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 "LambdaDelta-1/nf2/defs.ma".
19 include "LambdaDelta-1/pr2/clen.ma".
21 include "LambdaDelta-1/subst0/dec.ma".
23 include "LambdaDelta-1/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 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u |
46 (TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst)
47 u t) (THead (Bind Abst) t2 t) (H (THead (Bind Abst) t2 t) (pr2_head_1 c u t2
48 H0 (Bind Abst) t))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 c u
49 t0)) H0 u H1) in (eq_ind T u (\lambda (t0: T).(eq T u t0)) (refl_equal T u)
50 t2 H1))))) (\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind Abst) u) t
51 t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
52 (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _
53 _ t0) \Rightarrow t0])) (THead (Bind Abst) u t) (THead (Bind Abst) u t2) (H
54 (THead (Bind Abst) u t2) (let H_y \def (pr2_gen_cbind Abst c u t t2 H0) in
55 H_y))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 (CHead c (Bind
56 Abst) u) t t0)) H0 t H1) in (eq_ind T t (\lambda (t0: T).(eq T t t0))
57 (refl_equal T t) t2 H1))))))))).
60 \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Flat Cast) u
61 t)) \to (\forall (P: Prop).P))))
63 \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (nf2 c (THead
64 (Flat Cast) u t))).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) u t (H t
65 (pr2_free c (THead (Flat Cast) u t) t (pr0_tau t t (pr0_refl t) u))) P))))).
68 \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((nf2 c
69 (THead (Flat Appl) u (THead (Bind Abst) v t))) \to (\forall (P: Prop).P)))))
71 \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
72 ((\forall (t2: T).((pr2 c (THead (Flat Appl) u (THead (Bind Abst) v t)) t2)
73 \to (eq T (THead (Flat Appl) u (THead (Bind Abst) v t)) t2))))).(\lambda (P:
74 Prop).(let H0 \def (eq_ind T (THead (Flat Appl) u (THead (Bind Abst) v t))
75 (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
76 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
77 (match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
78 (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u t) (H (THead (Bind
79 Abbr) u t) (pr2_free c (THead (Flat Appl) u (THead (Bind Abst) v t)) (THead
80 (Bind Abbr) u t) (pr0_beta v u u (pr0_refl u) t t (pr0_refl t))))) in
81 (False_ind P H0))))))).
84 \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c
85 (THead (Flat f) u t)) \to (land (nf2 c u) (nf2 c t))))))
87 \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
88 ((\forall (t2: T).((pr2 c (THead (Flat f) u t) t2) \to (eq T (THead (Flat f)
89 u t) t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall
90 (t2: T).((pr2 c t t2) \to (eq T t t2))) (\lambda (t2: T).(\lambda (H0: (pr2 c
91 u t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return
92 (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
93 (THead _ t0 _) \Rightarrow t0])) (THead (Flat f) u t) (THead (Flat f) t2 t)
94 (H (THead (Flat f) t2 t) (pr2_head_1 c u t2 H0 (Flat f) t))) in H1)))
95 (\lambda (t2: T).(\lambda (H0: (pr2 c t t2)).(let H1 \def (f_equal T T
96 (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
97 \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) \Rightarrow t0]))
98 (THead (Flat f) u t) (THead (Flat f) u t2) (H (THead (Flat f) u t2)
99 (pr2_head_2 c u t t2 (Flat f) (pr2_cflat c t t2 H0 f u)))) in H1)))))))).
101 theorem nf2_gen__nf2_gen_aux:
102 \forall (b: B).(\forall (x: T).(\forall (u: T).(\forall (d: nat).((eq T
103 (THead (Bind b) u (lift (S O) d x)) x) \to (\forall (P: Prop).P)))))
105 \lambda (b: B).(\lambda (x: T).(T_ind (\lambda (t: T).(\forall (u:
106 T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to
107 (\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (d:
108 nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d (TSort n))) (TSort
109 n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead (Bind b) u (lift (S O)
110 d (TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
111 with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
112 _) \Rightarrow True])) I (TSort n) H) in (False_ind P H0))))))) (\lambda (n:
113 nat).(\lambda (u: T).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u
114 (lift (S O) d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind
115 T (THead (Bind b) u (lift (S O) d (TLRef n))) (\lambda (ee: T).(match ee in T
116 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
117 \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in
118 (False_ind P H0))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: ((\forall
119 (u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to
120 (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall (u:
121 T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t0)) t0) \to
122 (\forall (P: Prop).P)))))).(\lambda (u: T).(\lambda (d: nat).(\lambda (H1:
123 (eq T (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t
124 t0))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e
125 in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef
126 _) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u
127 (lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let H3 \def (f_equal T
128 T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
129 \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t1 _) \Rightarrow t1]))
130 (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let
131 H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
132 with [(TSort _) \Rightarrow (THead k ((let rec lref_map (f: ((nat \to nat)))
133 (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort
134 n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i
135 | false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0
136 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0:
137 nat).(plus x0 (S O))) d t) ((let rec lref_map (f: ((nat \to nat))) (d0: nat)
138 (t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort n) |
139 (TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i |
140 false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0 (lref_map
141 f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0: nat).(plus
142 x0 (S O))) (s k d) t0)) | (TLRef _) \Rightarrow (THead k ((let rec lref_map
143 (f: ((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort
144 n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0)
145 with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2)
146 \Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in
147 lref_map) (\lambda (x0: nat).(plus x0 (S O))) d t) ((let rec lref_map (f:
148 ((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n)
149 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with
150 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2)
151 \Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in
152 lref_map) (\lambda (x0: nat).(plus x0 (S O))) (s k d) t0)) | (THead _ _ t1)
153 \Rightarrow t1])) (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t
154 t0) H1) in (\lambda (_: (eq T u t)).(\lambda (H6: (eq K (Bind b) k)).(let H7
155 \def (eq_ind_r K k (\lambda (k0: K).(eq T (lift (S O) d (THead k0 t t0)) t0))
156 H4 (Bind b) H6) in (let H8 \def (eq_ind T (lift (S O) d (THead (Bind b) t
157 t0)) (\lambda (t1: T).(eq T t1 t0)) H7 (THead (Bind b) (lift (S O) d t) (lift
158 (S O) (S d) t0)) (lift_bind b t t0 (S O) d)) in (H0 (lift (S O) d t) (S d) H8
159 P)))))) H3)) H2))))))))))) x)).
161 theorem nf2_gen_abbr:
162 \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abbr) u
163 t)) \to (\forall (P: Prop).P))))
165 \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
166 T).((pr2 c (THead (Bind Abbr) u t) t2) \to (eq T (THead (Bind Abbr) u t)
167 t2))))).(\lambda (P: Prop).(let H_x \def (dnf_dec u t O) in (let H0 \def H_x
168 in (ex_ind T (\lambda (v: T).(or (subst0 O u t (lift (S O) O v)) (eq T t
169 (lift (S O) O v)))) P (\lambda (x: T).(\lambda (H1: (or (subst0 O u t (lift
170 (S O) O x)) (eq T t (lift (S O) O x)))).(or_ind (subst0 O u t (lift (S O) O
171 x)) (eq T t (lift (S O) O x)) P (\lambda (H2: (subst0 O u t (lift (S O) O
172 x))).(let H3 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
173 (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _
174 _ t0) \Rightarrow t0])) (THead (Bind Abbr) u t) (THead (Bind Abbr) u (lift (S
175 O) O x)) (H (THead (Bind Abbr) u (lift (S O) O x)) (pr2_free c (THead (Bind
176 Abbr) u t) (THead (Bind Abbr) u (lift (S O) O x)) (pr0_delta u u (pr0_refl u)
177 t t (pr0_refl t) (lift (S O) O x) H2)))) in (let H4 \def (eq_ind T t (\lambda
178 (t0: T).(subst0 O u t0 (lift (S O) O x))) H2 (lift (S O) O x) H3) in
179 (subst0_refl u (lift (S O) O x) O H4 P)))) (\lambda (H2: (eq T t (lift (S O)
180 O x))).(let H3 \def (eq_ind T t (\lambda (t0: T).(\forall (t2: T).((pr2 c
181 (THead (Bind Abbr) u t0) t2) \to (eq T (THead (Bind Abbr) u t0) t2)))) H
182 (lift (S O) O x) H2) in (nf2_gen__nf2_gen_aux Abbr x u O (H3 x (pr2_free c
183 (THead (Bind Abbr) u (lift (S O) O x)) x (pr0_zeta Abbr not_abbr_abst x x
184 (pr0_refl x) u))) P))) H1))) H0))))))).
186 theorem nf2_gen_void:
187 \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Void) u
188 (lift (S O) O t))) \to (\forall (P: Prop).P))))
190 \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
191 T).((pr2 c (THead (Bind Void) u (lift (S O) O t)) t2) \to (eq T (THead (Bind
192 Void) u (lift (S O) O t)) t2))))).(\lambda (P: Prop).(nf2_gen__nf2_gen_aux
193 Void t u O (H t (pr2_free c (THead (Bind Void) u (lift (S O) O t)) t
194 (pr0_zeta Void (sym_not_eq B Abst Void not_abst_void) t t (pr0_refl t) u)))