(* This file was automatically generated: do not edit *********************)
+include "LambdaDelta-1/sn3/defs.ma".
-
-include "sn3/defs.ma".
-
-include "pr3/props.ma".
+include "LambdaDelta-1/pr3/props.ma".
theorem sn3_gen_bind:
\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c
\def
\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
(sn3 c (THead (Bind b) u t))).(insert_eq T (THead (Bind b) u t) (\lambda (t0:
-T).(sn3 c t0)) (land (sn3 c u) (sn3 (CHead c (Bind b) u) t)) (\lambda (y:
-T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T y (THead
-(Bind b) u t0)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t0)))) (unintro
-T u (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind b) t0 x)) \to
-(land (sn3 c t0) (sn3 (CHead c (Bind b) t0) x))))) (sn3_ind c (\lambda (t0:
-T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Bind b) x x0)) \to
+T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 (CHead c (Bind b) u) t)))
+(\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T
+y (THead (Bind b) u t0)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t0))))
+(unintro T u (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind b) t0 x))
+\to (land (sn3 c t0) (sn3 (CHead c (Bind b) t0) x))))) (sn3_ind c (\lambda
+(t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Bind b) x x0)) \to
(land (sn3 c x) (sn3 (CHead c (Bind b) x) x0)))))) (\lambda (t1: T).(\lambda
(H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3
c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2)
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 (and_ind (sn3 c t2)
+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:
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 (and_ind (sn3 c x) (sn3
+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))))).
\def
\lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
(sn3 c (THead (Flat f) u t))).(insert_eq T (THead (Flat f) u t) (\lambda (t0:
-T).(sn3 c t0)) (land (sn3 c u) (sn3 c t)) (\lambda (y: T).(\lambda (H0: (sn3
-c y)).(unintro T t (\lambda (t0: T).((eq T y (THead (Flat f) u t0)) \to (land
-(sn3 c u) (sn3 c t0)))) (unintro T u (\lambda (t0: T).(\forall (x: T).((eq T
-y (THead (Flat f) t0 x)) \to (land (sn3 c t0) (sn3 c x))))) (sn3_ind c
-(\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat f) x
-x0)) \to (land (sn3 c x) (sn3 c x0)))))) (\lambda (t1: T).(\lambda (H1:
-((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1
-t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to
-(\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall (x0:
-T).((eq T t2 (THead (Flat f) x x0)) \to (land (sn3 c x) (sn3 c
-x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1 (THead
-(Flat f) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2:
-T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to
-(\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Flat f) x1 x2)) \to (land
-(sn3 c x1) (sn3 c x2))))))))) H2 (THead (Flat f) x x0) H3) in (let H5 \def
-(eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall
-(P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead (Flat f) x x0)
-H3) in (conj (sn3 c x) (sn3 c x0) (sn3_sing c x (\lambda (t2: 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
-(and_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).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 c t))) (\lambda (y:
+T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T y (THead
+(Flat f) u t0)) \to (land (sn3 c u) (sn3 c t0)))) (unintro T u (\lambda (t0:
+T).(\forall (x: T).((eq T y (THead (Flat f) t0 x)) \to (land (sn3 c t0) (sn3
+c x))))) (sn3_ind c (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T
+t0 (THead (Flat f) x x0)) \to (land (sn3 c x) (sn3 c x0)))))) (\lambda (t1:
+T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
+(\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat f) x x0)) \to (land
+(sn3 c x) (sn3 c x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3:
+(eq T t1 (THead (Flat f) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0:
+T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c
+t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Flat f) x1
+x2)) \to (land (sn3 c x1) (sn3 c x2))))))))) H2 (THead (Flat f) x x0) H3) in
+(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2)
+\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead
+(Flat f) x x0) H3) in (conj (sn3 c x) (sn3 c x0) (sn3_sing c x (\lambda (t2:
+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
+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 (and_ind (sn3 c x) (sn3 c t2) (sn3 c t2) (\lambda (_: (sn3 c
+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:
T).(\forall (t: T).((sn3 c (THead k0 u t)) \to (sn3 c u)))))) (\lambda (b:
B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
(Bind b) u t))).(let H_x \def (sn3_gen_bind b c u t H) in (let H0 \def H_x in
-(and_ind (sn3 c u) (sn3 (CHead c (Bind b) u) t) (sn3 c u) (\lambda (H1: (sn3
+(land_ind (sn3 c u) (sn3 (CHead c (Bind b) u) t) (sn3 c u) (\lambda (H1: (sn3
c u)).(\lambda (_: (sn3 (CHead c (Bind b) u) t)).H1)) H0)))))))) (\lambda (f:
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
-(and_ind (sn3 c u) (sn3 c t) (sn3 c u) (\lambda (H1: (sn3 c u)).(\lambda (_:
+(land_ind (sn3 c u) (sn3 c t) (sn3 c u) (\lambda (H1: (sn3 c u)).(\lambda (_:
(sn3 c t)).H1)) H0)))))))) k).
theorem sn3_gen_cflat:
\def
\lambda (c1: C).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda
(H: (sn3 c1 (lift h d t))).(insert_eq T (lift h d t) (\lambda (t0: T).(sn3 c1
-t0)) (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t))) (\lambda (y:
-T).(\lambda (H0: (sn3 c1 y)).(unintro T t (\lambda (t0: T).((eq T y (lift h d
-t0)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t0))))) (sn3_ind c1
-(\lambda (t0: T).(\forall (x: T).((eq T t0 (lift h d x)) \to (\forall (c2:
-C).((drop h d c1 c2) \to (sn3 c2 x)))))) (\lambda (t1: T).(\lambda (H1:
-((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t1
-t2) \to (sn3 c1 t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to
-(\forall (P: Prop).P))) \to ((pr3 c1 t1 t2) \to (\forall (x: T).((eq T t2
-(lift h d x)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2
-x)))))))))).(\lambda (x: T).(\lambda (H3: (eq T t1 (lift h d x))).(\lambda
-(c2: C).(\lambda (H4: (drop h d c1 c2)).(let H5 \def (eq_ind T t1 (\lambda
-(t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to
-((pr3 c1 t0 t2) \to (\forall (x0: T).((eq T t2 (lift h d x0)) \to (\forall
-(c3: C).((drop h d c1 c3) \to (sn3 c3 x0))))))))) H2 (lift h d x) H3) in (let
-H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to
-(\forall (P: Prop).P))) \to ((pr3 c1 t0 t2) \to (sn3 c1 t2))))) H1 (lift h d
-x) H3) in (sn3_sing c2 x (\lambda (t2: T).(\lambda (H7: (((eq T x t2) \to
-(\forall (P: Prop).P)))).(\lambda (H8: (pr3 c2 x t2)).(H5 (lift h d t2)
-(\lambda (H9: (eq T (lift h d x) (lift h d t2))).(\lambda (P: Prop).(let H10
-\def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c2 x t0)) H8 x (lift_inj x t2 h d
-H9)) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to
+t0)) (\lambda (_: T).(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t))))
+(\lambda (y: T).(\lambda (H0: (sn3 c1 y)).(unintro T t (\lambda (t0: T).((eq
+T y (lift h d t0)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t0)))))
+(sn3_ind c1 (\lambda (t0: T).(\forall (x: T).((eq T t0 (lift h d x)) \to
+(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 x)))))) (\lambda (t1:
+T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 c1 t1 t2) \to (sn3 c1 t2)))))).(\lambda (H2: ((\forall
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t1 t2) \to
+(\forall (x: T).((eq T t2 (lift h d x)) \to (\forall (c2: C).((drop h d c1
+c2) \to (sn3 c2 x)))))))))).(\lambda (x: T).(\lambda (H3: (eq T t1 (lift h d
+x))).(\lambda (c2: C).(\lambda (H4: (drop h d c1 c2)).(let H5 \def (eq_ind T
+t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 c1 t0 t2) \to (\forall (x0: T).((eq T t2 (lift h d x0))
+\to (\forall (c3: C).((drop h d c1 c3) \to (sn3 c3 x0))))))))) H2 (lift h d
+x) H3) in (let H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq
+T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t0 t2) \to (sn3 c1 t2)))))
+H1 (lift h d x) H3) in (sn3_sing c2 x (\lambda (t2: T).(\lambda (H7: (((eq T
+x t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr3 c2 x t2)).(H5 (lift h d
+t2) (\lambda (H9: (eq T (lift h d x) (lift h d t2))).(\lambda (P: Prop).(let
+H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c2 x t0)) H8 x (lift_inj x t2 h
+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))))).
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