(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (_: (le j
+i)).(refl_equal nat (minus (s (Flat f) i) j)))))) k).
+
+theorem minus_s_s:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (s k i) (s
+k j)) (minus i j))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).(eq nat (minus (s k0 i) (s k0 j)) (minus i j))))) (\lambda (_:
+B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i j)))))
+(\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i
+j))))) k).
+
+theorem s_le:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le i j) \to (le (s k i)
+(s k j)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).((le i j) \to (le (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i:
+nat).(\lambda (j: nat).(\lambda (H: (le i j)).(le_n_S i j H))))) (\lambda (_:
+F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le i j)).H)))) k).
+
+theorem s_lt:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt i j) \to (lt (s k i)
+(s k j)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).((lt i j) \to (lt (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i:
+nat).(\lambda (j: nat).(\lambda (H: (lt i j)).(le_n_S (S i) j H))))) (\lambda
+(_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt i j)).H)))) k).
+
+theorem s_inj:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (s k i) (s k j))
+\to (eq nat i j))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).((eq nat (s k0 i) (s k0 j)) \to (eq nat i j))))) (\lambda (b:
+B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (eq nat (s (Bind b) i) (s
+(Bind b) j))).(eq_add_S i j H))))) (\lambda (f: F).(\lambda (i: nat).(\lambda
+(j: nat).(\lambda (H: (eq nat (s (Flat f) i) (s (Flat f) j))).H)))) k).
+
+theorem s_inc:
+ \forall (k: K).(\forall (i: nat).(le i (s k i)))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(le i (s k0 i))))
+(\lambda (b: B).(\lambda (i: nat).(le_S_n i (s (Bind b) i) (le_S (S i) (s
+(Bind b) i) (le_n (s (Bind b) i)))))) (\lambda (f: F).(\lambda (i: nat).(le_n
+(s (Flat f) i)))) k).
+
+theorem s_arith0:
+ \forall (k: K).(\forall (i: nat).(eq nat (minus (s k i) (s k O)) i))
+\def
+ \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (minus i O) (\lambda (n:
+nat).(eq nat n i)) (eq_ind nat i (\lambda (n: nat).(eq nat n i)) (refl_equal
+nat i) (minus i O) (minus_n_O i)) (minus (s k i) (s k O)) (minus_s_s k i O))).
+
+theorem s_arith1:
+ \forall (b: B).(\forall (i: nat).(eq nat (minus (s (Bind b) i) (S O)) i))
+\def
+ \lambda (_: B).(\lambda (i: nat).(eq_ind nat i (\lambda (n: nat).(eq nat n
+i)) (refl_equal nat i) (minus i O) (minus_n_O i))).
+
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