(* 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
\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
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
(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
(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
(\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 *)