update in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / ground_1 / blt / props.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_1/blt/props.ma b/matita/matita/contribs/lambdadelta/ground_1/blt/props.ma
index 689d6b45e2e96a20383fedf95da8dba1034848f6..7a6c3f27a12b90c8f972157e8cb1b019fa84fb64 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_1/blt/props.ma
+++ b/matita/matita/contribs/lambdadelta/ground_1/blt/props.ma
@@ -16,27 +16,26 @@
 
 include "ground_1/blt/defs.ma".
 
-theorem lt_blt:
+lemma lt_blt:
  \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq bool (blt y x) true)))
 \def
  \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to 
 (eq bool (blt y n) true)))) (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0 
-\def (match H in le with [le_n \Rightarrow (\lambda (H0: (eq nat (S y) 
-O)).(let H1 \def (eq_ind nat (S y) (\lambda (e: nat).(match e in nat with [O 
-\Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq bool 
-(blt y O) true) H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) 
-O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: nat).(match e in nat with [O 
-\Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S 
-y) m) \to (eq bool (blt y O) true)) H2)) H0))]) in (H0 (refl_equal nat O))))) 
-(\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq bool (blt 
-y n) true))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((lt n0 (S n)) 
-\to (eq bool (blt n0 (S n)) true))) (\lambda (_: (lt O (S n))).(refl_equal 
-bool true)) (\lambda (n0: nat).(\lambda (_: (((lt n0 (S n)) \to (eq bool 
-(match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m n)]) 
-true)))).(\lambda (H1: (lt (S n0) (S n))).(H n0 (le_S_n (S n0) n H1))))) 
-y)))) x).
+\def (match H with [le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let H1 
+\def (eq_ind nat (S y) (\lambda (e: nat).(match e with [O \Rightarrow False | 
+(S _) \Rightarrow True])) I O H0) in (False_ind (eq bool (blt y O) true) 
+H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def 
+(eq_ind nat (S m) (\lambda (e: nat).(match e with [O \Rightarrow False | (S 
+_) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq bool (blt 
+y O) true)) H2)) H0))]) in (H0 (refl_equal nat O))))) (\lambda (n: 
+nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq bool (blt y n) 
+true))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((lt n0 (S n)) \to 
+(eq bool (blt n0 (S n)) true))) (\lambda (_: (lt O (S n))).(refl_equal bool 
+true)) (\lambda (n0: nat).(\lambda (_: (((lt n0 (S n)) \to (eq bool (match n0 
+with [O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true)))).(\lambda 
+(H1: (lt (S n0) (S n))).(H n0 (le_S_n (S n0) n H1))))) y)))) x).
 
-theorem le_bge:
+lemma le_bge:
  \forall (x: nat).(\forall (y: nat).((le x y) \to (eq bool (blt y x) false)))
 \def
  \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to 
@@ -44,36 +43,35 @@ theorem le_bge:
 y)).(refl_equal bool false))) (\lambda (n: nat).(\lambda (H: ((\forall (y: 
 nat).((le n y) \to (eq bool (blt y n) false))))).(\lambda (y: nat).(nat_ind 
 (\lambda (n0: nat).((le (S n) n0) \to (eq bool (blt n0 (S n)) false))) 
-(\lambda (H0: (le (S n) O)).(let H1 \def (match H0 in le with [le_n 
-\Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def (eq_ind nat (S n) 
-(\lambda (e: nat).(match e in nat with [O \Rightarrow False | (S _) 
-\Rightarrow True])) I O H1) in (False_ind (eq bool (blt O (S n)) false) H2))) 
-| (le_S m H1) \Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H3 \def 
-(eq_ind nat (S m) (\lambda (e: nat).(match e in nat with [O \Rightarrow False 
-| (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S n) m) \to (eq bool 
-(blt O (S n)) false)) H3)) H1))]) in (H1 (refl_equal nat O)))) (\lambda (n0: 
-nat).(\lambda (_: (((le (S n) n0) \to (eq bool (blt n0 (S n)) 
-false)))).(\lambda (H1: (le (S n) (S n0))).(H n0 (le_S_n n n0 H1))))) y)))) 
-x).
+(\lambda (H0: (le (S n) O)).(let H1 \def (match H0 with [le_n \Rightarrow 
+(\lambda (H1: (eq nat (S n) O)).(let H2 \def (eq_ind nat (S n) (\lambda (e: 
+nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) 
+in (False_ind (eq bool (blt O (S n)) false) H2))) | (le_S m H1) \Rightarrow 
+(\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e: 
+nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) 
+in (False_ind ((le (S n) m) \to (eq bool (blt O (S n)) false)) H3)) H1))]) in 
+(H1 (refl_equal nat O)))) (\lambda (n0: nat).(\lambda (_: (((le (S n) n0) \to 
+(eq bool (blt n0 (S n)) false)))).(\lambda (H1: (le (S n) (S n0))).(H n0 
+(le_S_n n n0 H1))))) y)))) x).
 
-theorem blt_lt:
+lemma blt_lt:
  \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) true) \to (lt y x)))
 \def
  \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt 
 y n) true) \to (lt y n)))) (\lambda (y: nat).(\lambda (H: (eq bool (blt y O) 
-true)).(let H0 \def (match H in eq with [refl_equal \Rightarrow (\lambda (H0: 
-(eq bool (blt y O) true)).(let H1 \def (eq_ind bool (blt y O) (\lambda (e: 
-bool).(match e in bool with [true \Rightarrow False | false \Rightarrow 
-True])) I true H0) in (False_ind (lt y O) H1)))]) in (H0 (refl_equal bool 
-true))))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq bool (blt y 
-n) true) \to (lt y n))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((eq 
-bool (blt n0 (S n)) true) \to (lt n0 (S n)))) (\lambda (_: (eq bool true 
-true)).(le_S_n (S O) (S n) (le_n_S (S O) (S n) (le_n_S O n (le_O_n n))))) 
-(\lambda (n0: nat).(\lambda (_: (((eq bool (match n0 with [O \Rightarrow true 
-| (S m) \Rightarrow (blt m n)]) true) \to (lt n0 (S n))))).(\lambda (H1: (eq 
-bool (blt n0 n) true)).(lt_n_S n0 n (H n0 H1))))) y)))) x).
+true)).(let H0 \def (match H with [refl_equal \Rightarrow (\lambda (H0: (eq 
+bool (blt y O) true)).(let H1 \def (eq_ind bool (blt y O) (\lambda (e: 
+bool).(match e with [true \Rightarrow False | false \Rightarrow True])) I 
+true H0) in (False_ind (lt y O) H1)))]) in (H0 (refl_equal bool true))))) 
+(\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq bool (blt y n) true) 
+\to (lt y n))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((eq bool (blt 
+n0 (S n)) true) \to (lt n0 (S n)))) (\lambda (_: (eq bool true true)).(le_S_n 
+(S O) (S n) (le_n_S (S O) (S n) (le_n_S O n (le_O_n n))))) (\lambda (n0: 
+nat).(\lambda (_: (((eq bool (match n0 with [O \Rightarrow true | (S m) 
+\Rightarrow (blt m n)]) true) \to (lt n0 (S n))))).(\lambda (H1: (eq bool 
+(blt n0 n) true)).(lt_n_S n0 n (H n0 H1))))) y)))) x).
 
-theorem bge_le:
+lemma bge_le:
  \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) false) \to (le x y)))
 \def
  \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt 
@@ -81,12 +79,12 @@ y n) false) \to (le n y)))) (\lambda (y: nat).(\lambda (_: (eq bool (blt y O)
 false)).(le_O_n y))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq 
 bool (blt y n) false) \to (le n y))))).(\lambda (y: nat).(nat_ind (\lambda 
 (n0: nat).((eq bool (blt n0 (S n)) false) \to (le (S n) n0))) (\lambda (H0: 
-(eq bool (blt O (S n)) false)).(let H1 \def (match H0 in eq with [refl_equal 
+(eq bool (blt O (S n)) false)).(let H1 \def (match H0 with [refl_equal 
 \Rightarrow (\lambda (H1: (eq bool (blt O (S n)) false)).(let H2 \def (eq_ind 
-bool (blt O (S n)) (\lambda (e: bool).(match e in bool with [true \Rightarrow 
-True | false \Rightarrow False])) I false H1) in (False_ind (le (S n) O) 
-H2)))]) in (H1 (refl_equal bool false)))) (\lambda (n0: nat).(\lambda (_: 
-(((eq bool (blt n0 (S n)) false) \to (le (S n) n0)))).(\lambda (H1: (eq bool 
-(blt (S n0) (S n)) false)).(le_S_n (S n) (S n0) (le_n_S (S n) (S n0) (le_n_S 
-n n0 (H n0 H1))))))) y)))) x).
+bool (blt O (S n)) (\lambda (e: bool).(match e with [true \Rightarrow True | 
+false \Rightarrow False])) I false H1) in (False_ind (le (S n) O) H2)))]) in 
+(H1 (refl_equal bool false)))) (\lambda (n0: nat).(\lambda (_: (((eq bool 
+(blt n0 (S n)) false) \to (le (S n) n0)))).(\lambda (H1: (eq bool (blt (S n0) 
+(S n)) false)).(le_S_n (S n) (S n0) (le_n_S (S n) (S n0) (le_n_S n n0 (H n0 
+H1))))))) y)))) x).