(* This file was automatically generated: do not edit *********************)
-include "LambdaDelta-1/s/defs.ma".
+include "Basic-1/s/defs.ma".
theorem s_S:
\forall (k: K).(\forall (i: nat).(eq nat (s k (S i)) (S (s k i))))
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).
+(* COMMENTS
+Initial nodes: 65
+END *)
theorem s_plus:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j))
(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).
+(* COMMENTS
+Initial nodes: 79
+END *)
theorem s_plus_sym:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j))
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).
+(* COMMENTS
+Initial nodes: 117
+END *)
theorem s_minus:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).((le j i) \to (eq nat (s
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).
+(* COMMENTS
+Initial nodes: 137
+END *)
theorem minus_s_s:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (s k i) (s
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).
+(* COMMENTS
+Initial nodes: 67
+END *)
theorem s_le:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).((le i j) \to (le (s k i)
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).
+(* COMMENTS
+Initial nodes: 65
+END *)
theorem s_lt:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt i j) \to (lt (s k i)
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).
+(* COMMENTS
+Initial nodes: 67
+END *)
theorem s_inj:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (s k i) (s k j))
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).
+(* COMMENTS
+Initial nodes: 97
+END *)
theorem s_inc:
\forall (k: K).(\forall (i: nat).(le i (s k 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).
+(* COMMENTS
+Initial nodes: 73
+END *)
theorem s_arith0:
\forall (k: K).(\forall (i: nat).(eq nat (minus (s k i) (s k O)) i))
\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))).
+(* COMMENTS
+Initial nodes: 77
+END *)
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))).
+(* COMMENTS
+Initial nodes: 35
+END *)