+ \lambda (k: K).(\lambda (i: nat).(\lambda (j: nat).(let TMP_1 \def (r k i)
+in (let TMP_2 \def (S TMP_1) in (let TMP_7 \def (\lambda (n: nat).(let TMP_3
+\def (S j) in (let TMP_4 \def (minus n TMP_3) in (let TMP_5 \def (r k i) in
+(let TMP_6 \def (minus TMP_5 j) in (eq nat TMP_4 TMP_6)))))) in (let TMP_8
+\def (r k i) in (let TMP_9 \def (minus TMP_8 j) in (let TMP_10 \def
+(refl_equal nat TMP_9) in (let TMP_11 \def (S i) in (let TMP_12 \def (r k
+TMP_11) in (let TMP_13 \def (r_S k i) in (eq_ind_r nat TMP_2 TMP_7 TMP_10
+TMP_12 TMP_13)))))))))))).
+
+theorem 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).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(\forall
+(j: nat).((le (S i) (s k0 j)) \to (let TMP_1 \def (r k0 i) in (le TMP_1
+j)))))) in (let TMP_3 \def (\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)))))
+in (let TMP_4 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j:
+nat).(\lambda (H: (le (S i) j)).H)))) in (K_ind TMP_2 TMP_3 TMP_4 k)))).
+
+theorem 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).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(\forall
+(j: nat).((le (s k0 j) (S i)) \to (let TMP_1 \def (r k0 i) in (le j
+TMP_1)))))) in (let TMP_3 \def (\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)))))
+in (let TMP_4 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j:
+nat).(\lambda (H: (le j (S i))).H)))) in (K_ind TMP_2 TMP_3 TMP_4 k)))).
+
+theorem 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).(let TMP_6 \def (\lambda (k0: K).(\forall (i: nat).(\forall
+(j: nat).(let TMP_1 \def (S i) in (let TMP_2 \def (s k0 j) in (let TMP_3 \def
+(minus TMP_1 TMP_2) in (let TMP_4 \def (r k0 i) in (let TMP_5 \def (minus
+TMP_4 j) in (eq nat TMP_3 TMP_5))))))))) in (let TMP_10 \def (\lambda (b:
+B).(\lambda (i: nat).(\lambda (j: nat).(let TMP_7 \def (Bind b) in (let TMP_8
+\def (r TMP_7 i) in (let TMP_9 \def (minus TMP_8 j) in (refl_equal nat
+TMP_9))))))) in (let TMP_14 \def (\lambda (f: F).(\lambda (i: nat).(\lambda
+(j: nat).(let TMP_11 \def (Flat f) in (let TMP_12 \def (r TMP_11 i) in (let
+TMP_13 \def (minus TMP_12 j) in (refl_equal nat TMP_13))))))) in (K_ind TMP_6
+TMP_10 TMP_14 k)))).
+
+theorem 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).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(\forall
+(j: nat).((lt (s k0 j) (S i)) \to (let TMP_1 \def (r k0 i) in (lt j
+TMP_1)))))) in (let TMP_3 \def (\lambda (_: B).(\lambda (i: nat).(\lambda (j:
+nat).(\lambda (H: (lt (S j) (S i))).(lt_S_n j i H))))) in (let TMP_4 \def
+(\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt j (S
+i))).H)))) in (K_ind TMP_2 TMP_3 TMP_4 k)))).
+
+theorem 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).(let TMP_6 \def (\lambda (k0: K).(\forall (i: nat).(\forall
+(j: nat).(let TMP_1 \def (r k0 i) in (let TMP_2 \def (S j) in (let TMP_3 \def
+(minus TMP_1 TMP_2) in (let TMP_4 \def (s k0 j) in (let TMP_5 \def (minus i
+TMP_4) in (eq nat TMP_3 TMP_5))))))))) in (let TMP_10 \def (\lambda (b:
+B).(\lambda (i: nat).(\lambda (j: nat).(let TMP_7 \def (Bind b) in (let TMP_8
+\def (s TMP_7 j) in (let TMP_9 \def (minus i TMP_8) in (refl_equal nat
+TMP_9))))))) in (let TMP_14 \def (\lambda (f: F).(\lambda (i: nat).(\lambda
+(j: nat).(let TMP_11 \def (Flat f) in (let TMP_12 \def (s TMP_11 j) in (let
+TMP_13 \def (minus i TMP_12) in (refl_equal nat TMP_13))))))) in (K_ind TMP_6
+TMP_10 TMP_14 k)))).
+
+theorem 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).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(\forall
+(j: nat).((eq nat (S i) (s k0 j)) \to (let TMP_1 \def (r k0 i) in (eq nat
+TMP_1 j)))))) in (let TMP_3 \def (\lambda (_: B).(\lambda (i: nat).(\lambda
+(j: nat).(\lambda (H: (eq nat (S i) (S j))).(eq_add_S i j H))))) in (let
+TMP_4 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H:
+(eq nat (S i) j)).H)))) in (K_ind TMP_2 TMP_3 TMP_4 k)))).