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 80ebd666e2e290e36103be9b709e77b6d6c97d52..6dc07a0e181260407f027d8dfc1164f51b1f0a9c 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/r/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/r/props.ma
@@ -18,7 +18,7 @@ include "basic_1/r/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 
@@ -26,7 +26,7 @@ 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).
 
-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
@@ -36,7 +36,7 @@ nat).(eq nat (r k0 (plus i j)) (plus (r k0 i) j))))) (\lambda (b: B).(\lambda
 (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (r 
 (Flat f) i) j))))) k).
 
-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
@@ -45,7 +45,7 @@ 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).
 
-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
@@ -54,7 +54,7 @@ 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)))).
 
-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
@@ -68,14 +68,14 @@ nat).(refl_equal nat i))))))) (\lambda (f: F).(\lambda (P: Prop).(\lambda (_:
 ((((\forall (i: nat).(eq nat (r (Flat f) i) (S i)))) \to P))).(H0 (\lambda 
 (i: nat).(refl_equal nat (S i)))))))) k).
 
-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).
 
-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: 
@@ -83,7 +83,7 @@ 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))).
 
-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
@@ -91,7 +91,7 @@ i)) (S j)) (minus (r k i) j))))
 (\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)))).
 
-theorem r_arith2:
+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
@@ -101,7 +101,7 @@ nat).((le (S i) (s k0 j)) \to (le (r k0 i) j))))) (\lambda (_: B).(\lambda
 (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).
 
-theorem r_arith3:
+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
@@ -111,7 +111,7 @@ nat).((le (s k0 j) (S i)) \to (le j (r k0 i)))))) (\lambda (_: B).(\lambda
 (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).
 
-theorem r_arith4:
+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
@@ -121,7 +121,7 @@ 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).
 
-theorem r_arith5:
+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
@@ -131,7 +131,7 @@ nat).((lt (s k0 j) (S i)) \to (lt j (r k0 i)))))) (\lambda (_: B).(\lambda
 (\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt j (S 
 i))).H)))) k).
 
-theorem r_arith6:
+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
@@ -141,7 +141,7 @@ 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).
 
-theorem r_arith7:
+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