+
+lemma r_arith2:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le (S i) (s k j)) \to
+(le (r k i) j))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).((le (S i) (s k0 j)) \to (le (r k0 i) j))))) (\lambda (_: B).(\lambda
+(i: nat).(\lambda (j: nat).(\lambda (H: (le (S i) (S j))).(let H_y \def
+(le_S_n i j H) in H_y))))) (\lambda (_: F).(\lambda (i: nat).(\lambda (j:
+nat).(\lambda (H: (le (S i) j)).H)))) k).
+
+lemma r_arith3:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le (s k j) (S i)) \to
+(le j (r k i)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).((le (s k0 j) (S i)) \to (le j (r k0 i)))))) (\lambda (_: B).(\lambda
+(i: nat).(\lambda (j: nat).(\lambda (H: (le (S j) (S i))).(let H_y \def
+(le_S_n j i H) in H_y))))) (\lambda (_: F).(\lambda (i: nat).(\lambda (j:
+nat).(\lambda (H: (le j (S i))).H)))) k).
+
+lemma r_arith4:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (S i) (s k
+j)) (minus (r k i) j))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).(eq nat (minus (S i) (s k0 j)) (minus (r k0 i) j))))) (\lambda (b:
+B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus (r (Bind b) i)
+j))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat
+(minus (r (Flat f) i) j))))) k).
+
+lemma r_arith5:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt (s k j) (S i)) \to
+(lt j (r k i)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).((lt (s k0 j) (S i)) \to (lt j (r k0 i)))))) (\lambda (_: B).(\lambda
+(i: nat).(\lambda (j: nat).(\lambda (H: (lt (S j) (S i))).(lt_S_n j i H)))))
+(\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt j (S
+i))).H)))) k).
+
+lemma r_arith6:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k i) (S
+j)) (minus i (s k j)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).(eq nat (minus (r k0 i) (S j)) (minus i (s k0 j)))))) (\lambda (b:
+B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i (s (Bind b)
+j)))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat
+(minus i (s (Flat f) j)))))) k).
+
+lemma r_arith7:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (S i) (s k j))
+\to (eq nat (r k i) j))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).((eq nat (S i) (s k0 j)) \to (eq nat (r k0 i) j))))) (\lambda (_:
+B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (eq nat (S i) (S
+j))).(eq_add_S i j H))))) (\lambda (_: F).(\lambda (i: nat).(\lambda (j:
+nat).(\lambda (H: (eq nat (S i) j)).H)))) k).