1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 include "basic_1A/nf2/defs.ma".
19 include "basic_1A/pr2/clen.ma".
21 include "basic_1A/subst0/dec.ma".
23 include "basic_1A/T/props.ma".
26 \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c
27 (CHead d (Bind Abbr) u)) \to ((nf2 c (TLRef i)) \to (\forall (P: Prop).P))))))
29 \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
30 (H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H0: ((\forall (t2: T).((pr2
31 c (TLRef i) t2) \to (eq T (TLRef i) t2))))).(\lambda (P:
32 Prop).(lift_gen_lref_false (S i) O i (le_O_n i) (le_n (plus O (S i))) u (H0
33 (lift (S i) O u) (pr2_delta c d u i H (TLRef i) (TLRef i) (pr0_refl (TLRef
34 i)) (lift (S i) O u) (subst0_lref u i))) P))))))).
37 \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abst) u
38 t)) \to (land (nf2 c u) (nf2 (CHead c (Bind Abst) u) t)))))
40 \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
41 T).((pr2 c (THead (Bind Abst) u t) t2) \to (eq T (THead (Bind Abst) u t)
42 t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall (t2:
43 T).((pr2 (CHead c (Bind Abst) u) t t2) \to (eq T t t2))) (\lambda (t2:
44 T).(\lambda (H0: (pr2 c u t2)).(let H1 \def (f_equal T T (\lambda (e:
45 T).(match e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead
46 _ t0 _) \Rightarrow t0])) (THead (Bind Abst) u t) (THead (Bind Abst) t2 t) (H
47 (THead (Bind Abst) t2 t) (pr2_head_1 c u t2 H0 (Bind Abst) t))) in (let H2
48 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 c u t0)) H0 u H1) in (eq_ind T u
49 (\lambda (t0: T).(eq T u t0)) (refl_equal T u) t2 H1))))) (\lambda (t2:
50 T).(\lambda (H0: (pr2 (CHead c (Bind Abst) u) t t2)).(let H1 \def (f_equal T
51 T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t | (TLRef _)
52 \Rightarrow t | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abst) u t)
53 (THead (Bind Abst) u t2) (H (THead (Bind Abst) u t2) (let H_y \def
54 (pr2_gen_cbind Abst c u t t2 H0) in H_y))) in (let H2 \def (eq_ind_r T t2
55 (\lambda (t0: T).(pr2 (CHead c (Bind Abst) u) t t0)) H0 t H1) in (eq_ind T t
56 (\lambda (t0: T).(eq T t t0)) (refl_equal T t) t2 H1))))))))).
59 \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Flat Cast) u
60 t)) \to (\forall (P: Prop).P))))
62 \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (nf2 c (THead
63 (Flat Cast) u t))).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) u t (H t
64 (pr2_free c (THead (Flat Cast) u t) t (pr0_tau t t (pr0_refl t) u))) P))))).
67 \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((nf2 c
68 (THead (Flat Appl) u (THead (Bind Abst) v t))) \to (\forall (P: Prop).P)))))
70 \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
71 ((\forall (t2: T).((pr2 c (THead (Flat Appl) u (THead (Bind Abst) v t)) t2)
72 \to (eq T (THead (Flat Appl) u (THead (Bind Abst) v t)) t2))))).(\lambda (P:
73 Prop).(let H0 \def (eq_ind T (THead (Flat Appl) u (THead (Bind Abst) v t))
74 (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
75 \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _)
76 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u t)
77 (H (THead (Bind Abbr) u t) (pr2_free c (THead (Flat Appl) u (THead (Bind
78 Abst) v t)) (THead (Bind Abbr) u t) (pr0_beta v u u (pr0_refl u) t t
79 (pr0_refl t))))) in (False_ind P H0))))))).
82 \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c
83 (THead (Flat f) u t)) \to (land (nf2 c u) (nf2 c t))))))
85 \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
86 ((\forall (t2: T).((pr2 c (THead (Flat f) u t) t2) \to (eq T (THead (Flat f)
87 u t) t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall
88 (t2: T).((pr2 c t t2) \to (eq T t t2))) (\lambda (t2: T).(\lambda (H0: (pr2 c
89 u t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _)
90 \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0]))
91 (THead (Flat f) u t) (THead (Flat f) t2 t) (H (THead (Flat f) t2 t)
92 (pr2_head_1 c u t2 H0 (Flat f) t))) in H1))) (\lambda (t2: T).(\lambda (H0:
93 (pr2 c t t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e with [(TSort
94 _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) \Rightarrow t0]))
95 (THead (Flat f) u t) (THead (Flat f) u t2) (H (THead (Flat f) u t2)
96 (pr2_head_2 c u t t2 (Flat f) (pr2_cflat c t t2 H0 f u)))) in H1)))))))).
98 fact nf2_gen__nf2_gen_aux:
99 \forall (b: B).(\forall (x: T).(\forall (u: T).(\forall (d: nat).((eq T
100 (THead (Bind b) u (lift (S O) d x)) x) \to (\forall (P: Prop).P)))))
102 \lambda (b: B).(\lambda (x: T).(T_ind (\lambda (t: T).(\forall (u:
103 T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to
104 (\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (d:
105 nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d (TSort n))) (TSort
106 n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead (Bind b) u (lift (S O)
107 d (TSort n))) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False |
108 (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n)
109 H) in (False_ind P H0))))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (d:
110 nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d (TLRef n))) (TLRef
111 n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead (Bind b) u (lift (S O)
112 d (TLRef n))) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False |
113 (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n)
114 H) in (False_ind P H0))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (_:
115 ((\forall (u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t))
116 t) \to (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall
117 (u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t0)) t0) \to
118 (\forall (P: Prop).P)))))).(\lambda (u: T).(\lambda (d: nat).(\lambda (H1:
119 (eq T (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t
120 t0))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e
121 with [(TSort _) \Rightarrow (Bind b) | (TLRef _) \Rightarrow (Bind b) |
122 (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u (lift (S O) d (THead k t
123 t0))) (THead k t t0) H1) in ((let H3 \def (f_equal T T (\lambda (e: T).(match
124 e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t1 _)
125 \Rightarrow t1])) (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t
126 t0) H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e with [(TSort
127 _) \Rightarrow (THead k (lref_map (\lambda (x0: nat).(plus x0 (S O))) d t)
128 (lref_map (\lambda (x0: nat).(plus x0 (S O))) (s k d) t0)) | (TLRef _)
129 \Rightarrow (THead k (lref_map (\lambda (x0: nat).(plus x0 (S O))) d t)
130 (lref_map (\lambda (x0: nat).(plus x0 (S O))) (s k d) t0)) | (THead _ _ t1)
131 \Rightarrow t1])) (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t
132 t0) H1) in (\lambda (_: (eq T u t)).(\lambda (H6: (eq K (Bind b) k)).(let H7
133 \def (eq_ind_r K k (\lambda (k0: K).(eq T (lift (S O) d (THead k0 t t0)) t0))
134 H4 (Bind b) H6) in (let H8 \def (eq_ind T (lift (S O) d (THead (Bind b) t
135 t0)) (\lambda (t1: T).(eq T t1 t0)) H7 (THead (Bind b) (lift (S O) d t) (lift
136 (S O) (S d) t0)) (lift_bind b t t0 (S O) d)) in (H0 (lift (S O) d t) (S d) H8
137 P)))))) H3)) H2))))))))))) x)).
140 \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abbr) u
141 t)) \to (\forall (P: Prop).P))))
143 \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
144 T).((pr2 c (THead (Bind Abbr) u t) t2) \to (eq T (THead (Bind Abbr) u t)
145 t2))))).(\lambda (P: Prop).(let H_x \def (dnf_dec u t O) in (let H0 \def H_x
146 in (ex_ind T (\lambda (v: T).(or (subst0 O u t (lift (S O) O v)) (eq T t
147 (lift (S O) O v)))) P (\lambda (x: T).(\lambda (H1: (or (subst0 O u t (lift
148 (S O) O x)) (eq T t (lift (S O) O x)))).(or_ind (subst0 O u t (lift (S O) O
149 x)) (eq T t (lift (S O) O x)) P (\lambda (H2: (subst0 O u t (lift (S O) O
150 x))).(let H3 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _)
151 \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) \Rightarrow t0]))
152 (THead (Bind Abbr) u t) (THead (Bind Abbr) u (lift (S O) O x)) (H (THead
153 (Bind Abbr) u (lift (S O) O x)) (pr2_free c (THead (Bind Abbr) u t) (THead
154 (Bind Abbr) u (lift (S O) O x)) (pr0_delta u u (pr0_refl u) t t (pr0_refl t)
155 (lift (S O) O x) H2)))) in (let H4 \def (eq_ind T t (\lambda (t0: T).(subst0
156 O u t0 (lift (S O) O x))) H2 (lift (S O) O x) H3) in (subst0_refl u (lift (S
157 O) O x) O H4 P)))) (\lambda (H2: (eq T t (lift (S O) O x))).(let H3 \def
158 (eq_ind T t (\lambda (t0: T).(\forall (t2: T).((pr2 c (THead (Bind Abbr) u
159 t0) t2) \to (eq T (THead (Bind Abbr) u t0) t2)))) H (lift (S O) O x) H2) in
160 (nf2_gen__nf2_gen_aux Abbr x u O (H3 x (pr2_free c (THead (Bind Abbr) u (lift
161 (S O) O x)) x (pr0_zeta Abbr not_abbr_abst x x (pr0_refl x) u))) P))) H1)))
165 \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Void) u
166 (lift (S O) O t))) \to (\forall (P: Prop).P))))
168 \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
169 T).((pr2 c (THead (Bind Void) u (lift (S O) O t)) t2) \to (eq T (THead (Bind
170 Void) u (lift (S O) O t)) t2))))).(\lambda (P: Prop).(nf2_gen__nf2_gen_aux
171 Void t u O (H t (pr2_free c (THead (Bind Void) u (lift (S O) O t)) t
172 (pr0_zeta Void not_void_abst t t (pr0_refl t) u))) P))))).