(* This file was automatically generated: do not edit *********************)
-include "Basic-1/r/defs.ma".
+include "basic_1/r/defs.ma".
-include "Basic-1/s/defs.ma".
+include "basic_1/s/defs.ma".
-theorem r_S:
+lemma r_S:
\forall (k: K).(\forall (i: nat).(eq nat (r k (S i)) (S (r k i))))
\def
\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (r k0 (S
i)) (S (r k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (r
(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (r (Flat
f) i))))) k).
-(* COMMENTS
-Initial nodes: 65
-END *)
-theorem r_plus:
+lemma r_plus:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j))
(plus (r k i) j))))
\def
(i: nat).(\lambda (j: nat).(refl_equal nat (plus (r (Bind b) i) j)))))
(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (r
(Flat f) i) j))))) k).
-(* COMMENTS
-Initial nodes: 79
-END *)
-theorem r_plus_sym:
+lemma r_plus_sym:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j))
(plus i (r k j)))))
\def
nat).(eq nat (r k0 (plus i j)) (plus i (r k0 j)))))) (\lambda (_: B).(\lambda
(i: nat).(\lambda (j: nat).(refl_equal nat (plus i j))))) (\lambda (_:
F).(\lambda (i: nat).(\lambda (j: nat).(plus_n_Sm i j)))) k).
-(* COMMENTS
-Initial nodes: 63
-END *)
-theorem r_minus:
+lemma r_minus:
\forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (k: K).(eq nat
(minus (r k i) (S n)) (r k (minus i (S n)))))))
\def
K).(K_ind (\lambda (k0: K).(eq nat (minus (r k0 i) (S n)) (r k0 (minus i (S
n))))) (\lambda (_: B).(refl_equal nat (minus i (S n)))) (\lambda (_:
F).(minus_x_Sy i n H)) k)))).
-(* COMMENTS
-Initial nodes: 69
-END *)
-theorem r_dis:
+lemma r_dis:
\forall (k: K).(\forall (P: Prop).(((((\forall (i: nat).(eq nat (r k i) i)))
\to P)) \to (((((\forall (i: nat).(eq nat (r k i) (S i)))) \to P)) \to P)))
\def
((((\forall (i: nat).(eq nat (r (Flat f) i) i))) \to P))).(\lambda (H0:
((((\forall (i: nat).(eq nat (r (Flat f) i) (S i)))) \to P))).(H0 (\lambda
(i: nat).(refl_equal nat (S i)))))))) k).
-(* COMMENTS
-Initial nodes: 151
-END *)
-theorem s_r:
+lemma s_r:
\forall (k: K).(\forall (i: nat).(eq nat (s k (r k i)) (S i)))
\def
\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (r k0
i)) (S i)))) (\lambda (_: B).(\lambda (i: nat).(refl_equal nat (S i))))
(\lambda (_: F).(\lambda (i: nat).(refl_equal nat (S i)))) k).
-(* COMMENTS
-Initial nodes: 51
-END *)
-theorem r_arith0:
+lemma r_arith0:
\forall (k: K).(\forall (i: nat).(eq nat (minus (r k (S i)) (S O)) (r k i)))
\def
\lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (S (r k i)) (\lambda (n:
nat).(eq nat (minus n (S O)) (r k i))) (eq_ind_r nat (r k i) (\lambda (n:
nat).(eq nat n (r k i))) (refl_equal nat (r k i)) (minus (S (r k i)) (S O))
(minus_Sx_SO (r k i))) (r k (S i)) (r_S k i))).
-(* COMMENTS
-Initial nodes: 105
-END *)
-theorem r_arith1:
+lemma r_arith1:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k (S
i)) (S j)) (minus (r k i) j))))
\def
\lambda (k: K).(\lambda (i: nat).(\lambda (j: nat).(eq_ind_r nat (S (r k i))
(\lambda (n: nat).(eq nat (minus n (S j)) (minus (r k i) j))) (refl_equal nat
(minus (r k i) j)) (r k (S i)) (r_S k i)))).
-(* COMMENTS
-Initial nodes: 69
-END *)
+
+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).