update in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_1 / r / props.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_1/r/props.ma b/matita/matita/contribs/lambdadelta/basic_1/r/props.ma
index 0815aaf5e8f042a0c78610639874b09323c36dc9..6dc07a0e181260407f027d8dfc1164f51b1f0a9c 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/r/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/r/props.ma
@@ -14,22 +14,19 @@
 
 (* 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
@@ -38,11 +35,8 @@ nat).(eq nat (r k0 (plus i j)) (plus (r k0 i) j))))) (\lambda (b: B).(\lambda
 (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
@@ -50,11 +44,8 @@ theorem r_plus_sym:
 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
@@ -62,11 +53,8 @@ theorem r_minus:
 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
@@ -79,39 +67,87 @@ nat).(refl_equal nat i))))))) (\lambda (f: F).(\lambda (P: Prop).(\lambda (_:
 ((((\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).