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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/fwd".
19 include "nf2/defs.ma".
21 include "pr2/clen.ma".
23 theorem nf2_gen_base__aux:
24 \forall (k: K).(\forall (t: T).(\forall (u: T).((eq T (THead k u t) t) \to
25 (\forall (P: Prop).P))))
27 \lambda (k: K).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (u: T).((eq
28 T (THead k u t0) t0) \to (\forall (P: Prop).P)))) (\lambda (n: nat).(\lambda
29 (u: T).(\lambda (H: (eq T (THead k u (TSort n)) (TSort n))).(\lambda (P:
30 Prop).(let H0 \def (eq_ind T (THead k u (TSort n)) (\lambda (ee: T).(match ee
31 in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
32 _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H) in
33 (False_ind P H0)))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (H: (eq T
34 (THead k u (TLRef n)) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind T
35 (THead k u (TLRef n)) (\lambda (ee: T).(match ee in T return (\lambda (_:
36 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
37 (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in (False_ind P H0))))))
38 (\lambda (k0: K).(\lambda (t0: T).(\lambda (_: ((\forall (u: T).((eq T (THead
39 k u t0) t0) \to (\forall (P: Prop).P))))).(\lambda (t1: T).(\lambda (H0:
40 ((\forall (u: T).((eq T (THead k u t1) t1) \to (\forall (P:
41 Prop).P))))).(\lambda (u: T).(\lambda (H1: (eq T (THead k u (THead k0 t0 t1))
42 (THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e:
43 T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k |
44 (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u (THead
45 k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H3 \def (f_equal T T (\lambda (e:
46 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u |
47 (TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2])) (THead k u (THead
48 k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H4 \def (f_equal T T (\lambda (e:
49 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead
50 k0 t0 t1) | (TLRef _) \Rightarrow (THead k0 t0 t1) | (THead _ _ t2)
51 \Rightarrow t2])) (THead k u (THead k0 t0 t1)) (THead k0 t0 t1) H1) in
52 (\lambda (_: (eq T u t0)).(\lambda (H6: (eq K k k0)).(let H7 \def (eq_ind K k
53 (\lambda (k1: K).(\forall (u0: T).((eq T (THead k1 u0 t1) t1) \to (\forall
54 (P0: Prop).P0)))) H0 k0 H6) in (H7 t0 H4 P))))) H3)) H2)))))))))) t)).
57 \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c
58 (CHead d (Bind Abbr) u)) \to ((nf2 c (TLRef i)) \to (\forall (P: Prop).P))))))
60 \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
61 (H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H0: ((\forall (t2: T).((pr2
62 c (TLRef i) t2) \to (eq T (TLRef i) t2))))).(\lambda (P:
63 Prop).(lift_gen_lref_false (S i) O i (le_O_n i) (le_n (plus O (S i))) u (H0
64 (lift (S i) O u) (pr2_delta c d u i H (TLRef i) (TLRef i) (pr0_refl (TLRef
65 i)) (lift (S i) O u) (subst0_lref u i))) P))))))).
68 \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abst) u
69 t)) \to (land (nf2 c u) (nf2 (CHead c (Bind Abst) u) t)))))
71 \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
72 T).((pr2 c (THead (Bind Abst) u t) t2) \to (eq T (THead (Bind Abst) u t)
73 t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall (t2:
74 T).((pr2 (CHead c (Bind Abst) u) t t2) \to (eq T t t2))) (\lambda (t2:
75 T).(\lambda (H0: (pr2 c u t2)).(let H1 \def (f_equal T T (\lambda (e:
76 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u |
77 (TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst)
78 u t) (THead (Bind Abst) t2 t) (H (THead (Bind Abst) t2 t) (pr2_head_1 c u t2
79 H0 (Bind Abst) t))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 c u
80 t0)) H0 u H1) in (eq_ind T u (\lambda (t0: T).(eq T u t0)) (refl_equal T u)
81 t2 H1))))) (\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind Abst) u) t
82 t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
83 (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _
84 _ t0) \Rightarrow t0])) (THead (Bind Abst) u t) (THead (Bind Abst) u t2) (H
85 (THead (Bind Abst) u t2) (let H_y \def (pr2_gen_cbind Abst c u t t2 H0) in
86 H_y))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 (CHead c (Bind
87 Abst) u) t t0)) H0 t H1) in (eq_ind T t (\lambda (t0: T).(eq T t t0))
88 (refl_equal T t) t2 H1))))))))).
91 \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Flat Cast) u
92 t)) \to (\forall (P: Prop).P))))
94 \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (nf2 c (THead
95 (Flat Cast) u t))).(\lambda (P: Prop).(nf2_gen_base__aux (Flat Cast) t u (H t
96 (pr2_free c (THead (Flat Cast) u t) t (pr0_epsilon t t (pr0_refl t) u)))
100 \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c
101 (THead (Flat f) u t)) \to (land (nf2 c u) (nf2 c t))))))
103 \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
104 ((\forall (t2: T).((pr2 c (THead (Flat f) u t) t2) \to (eq T (THead (Flat f)
105 u t) t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall
106 (t2: T).((pr2 c t t2) \to (eq T t t2))) (\lambda (t2: T).(\lambda (H0: (pr2 c
107 u t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return
108 (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
109 (THead _ t0 _) \Rightarrow t0])) (THead (Flat f) u t) (THead (Flat f) t2 t)
110 (H (THead (Flat f) t2 t) (pr2_head_1 c u t2 H0 (Flat f) t))) in H1)))
111 (\lambda (t2: T).(\lambda (H0: (pr2 c t t2)).(let H1 \def (f_equal T T
112 (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
113 \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) \Rightarrow t0]))
114 (THead (Flat f) u t) (THead (Flat f) u t2) (H (THead (Flat f) u t2)
115 (pr2_head_2 c u t t2 (Flat f) (pr2_cflat c t t2 H0 f u)))) in H1)))))))).