update in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_1 / sn3 / fwd.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/fwd.ma
index 68276fe9f7ef2c7212276ac9b990913798a8459b..fe603b59cd541577a1e9038385f90c331d60dc53 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/sn3/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/sn3/fwd.ma
@@ -14,11 +14,20 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/sn3/defs.ma".
+include "basic_1/sn3/defs.ma".
 
-include "Basic-1/pr3/props.ma".
+include "basic_1/pr3/props.ma".
 
-theorem sn3_gen_bind:
+implied rec lemma sn3_ind (c: C) (P: (T \to Prop)) (f: (\forall (t1: 
+T).(((\forall (t2: T).((((eq T t1 t2) \to (\forall (P0: Prop).P0))) \to ((pr3 
+c t1 t2) \to (sn3 c t2))))) \to (((\forall (t2: T).((((eq T t1 t2) \to 
+(\forall (P0: Prop).P0))) \to ((pr3 c t1 t2) \to (P t2))))) \to (P t1))))) 
+(t: T) (s0: sn3 c t) on s0: P t \def match s0 with [(sn3_sing t1 s1) 
+\Rightarrow (f t1 s1 (\lambda (t2: T).(\lambda (p: (((eq T t1 t2) \to 
+(\forall (P0: Prop).P0)))).(\lambda (p0: (pr3 c t1 t2)).((sn3_ind c P f) t2 
+(s1 t2 p p0))))))].
+
+lemma sn3_gen_bind:
  \forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
 (THead (Bind b) u t)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t))))))
 \def
@@ -47,34 +56,30 @@ x2)) \to (land (sn3 c x1) (sn3 (CHead c (Bind b) x1) x2))))))))) H2 (THead
 \to (\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x t2)).(let H8 \def (H4 
 (THead (Bind b) t2 x0) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind 
 b) t2 x0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | 
-(TLRef _) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind b) x 
-x0) (THead (Bind b) t2 x0) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: 
-T).(pr3 c x t0)) H7 x H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: 
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H6 x H9) in (H11 (refl_equal T 
-x) P)))))) (pr3_head_12 c x t2 H7 (Bind b) x0 x0 (pr3_refl (CHead c (Bind b) 
-t2) x0)) t2 x0 (refl_equal T (THead (Bind b) t2 x0))) in (land_ind (sn3 c t2) 
-(sn3 (CHead c (Bind b) t2) x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda 
-(_: (sn3 (CHead c (Bind b) t2) x0)).H9)) H8)))))) (sn3_sing (CHead c (Bind b) 
-x) x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P: 
-Prop).P)))).(\lambda (H7: (pr3 (CHead c (Bind b) x) x0 t2)).(let H8 \def (H4 
-(THead (Bind b) x t2) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind 
-b) x t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | 
-(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind b) x 
-x0) (THead (Bind b) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: 
-T).(pr3 (CHead c (Bind b) x) x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T 
-t2 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in 
-(H11 (refl_equal T x0) P)))))) (pr3_head_12 c x x (pr3_refl c x) (Bind b) x0 
-t2 H7) x t2 (refl_equal T (THead (Bind b) x t2))) in (land_ind (sn3 c x) (sn3 
-(CHead c (Bind b) x) t2) (sn3 (CHead c (Bind b) x) t2) (\lambda (_: (sn3 c 
-x)).(\lambda (H10: (sn3 (CHead c (Bind b) x) t2)).H10)) H8))))))))))))))) y 
-H0))))) H))))).
-(* COMMENTS
-Initial nodes: 1055
-END *)
+T).(match e with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead 
+_ t0 _) \Rightarrow t0])) (THead (Bind b) x x0) (THead (Bind b) t2 x0) H8) in 
+(let H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c x t0)) H7 x H9) in (let 
+H11 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: 
+Prop).P0))) H6 x H9) in (H11 (refl_equal T x) P)))))) (pr3_head_12 c x t2 H7 
+(Bind b) x0 x0 (pr3_refl (CHead c (Bind b) t2) x0)) t2 x0 (refl_equal T 
+(THead (Bind b) t2 x0))) in (land_ind (sn3 c t2) (sn3 (CHead c (Bind b) t2) 
+x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda (_: (sn3 (CHead c (Bind b) 
+t2) x0)).H9)) H8)))))) (sn3_sing (CHead c (Bind b) x) x0 (\lambda (t2: 
+T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P: Prop).P)))).(\lambda (H7: 
+(pr3 (CHead c (Bind b) x) x0 t2)).(let H8 \def (H4 (THead (Bind b) x t2) 
+(\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind b) x t2))).(\lambda 
+(P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) 
+(THead (Bind b) x x0) (THead (Bind b) x t2) H8) in (let H10 \def (eq_ind_r T 
+t2 (\lambda (t0: T).(pr3 (CHead c (Bind b) x) x0 t0)) H7 x0 H9) in (let H11 
+\def (eq_ind_r T t2 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: 
+Prop).P0))) H6 x0 H9) in (H11 (refl_equal T x0) P)))))) (pr3_head_12 c x x 
+(pr3_refl c x) (Bind b) x0 t2 H7) x t2 (refl_equal T (THead (Bind b) x t2))) 
+in (land_ind (sn3 c x) (sn3 (CHead c (Bind b) x) t2) (sn3 (CHead c (Bind b) 
+x) t2) (\lambda (_: (sn3 c x)).(\lambda (H10: (sn3 (CHead c (Bind b) x) 
+t2)).H10)) H8))))))))))))))) y H0))))) H))))).
 
-theorem sn3_gen_flat:
+lemma sn3_gen_flat:
  \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
 (THead (Flat f) u t)) \to (land (sn3 c u) (sn3 c t))))))
 \def
@@ -101,32 +106,28 @@ x2)) \to (land (sn3 c x1) (sn3 c x2))))))))) H2 (THead (Flat f) x x0) H3) in
 T).(\lambda (H6: (((eq T x t2) \to (\forall (P: Prop).P)))).(\lambda (H7: 
 (pr3 c x t2)).(let H8 \def (H4 (THead (Flat f) t2 x0) (\lambda (H8: (eq T 
 (THead (Flat f) x x0) (THead (Flat f) t2 x0))).(\lambda (P: Prop).(let H9 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat f) x x0) (THead (Flat f) t2 x0) H8) in (let 
-H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c x t0)) H7 x H9) in (let H11 
-\def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: 
-Prop).P0))) H6 x H9) in (H11 (refl_equal T x) P)))))) (pr3_head_12 c x t2 H7 
-(Flat f) x0 x0 (pr3_refl (CHead c (Flat f) t2) x0)) t2 x0 (refl_equal T 
-(THead (Flat f) t2 x0))) in (land_ind (sn3 c t2) (sn3 c x0) (sn3 c t2) 
-(\lambda (H9: (sn3 c t2)).(\lambda (_: (sn3 c x0)).H9)) H8)))))) (sn3_sing c 
-x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P: 
-Prop).P)))).(\lambda (H7: (pr3 c x0 t2)).(let H8 \def (H4 (THead (Flat f) x 
-t2) (\lambda (H8: (eq T (THead (Flat f) x x0) (THead (Flat f) x 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x | 
+(TLRef _) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Flat f) x 
+x0) (THead (Flat f) t2 x0) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: 
+T).(pr3 c x t0)) H7 x H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: 
+T).((eq T x t0) \to (\forall (P0: Prop).P0))) H6 x H9) in (H11 (refl_equal T 
+x) P)))))) (pr3_head_12 c x t2 H7 (Flat f) x0 x0 (pr3_refl (CHead c (Flat f) 
+t2) x0)) t2 x0 (refl_equal T (THead (Flat f) t2 x0))) in (land_ind (sn3 c t2) 
+(sn3 c x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda (_: (sn3 c x0)).H9)) 
+H8)))))) (sn3_sing c x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to 
+(\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x0 t2)).(let H8 \def (H4 (THead 
+(Flat f) x t2) (\lambda (H8: (eq T (THead (Flat f) x x0) (THead (Flat f) x 
 t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match e 
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) 
-\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat f) x x0) 
-(THead (Flat f) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: 
-T).(pr3 c x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: 
-T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in (H11 (refl_equal 
-T x0) P)))))) (pr3_thin_dx c x0 t2 H7 x f) x t2 (refl_equal T (THead (Flat f) 
-x t2))) in (land_ind (sn3 c x) (sn3 c t2) (sn3 c t2) (\lambda (_: (sn3 c 
-x)).(\lambda (H10: (sn3 c t2)).H10)) H8))))))))))))))) y H0))))) H))))).
-(* COMMENTS
-Initial nodes: 925
-END *)
+with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) 
+\Rightarrow t0])) (THead (Flat f) x x0) (THead (Flat f) x t2) H8) in (let H10 
+\def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c x0 t0)) H7 x0 H9) in (let H11 
+\def (eq_ind_r T t2 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: 
+Prop).P0))) H6 x0 H9) in (H11 (refl_equal T x0) P)))))) (pr3_thin_dx c x0 t2 
+H7 x f) x t2 (refl_equal T (THead (Flat f) x t2))) in (land_ind (sn3 c x) 
+(sn3 c t2) (sn3 c t2) (\lambda (_: (sn3 c x)).(\lambda (H10: (sn3 c 
+t2)).H10)) H8))))))))))))))) y H0))))) H))))).
 
-theorem sn3_gen_head:
+lemma sn3_gen_head:
  \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
 (THead k u t)) \to (sn3 c u)))))
 \def
@@ -140,11 +141,8 @@ F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
 (Flat f) u t))).(let H_x \def (sn3_gen_flat f c u t H) in (let H0 \def H_x in 
 (land_ind (sn3 c u) (sn3 c t) (sn3 c u) (\lambda (H1: (sn3 c u)).(\lambda (_: 
 (sn3 c t)).H1)) H0)))))))) k).
-(* COMMENTS
-Initial nodes: 191
-END *)
 
-theorem sn3_gen_cflat:
+lemma sn3_gen_cflat:
  \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 (CHead 
 c (Flat f) u) t) \to (sn3 c t)))))
 \def
@@ -157,11 +155,8 @@ t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to
 (sn3 c t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) 
 \to (\forall (P: Prop).P)))).(\lambda (H3: (pr3 c t1 t2)).(H1 t2 H2 
 (pr3_cflat c t1 t2 H3 f u))))))))) t H))))).
-(* COMMENTS
-Initial nodes: 175
-END *)
 
-theorem sn3_gen_lift:
+lemma sn3_gen_lift:
  \forall (c1: C).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((sn3 c1 
 (lift h d t)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t)))))))
 \def
@@ -191,7 +186,4 @@ d H9)) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to
 (\forall (P0: Prop).P0))) H7 x (lift_inj x t2 h d H9)) in (H11 (refl_equal T 
 x) P))))) (pr3_lift c1 c2 h d H4 x t2 H8) t2 (refl_equal T (lift h d t2)) c2 
 H4)))))))))))))) y H0)))) H))))).
-(* COMMENTS
-Initial nodes: 565
-END *)