we removed about 100 match-with costruction turning them into applications

[helm.git] / matita / contribs / LAMBDA-TYPES / Level-1 / LambdaDelta / clear / props.ma

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/clear/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/clear/props.ma
index d6c1350942acbc8a3a38f9ea3df687bae2a4bb26..d2e2aa7fc9df3fa77aabbc2700ad4db3547edd7b 100644 (file)
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/clear/props.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/clear/props.ma
@@ -26,12 +26,12 @@ theorem clear_clear:
 (CSort n) c2)).(clear_gen_sort c2 n H (clear c2 c2))))) (\lambda (c: 
 C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (clear c2 
 c2))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (H0: (clear 
-(CHead c k t) c2)).((match k in K return (\lambda (k0: K).((clear (CHead c k0 
-t) c2) \to (clear c2 c2))) with [(Bind b) \Rightarrow (\lambda (H1: (clear 
-(CHead c (Bind b) t) c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: 
-C).(clear c0 c0)) (clear_bind b c t) c2 (clear_gen_bind b c c2 t H1))) | 
-(Flat f) \Rightarrow (\lambda (H1: (clear (CHead c (Flat f) t) c2)).(H c2 
-(clear_gen_flat f c c2 t H1)))]) H0))))))) c1).
+(CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear (CHead c k0 t) c2) \to 
+(clear c2 c2))) (\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t) 
+c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(clear c0 c0)) 
+(clear_bind b c t) c2 (clear_gen_bind b c c2 t H1)))) (\lambda (f: 
+F).(\lambda (H1: (clear (CHead c (Flat f) t) c2)).(H c2 (clear_gen_flat f c 
+c2 t H1)))) k H0))))))) c1).
 
 theorem clear_mono:
  \forall (c: C).(\forall (c1: C).((clear c c1) \to (\forall (c2: C).((clear c 
@@ -44,17 +44,16 @@ C).(\lambda (H0: (clear (CSort n) c2)).(clear_gen_sort c2 n H0 (eq C c1
 c2))))))) (\lambda (c0: C).(\lambda (H: ((\forall (c1: C).((clear c0 c1) \to 
 (\forall (c2: C).((clear c0 c2) \to (eq C c1 c2))))))).(\lambda (k: 
 K).(\lambda (t: T).(\lambda (c1: C).(\lambda (H0: (clear (CHead c0 k t) 
-c1)).(\lambda (c2: C).(\lambda (H1: (clear (CHead c0 k t) c2)).((match k in K 
-return (\lambda (k0: K).((clear (CHead c0 k0 t) c1) \to ((clear (CHead c0 k0 
-t) c2) \to (eq C c1 c2)))) with [(Bind b) \Rightarrow (\lambda (H2: (clear 
-(CHead c0 (Bind b) t) c1)).(\lambda (H3: (clear (CHead c0 (Bind b) t) 
-c2)).(eq_ind_r C (CHead c0 (Bind b) t) (\lambda (c3: C).(eq C c1 c3)) 
-(eq_ind_r C (CHead c0 (Bind b) t) (\lambda (c3: C).(eq C c3 (CHead c0 (Bind 
-b) t))) (refl_equal C (CHead c0 (Bind b) t)) c1 (clear_gen_bind b c0 c1 t 
-H2)) c2 (clear_gen_bind b c0 c2 t H3)))) | (Flat f) \Rightarrow (\lambda (H2: 
-(clear (CHead c0 (Flat f) t) c1)).(\lambda (H3: (clear (CHead c0 (Flat f) t) 
-c2)).(H c1 (clear_gen_flat f c0 c1 t H2) c2 (clear_gen_flat f c0 c2 t 
-H3))))]) H0 H1))))))))) c).
+c1)).(\lambda (c2: C).(\lambda (H1: (clear (CHead c0 k t) c2)).(K_ind 
+(\lambda (k0: K).((clear (CHead c0 k0 t) c1) \to ((clear (CHead c0 k0 t) c2) 
+\to (eq C c1 c2)))) (\lambda (b: B).(\lambda (H2: (clear (CHead c0 (Bind b) 
+t) c1)).(\lambda (H3: (clear (CHead c0 (Bind b) t) c2)).(eq_ind_r C (CHead c0 
+(Bind b) t) (\lambda (c3: C).(eq C c1 c3)) (eq_ind_r C (CHead c0 (Bind b) t) 
+(\lambda (c3: C).(eq C c3 (CHead c0 (Bind b) t))) (refl_equal C (CHead c0 
+(Bind b) t)) c1 (clear_gen_bind b c0 c1 t H2)) c2 (clear_gen_bind b c0 c2 t 
+H3))))) (\lambda (f: F).(\lambda (H2: (clear (CHead c0 (Flat f) t) 
+c1)).(\lambda (H3: (clear (CHead c0 (Flat f) t) c2)).(H c1 (clear_gen_flat f 
+c0 c1 t H2) c2 (clear_gen_flat f c0 c2 t H3))))) k H0 H1))))))))) c).
 
 theorem clear_trans:
  \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (c2: C).((clear c 
@@ -67,15 +66,14 @@ C).(\lambda (_: (clear c c2)).(clear_gen_sort c n H (clear (CSort n)
 c2))))))) (\lambda (c: C).(\lambda (H: ((\forall (c0: C).((clear c c0) \to 
 (\forall (c2: C).((clear c0 c2) \to (clear c c2))))))).(\lambda (k: 
 K).(\lambda (t: T).(\lambda (c0: C).(\lambda (H0: (clear (CHead c k t) 
-c0)).(\lambda (c2: C).(\lambda (H1: (clear c0 c2)).((match k in K return 
-(\lambda (k0: K).((clear (CHead c k0 t) c0) \to (clear (CHead c k0 t) c2))) 
-with [(Bind b) \Rightarrow (\lambda (H2: (clear (CHead c (Bind b) t) 
-c0)).(let H3 \def (eq_ind C c0 (\lambda (c: C).(clear c c2)) H1 (CHead c 
-(Bind b) t) (clear_gen_bind b c c0 t H2)) in (eq_ind_r C (CHead c (Bind b) t) 
-(\lambda (c3: C).(clear (CHead c (Bind b) t) c3)) (clear_bind b c t) c2 
-(clear_gen_bind b c c2 t H3)))) | (Flat f) \Rightarrow (\lambda (H2: (clear 
-(CHead c (Flat f) t) c0)).(clear_flat c c2 (H c0 (clear_gen_flat f c c0 t H2) 
-c2 H1) f t))]) H0))))))))) c1).
+c0)).(\lambda (c2: C).(\lambda (H1: (clear c0 c2)).(K_ind (\lambda (k0: 
+K).((clear (CHead c k0 t) c0) \to (clear (CHead c k0 t) c2))) (\lambda (b: 
+B).(\lambda (H2: (clear (CHead c (Bind b) t) c0)).(let H3 \def (eq_ind C c0 
+(\lambda (c3: C).(clear c3 c2)) H1 (CHead c (Bind b) t) (clear_gen_bind b c 
+c0 t H2)) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(clear (CHead 
+c (Bind b) t) c3)) (clear_bind b c t) c2 (clear_gen_bind b c c2 t H3))))) 
+(\lambda (f: F).(\lambda (H2: (clear (CHead c (Flat f) t) c0)).(clear_flat c 
+c2 (H c0 (clear_gen_flat f c c0 t H2) c2 H1) f t))) k H0))))))))) c1).
 
 theorem clear_ctail:
  \forall (b: B).(\forall (c1: C).(\forall (c2: C).(\forall (u2: T).((clear c1 
@@ -87,41 +85,40 @@ C).(\forall (u2: T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k:
 K).(\forall (u1: T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) 
 u2)))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H: 
 (clear (CSort n) (CHead c2 (Bind b) u2))).(\lambda (k: K).(\lambda (u1: 
-T).(match k in K return (\lambda (k0: K).(clear (CHead (CSort n) k0 u1) 
-(CHead (CTail k0 u1 c2) (Bind b) u2))) with [(Bind b0) \Rightarrow 
-(clear_gen_sort (CHead c2 (Bind b) u2) n H (clear (CHead (CSort n) (Bind b0) 
-u1) (CHead (CTail (Bind b0) u1 c2) (Bind b) u2))) | (Flat f) \Rightarrow 
-(clear_gen_sort (CHead c2 (Bind b) u2) n H (clear (CHead (CSort n) (Flat f) 
-u1) (CHead (CTail (Flat f) u1 c2) (Bind b) u2)))]))))))) (\lambda (c: 
-C).(\lambda (H: ((\forall (c2: C).(\forall (u2: T).((clear c (CHead c2 (Bind 
-b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k u1 c) (CHead 
-(CTail k u1 c2) (Bind b) u2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda 
-(c2: C).(\lambda (u2: T).(\lambda (H0: (clear (CHead c k t) (CHead c2 (Bind 
-b) u2))).(\lambda (k0: K).(\lambda (u1: T).((match k in K return (\lambda 
-(k1: K).((clear (CHead c k1 t) (CHead c2 (Bind b) u2)) \to (clear (CHead 
-(CTail k0 u1 c) k1 t) (CHead (CTail k0 u1 c2) (Bind b) u2)))) with [(Bind b0) 
-\Rightarrow (\lambda (H1: (clear (CHead c (Bind b0) t) (CHead c2 (Bind b) 
-u2))).(let H2 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda 
-(_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c _ _) \Rightarrow c])) 
-(CHead c2 (Bind b) u2) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 
-(Bind b) u2) t H1)) in ((let H3 \def (f_equal C B (\lambda (e: C).(match e in 
-C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) 
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) 
-\Rightarrow b | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead c 
-(Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in ((let H4 
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) 
-with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 
+T).(K_ind (\lambda (k0: K).(clear (CHead (CSort n) k0 u1) (CHead (CTail k0 u1 
+c2) (Bind b) u2))) (\lambda (b0: B).(clear_gen_sort (CHead c2 (Bind b) u2) n 
+H (clear (CHead (CSort n) (Bind b0) u1) (CHead (CTail (Bind b0) u1 c2) (Bind 
+b) u2)))) (\lambda (f: F).(clear_gen_sort (CHead c2 (Bind b) u2) n H (clear 
+(CHead (CSort n) (Flat f) u1) (CHead (CTail (Flat f) u1 c2) (Bind b) u2)))) 
+k))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (u2: 
+T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: 
+T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) u2))))))))).(\lambda 
+(k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H0: (clear 
+(CHead c k t) (CHead c2 (Bind b) u2))).(\lambda (k0: K).(\lambda (u1: 
+T).(K_ind (\lambda (k1: K).((clear (CHead c k1 t) (CHead c2 (Bind b) u2)) \to 
+(clear (CHead (CTail k0 u1 c) k1 t) (CHead (CTail k0 u1 c2) (Bind b) u2)))) 
+(\lambda (b0: B).(\lambda (H1: (clear (CHead c (Bind b0) t) (CHead c2 (Bind 
+b) u2))).(let H2 \def (f_equal C C (\lambda (e: C).(match e in C return 
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) 
+\Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t) 
+(clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in ((let H3 \def (f_equal 
+C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) 
+\Rightarrow b | (CHead _ k1 _) \Rightarrow (match k1 in K return (\lambda (_: 
+K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead c2 
 (Bind b) u2) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) 
-u2) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C c2 c)).(eq_ind_r 
-T t (\lambda (t0: T).(clear (CHead (CTail k0 u1 c) (Bind b0) t) (CHead (CTail 
-k0 u1 c2) (Bind b) t0))) (eq_ind_r C c (\lambda (c0: C).(clear (CHead (CTail 
-k0 u1 c) (Bind b0) t) (CHead (CTail k0 u1 c0) (Bind b) t))) (eq_ind B b 
-(\lambda (b1: B).(clear (CHead (CTail k0 u1 c) (Bind b1) t) (CHead (CTail k0 
-u1 c) (Bind b) t))) (clear_bind b (CTail k0 u1 c) t) b0 H5) c2 H6) u2 H4)))) 
-H3)) H2))) | (Flat f) \Rightarrow (\lambda (H1: (clear (CHead c (Flat f) t) 
-(CHead c2 (Bind b) u2))).(clear_flat (CTail k0 u1 c) (CHead (CTail k0 u1 c2) 
-(Bind b) u2) (H c2 u2 (clear_gen_flat f c (CHead c2 (Bind b) u2) t H1) k0 u1) 
-f t))]) H0)))))))))) c1)).
+u2) t H1)) in ((let H4 \def (f_equal C T (\lambda (e: C).(match e in C return 
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t0) 
+\Rightarrow t0])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t) 
+(clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b 
+b0)).(\lambda (H6: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: T).(clear (CHead 
+(CTail k0 u1 c) (Bind b0) t) (CHead (CTail k0 u1 c2) (Bind b) t0))) (eq_ind_r 
+C c (\lambda (c0: C).(clear (CHead (CTail k0 u1 c) (Bind b0) t) (CHead (CTail 
+k0 u1 c0) (Bind b) t))) (eq_ind B b (\lambda (b1: B).(clear (CHead (CTail k0 
+u1 c) (Bind b1) t) (CHead (CTail k0 u1 c) (Bind b) t))) (clear_bind b (CTail 
+k0 u1 c) t) b0 H5) c2 H6) u2 H4)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1: 
+(clear (CHead c (Flat f) t) (CHead c2 (Bind b) u2))).(clear_flat (CTail k0 u1 
+c) (CHead (CTail k0 u1 c2) (Bind b) u2) (H c2 u2 (clear_gen_flat f c (CHead 
+c2 (Bind b) u2) t H1) k0 u1) f t))) k H0)))))))))) c1)).
 
 theorem clear_cle:
  \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (cle c2 c1)))
@@ -131,14 +128,13 @@ theorem clear_cle:
 (H: (clear (CSort n) c2)).(clear_gen_sort c2 n H (le (cweight c2) O))))) 
 (\lambda (c: C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (le (cweight 
 c2) (cweight c)))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: 
-C).(\lambda (H0: (clear (CHead c k t) c2)).((match k in K return (\lambda 
-(k0: K).((clear (CHead c k0 t) c2) \to (le (cweight c2) (plus (cweight c) 
-(tweight t))))) with [(Bind b) \Rightarrow (\lambda (H1: (clear (CHead c 
-(Bind b) t) c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(le 
-(cweight c0) (plus (cweight c) (tweight t)))) (le_n (plus (cweight c) 
-(tweight t))) c2 (clear_gen_bind b c c2 t H1))) | (Flat f) \Rightarrow 
-(\lambda (H1: (clear (CHead c (Flat f) t) c2)).(le_S_n (cweight c2) (plus 
-(cweight c) (tweight t)) (le_n_S (cweight c2) (plus (cweight c) (tweight t)) 
-(le_plus_trans (cweight c2) (cweight c) (tweight t) (H c2 (clear_gen_flat f c 
-c2 t H1))))))]) H0))))))) c1).
+C).(\lambda (H0: (clear (CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear 
+(CHead c k0 t) c2) \to (le (cweight c2) (plus (cweight c) (tweight t))))) 
+(\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C 
+(CHead c (Bind b) t) (\lambda (c0: C).(le (cweight c0) (plus (cweight c) 
+(tweight t)))) (le_n (plus (cweight c) (tweight t))) c2 (clear_gen_bind b c 
+c2 t H1)))) (\lambda (f: F).(\lambda (H1: (clear (CHead c (Flat f) t) 
+c2)).(le_S_n (cweight c2) (plus (cweight c) (tweight t)) (le_n_S (cweight c2) 
+(plus (cweight c) (tweight t)) (le_plus_trans (cweight c2) (cweight c) 
+(tweight t) (H c2 (clear_gen_flat f c c2 t H1))))))) k H0))))))) c1).